finite element solution of the stokes problem with dominating coriolis force abstract the objective of this paper is to present a finite element formulation to solve the stokes problem with coriolis force this force results in a skew symmetric term in the weak formulation of the problem that deteriorates the stability of the standard galerkin finite element method when the viscosity is small we show that the stability is worsened due to the presence of the pressure gradient to enforce the incompressibility of the flow the relevance of this effect depends on the relative importance of the viscous force and the coriolis force which is measured by the ekman number when it is small oscillations occur using the galerkin approach to overcome them we propose two different methods based on a consistent modification of the basic galerkin formulation both methods eliminate the oscillations keeping the accuracy of the formulation and enhancing its numerical stability

accuracy improvement for immersed boundary method using lagrangian velocity interpolation abstract the immersed boundary method ibm has been extensively utilized in fluid dynamics simulations with various numerical methods there are two key steps in ibm namely the force distribution and velocity interpolation steps and the kernel function plays important roles in both steps the purpose for the velocity interpolation is to enhance the no slip condition between fluid and boundary surface however with the kernel function designed to satisfy the basic requirements for the force distribution the boundary velocity calculation process in traditional ibm simulations is actually a weighted averaging operation instead of an interpolation this induces large errors in the calculated boundary velocity for general nonlinear velocity distributions in this paper we propose to release the kernel function s role from the boundary velocity calculation and replace it with the classical lagrangian interpolation schemes to obtain the boundary velocity from the local flow field this idea has been tested in several two dimensional periodic systems and both flat and curved surfaces and steady and unsteady flows have been considered our simulation results show that in general a 20 30 accuracy improvement in velocity and flow structure interaction can be achieved with no extra cost in computational complexity and efficiency thanks to the simple mathematical formulation and computational algorithm

extinction in population dynamics we study a generic reaction diffusion model for single species population dynamics that includes reproduction death and competition the population is assumed to be confined in a refuge beyond which conditions are so harsh that they lead to certain extinction standard continuum mean field models in one dimension yield a critical refuge length l c such that a population in a refuge larger than this is assured survival herein we extend the model to take into account the discreteness and finiteness of the population which leads us to a stochastic description we present a particular critical criterion for likely extinction namely that the standard deviation of the population be equal to the mean according to this criterion we find that while survival can no longer be guaranteed for any refuge size for sufficiently weak competition one can make the refuge large enough certainly larger than l c to cause extinction to be unlikely however beyond a certain value of the competition rate parameter it is no longer possible to escape a likelihood of extinction even in an infinite refuge these unavoidable fluctuations therefore have a severe impact on refuge design issues

a fourth order finite difference method for solving elliptic interface problems with the fft acceleration abstract in this paper a new cartesian grid finite difference method is introduced based on the fourth order accurate matched interface and boundary mib method and fast fourier transform fft the proposed augmented mib method consists of two major parts for solving elliptic interface problems in two dimensions for the interior part in treating a smoothly curved interface zeroth and first order jump conditions are enforced repeatedly by the mib scheme to generate fictitious values near the interface for the exterior part two layers of zero padding solutions are introduced beyond the original rectangular domain so that the fft inversion becomes feasible different types of boundary conditions including dirichlet neumann robin and their mix combinations can be imposed via the mib scheme to generate fictitious values near boundaries based on fictitious values at both interfaces and boundaries the augmented mib method reconstructs cartesian derivative jumps as auxiliary variables then by treating such variables as unknowns an enlarged linear system is obtained in the schur complement solution of such system the fft algorithm will not sense the solution discontinuities so that the discrete laplacian can be efficiently inverted therefore the fft based augmented mib not only achieves a fourth order of accuracy in dealing with interfaces and boundaries but also produces an overall complexity of o n 2 log n for a n n uniform grid moreover the augmented mib scheme can provide fourth order accurate approximations to solution gradients and fluxes

on inf sup stabilized finite element methods for transient problems we consider the behavior of inf sup stabilization in the context of transient problems with multiple time scales our motivation for studying this setting is provided by reacting flows problems for which small time steps are necessary in the integration process we show that for algorithms defined through a process wherein spatial and temporal discretizations are separated the coupling of implicit time integration with spatial inf sup stabilization may lead to anomalous pressure behavior including the onset of spurious oscillations for very small time steps effectively this coupling introduces a stability criterion resulting in a dependence between the spatial grid size and the time step we illustrate our theoretical results by numerical examples that demonstrate the stability criterion 2004 elsevier b v all rights reserved

characteristic stabilized finite element method for the transient navier stokes equations abstract based on the lowest equal order conforming finite element subspace xh mh i e p1 p1 or q1 q1 elements a characteristic stabilized finite element method for transient navier stokes problem is proposed the proposed method has a number of attractive computational properties parameter free avoiding higher order derivatives or edge based data structures and averting the difficulties caused by trilinear terms existence uniqueness and error estimates of the approximate solution are proved by applying the technique of characteristic finite element method finally a serious of numerical experiments are given to show that this method is highly efficient for transient navier stokes problem

optimal isoparametric finite elements and error estimates for domains involving curved boundaries we describe a practical procedure for the triangulation of arbitrary n dimensional regular bounded domains by isoparametric simplicial elements preserving the optimal order of accuracy along similar lines we prove general error estimates for the finite element solution of second order elliptic problems various kinds of nonhomogeneous boundary conditions can be dealt with such as dirichlet neumann and robin boundary conditions

the berry keating operator on l 2 mathbb r _ gt rm d x and on compact quantum graphs with general self adjoint realizations the berry keating operator berry and keating 1999 siam rev 41 236 governing the schrodinger dynamics is discussed in the hilbert space and on compact quantum graphs it has been proved that the spectrum of hbk defined on is purely continuous and thus this quantization of hbk cannot yield the hypothetical hilbert polya operator possessing as eigenvalues the nontrivial zeros of the riemann zeta function a complete classification of all self adjoint extensions of hbk acting on compact quantum graphs is given together with the corresponding secular equation in form of a determinant whose zeros determine the discrete spectrum of hbk in addition an exact trace formula and the weyl asymptotics of the eigenvalue counting function are derived furthermore we introduce the squared berry keating operator which is a special case of the black scholes operator used in financial theory of option pricing again all self adjoint extensions the corresponding secular equation the trace formula and the weyl asymptotics are derived for h2bk on compact quantum graphs while the spectra of both hbk and h2bk on any compact quantum graph are discrete their weyl asymptotics demonstrate that neither hbk nor h2bk can yield as eigenvalues the nontrivial riemann zeros some simple examples are worked out in detail

properties of discrete delta functions and local convergence of the immersed boundary method many problems involving internal interfaces can be formulated as partial differential equations with singular source terms numerical approximation to such problems on a regular grid necessitates suitable regularizations of delta functions we study the convergence properties of such discretizations for constant coefficient elliptic problems using the immersed boundary method as an example we show how the order of the differential operator order of the finite difference discretization and properties of the discrete delta function all influence the local convergence behavior in particular we show how a recently introduced property of discrete delta functions the smoothing order is important in the determination of local convergence rates

a penalization method to take into account obstacles in incompressible viscous flows from the navier stokes brinkman model a penalization method has been derived by several authors to compute incompressible navier stokes equations around obstacles in this paper convergence theorems and error estimates are derived for two kinds of penalization the first one corresponds to l 2 penalization inducing a darcy equation in the solid body the second one corresponds to a h 1 penalization and induces a brinkman equation in the body numerical tests are performed to confirm the efficiency and accuracy of the method

the variational multiscale method a paradigm for computational mechanics abstract we present a general treatment of the variational multiscale method in the context of an abstract dirichlet problem we show how the exact theory represents a paradigm for subgrid scale models and a posteriori error estimation we examine hierarchical p methods and bubbles in order to understand and ultimately approximate the fine scale green s function which appears in the theory we review relationships between residual free bubbles element green s functions and stabilized methods these suggest the applicability of the methodology to physically interesting problems in fluid mechanics acoustics and electromagnetics

high order unconditionally stable fc ad solvers for general smooth domains ii elliptic parabolic and hyperbolic pdes theoretical considerations a new pde solver was introduced recently in part i of this two paper sequence on the basis of two main concepts the well known alternating direction implicit adi approach on one hand and a certain fourier continuation fc method for the resolution of the gibbs phenomenon on the other unlike previous alternating direction methods of order higher than one which only deliver unconditional stability for rectangular domains the new high order fc ad fourier continuation alternating direction algorithm yields unconditional stability for general domains at an o nlog n cost per time step for an n point spatial discretization grid in the present contribution we provide an overall theoretical discussion of the fc ad approach and we extend the fc ad methodology to linear hyperbolic pdes in particular we study the convergence properties of the newly introduced fc gram fourier continuation method for both approximation of general functions and solution of the alternating direction odes we also present for parabolic pdes on general domains and thus for our associated elliptic solvers a stability criterion which when satisfied ensures unconditional stability of the fc ad algorithm use of this criterion in conjunction with numerical evaluation of a series of singular values of the alternating direction discrete one dimensional operators suggests clearly that the fifth order accurate class of parabolic and elliptic fc ad solvers we propose is indeed unconditionally stable for all smooth spatial domains and for arbitrarily fine discretizations to illustrate the fc ad methodology in the hyperbolic pde context finally we present an example concerning the wave equation demonstrating sixth order spatial and fourth order temporal accuracy as well as a complete absence of the debilitating dispersion error also known as pollution error that arises as finite difference and finite element solvers are applied to solution of wave propagation problems

accuracy enhancement for non isoparametric finite element simulations in curved domains application to fluid flow abstract among a few known techniques the isoparametric version of the finite element method for meshes consisting of curved triangles or tetrahedra is the one most widely employed to solve pdes with essential conditions prescribed on curved boundaries it allows to recover optimal approximation properties that hold for elements of order greater than one in the energy norm for polytopic domains however besides a geometric complexity this technique requires the manipulation of rational functions and the use of numerical integration we consider a simple alternative to deal with dirichlet boundary conditions that bypasses these drawbacks without eroding qualitative approximation properties in the present work we first recall the main principle this technique is based upon by taking as a model the solution of the poisson equation with quadratic lagrange finite elements then we show that it extends very naturally to viscous incompressible flow problems although the technique applies to any higher order velocity pressure pairing as an illustration a thorough study thereof is conducted in the framework of the stokes system solved by the classical taylor hood method

fourier spectral embedded boundary solution of the poisson s and laplace equations with dirichlet boundary conditions a fourier spectral embedded boundary method for solution of the poisson s equation with dirichlet boundary conditions and arbitrary forcing functions including zero forcing function is presented in this paper this iterative method begins by transformation of the dirichlet boundary conditions from the physical boundaries to some corresponding regular grid points which are called the numerical boundaries using a second order interpolation method then the transformed boundary conditions and the forcing function are extended to a square smoothly and periodically via multiplying them by some suitable error functions instead of direct solution of the resulting extended poisson s problem it is suggested to define and solve an equivalent transient diffusion problem on the regular domain until achievement of the steady solution which is considered as the solution of the original problem without need of any numerical time integration method time advancement of the solution is obtained directly from the exact solution of the transient problem in the fourier space consequently timestep sizes can be chosen without stability limitations which it means higher rates of convergence in comparison with the classical relaxation methods the method is presented in details for one and two dimensional problems and a new emerged phenomenon which is called the saturation state is illustrated both in the physical and spectral spaces the numerical experiments have been performed on the one and two dimensional irregular domains to show the accuracy of the method and its superiority from the rate of convergence viewpoint to the other classical relaxation methods capability of the method in dealing with complex geometries and in presence of discontinuity at the boundaries has been shown via some numerical experiments on a four leaf shape geometry

three dimensional finite element methods for the stokes problem the tetrahedral finite element approximation of the stokes problem is analyzed by means of polynomials piecewise of degree k 1 for the velocity and continuous piecewise of degree k for the pressure a stability result is given for every k ge 1

an efficient method for the incompressible navier stokes equations on irregular domains with no slip boundary conditions high order up to the boundary common efficient schemes for the incompressible navier stokes equations such as projection or fractional step methods have limited temporal accuracy as a result of matrix splitting errors or introduce errors near the domain boundaries which destroy uniform convergence to the solution in this paper we recast the incompressible constant density navier stokes equations with the velocity prescribed at the boundary as an equivalent system for the primary variables velocity and pressure equation for the pressure the key difference from the usual approaches occurs at the boundaries where we use boundary conditions that unequivocally allow the pressure to be recovered from knowledge of the velocity at any fixed time this avoids the common difficulty of an apparently over determined poisson problem since in this alternative formulation the pressure can be accurately and efficiently recovered from the velocity the recast equations are ideal for numerical marching methods the new system can be discretized using a variety of methods including semi implicit treatments of viscosity and in principle to any desired order of accuracy in this work we illustrate the approach with a 2 d second order finite difference scheme on a cartesian grid and devise an algorithm to solve the equations on domains with curved non conforming boundaries including a case with a non trivial topology a circular obstruction inside the domain this algorithm achieves second order accuracy in the l norm for both the velocity and the pressure the scheme has a natural extension to 3 d

imex extensions of linear multistep methods with general monotonicity and boundedness properties for solving hyperbolic systems with stiff sources or relaxation terms time stepping methods should combine favorable monotonicity properties for shocks and steep solution gradients with good stability properties for stiff terms in this paper we consider implicit explicit imex multistep methods suitable methods will be constructed based on explicit methods with general monotonicity and boundedness properties for hyperbolic equations numerical comparisons are made with several implicit explicit runge kutta methods

a taxonomy of consistently stabilized finite element methods for the stokes problem stabilized mixed methods can circumvent the restrictive inf sup condition without introducing penalty errors for properly chosen stabilization parameters these methods are well posed for all conforming velocity pressure pairs however their variational forms have widely varying properties first stabilization offers a choice between weakly or strongly coercive bilinear forms that give rise to linear systems with identical solutions but very different matrix properties second coercivity may be conditional upon a proper choice of a stabilizing parameter here we focus on how these two aspects of stabilized methods affect their accuracy and efficient iterative solution we present results that indicate a preference of krylov subspace solvers for strongly coercive formulations stability criteria obtained by finite element and algebraic analyses are compared with numerical experiments while for two popular classes of stabilized methods sufficient stability bounds correlate well with numerical stability our experiments indicate the intriguing possibility that the pressure stabilized galerkin method is unconditionally stable

inverse spectral problem for quantum graphs the inverse spectral problem for the laplace operator on a finite metric graph is investigated it is shown that this problem has a unique solution for graphs with rationally independent edges and without vertices having valence 2 to prove the result a trace formula connecting the spectrum of the laplace operator with the set of periodic orbits for the metric graph is established

an immersed boundary method with formal second order accuracy and reduced numerical viscosity a formally second order accurate immersed boundary method is presented and tested in this paper we apply this new scheme to simulate the flow past a circular cylinder and study the effect of numerical viscosity on the accuracy of the computation by comparing the numerical results with those of a first order method the numerical evidence shows that the new scheme has less numerical viscosity and is therefore a better choice for the simulation of high reynolds number flows with immersed boundaries

stabilized discontinuous finite element approximations for stokes equations in this paper we derive two stabilized discontinuous finite element formulations symmetric and nonsymmetric for the stokes equations and the equations of the linear elasticity for almost incompressible materials these methods are derived via stabilization of a saddle point system where the continuity of the normal and tangential components of the velocity displacements are imposed in a weak sense via lagrange multipliers for both methods almost all reasonable pair of discontinuous finite element spaces can be used to approximate the velocity and the pressure optimal error estimate for the approximation of both the velocity of the symmetric formulation and pressure in l2 norm are obtained as well as one in a mesh dependent norm for the velocity in both symmetric and nonsymmetric formulations

stabilized finite element method for viscoplastic flow formulation and a simple progressive solution strategy abstract this paper presents a stabilized finite element formulation for steady state viscoplastic flow and a simple strategy for solving the resulting non linear equations with a newton raphson algorithm an eulerian stabilized finite element formulation is presented where mesh dependent terms are added element wise to enhance the stability of the mixed finite element formulation a local reconstruction method is used for computing derivatives of the stress field needed when higher order elements are used linearization of the weak form is derived to enable a newton raphson solution procedure of the resulting non linear equations in order to get convergence in the newton raphson algorithm a trial solution is needed which is within the radius of convergence an effective strategy for progressively moving inside the radius of convergence for highly non linear viscoplastic constitutive equations typical of metals is presented numerical experiments using the stabilization method with both linear and quadratic shape functions for the velocity and pressure fields in viscoplastic flow problems show that the stabilized method and the progressive convergence strategy are effective in non linear steady forming problems finally conclusions are inferred and extensions of this work are discussed

a fast immersed boundary method for external incompressible viscous flows using lattice green s functions a new parallel computationally efficient immersed boundary method for solving three dimensional viscous incompressible flows on unbounded domains is presented immersed surfaces with prescribed motions are generated using the interpolation and regularization operators obtained from the discrete delta function approach of the original peskin s immersed boundary method unlike peskin s method boundary forces are regarded as lagrange multipliers that are used to satisfy the no slip condition the incompressible navier stokes equations are discretized on an unbounded staggered cartesian grid and are solved in a finite number of operations using lattice green s function techniques these techniques are used to automatically enforce the natural free space boundary conditions and to implement a novel block wise adaptive grid that significantly reduces the run time cost of solutions by limiting operations to grid cells in the immediate vicinity and near wake region of the immersed surface these techniques also enable the construction of practical discrete viscous integrating factors that are used in combination with specialized half explicit runge kutta schemes to accurately and efficiently solve the differential algebraic equations describing the discrete momentum equation incompressibility constraint and no slip constraint linear systems of equations resulting from the time integration scheme are efficiently solved using an approximation free nested projection technique the algebraic properties of the discrete operators are used to reduce projection steps to simple discrete elliptic problems e g discrete poisson problems that are compatible with recent parallel fast multipole methods for difference equations numerical experiments on low aspect ratio flat plates and spheres at reynolds numbers up to 3700 are used to verify the accuracy and physical fidelity of the formulation

unconditionally stable mixed finite element methods for darcy flow unconditionally stable finite element methods for darcy flow are derived by adding least squares residual forms of the governing equations to the classical mixed formulations the proposed methods are free of mesh dependent stabilization parameters and allow the use of the classical continuous lagrangian finite element spaces of any order for the velocity and the potential stability convergence and error estimates are derived and numerical experiments are presented to demonstrate the flexibility of the proposed finite element formulations and to confirm the predicted rates of convergence

derivation and validation of a novel implicit second order accurate immersed boundary method a novel implicit second order accurate immersed boundary method ibm for simulating the flow around arbitrary stationary bodies is developed implemented and validated in this paper the ibm is used to efficiently take into account the existence of bodies within the fluid domain the flow domain consist of simple cartesian cells whereas the body can be arbitrary at the triangulated interface of the body and the fluid the immersed boundary the coefficients obtained from discretizing the navier stokes equations are closed with a second order accurate interpolation arising from the immersed boundary condition employed at the interface two different conditions are developed in this paper and it is shown that for the mirroring method the resulting coefficients lead to a well posed and diagonally dominant system which can be efficiently solved with a preconditioned krylov sub space solver the immersed boundary condition generates a fictitious reversed velocity field inside the immersed boundary which is excluded from the continuity equation to account for the presence of the ib in the pressure correction equation resulting in no mass flux over the ib the force acting on the object from the fluid is determined by integrating the pressure and the viscous forces over the object the method is validated by simulating the flow around a sphere for a range of re numbers it is shown that the drag is very well in agreement with experimental data accuracy and convergence studies are employed proving the second order accuracy of the method and showing the superiority in convergence rate compared to other ibm finally the drag force of a cluster of non spherical particles is employed to show the generality and potential of the method

stabilized mixed methods for the stokes problem the solution of the stokes problem is approximated by three stabilized mixed methods one due to hughes balestra and franca and the other two being variants of this procedure in each case the bilinear form associated with the saddle point problem of the standard mixed formulation is modified to become coercive over the finite element space error estimates are derived for each procedure

an absolutely stabilized finite element method for the stokes problem on presente une formulation a elements finis absolument stabilisee pour le probleme de stokes on etablit des estimations d erreurs optimales dans la norme l 2 pour l approximation des champs de vitesses et de pressions

investigations on two kinds of two level stabilized finite element methods for the stationary navier stokes equations abstract this article considers two kinds of two level stabilized finite element methods based on local gauss integral technique for the two dimensional stationary navier stokes equations approximated by the lowest equal order elements which do not satisfy the inf sup condition the two level methods consist of solving a small non linear system on the coarse mesh and then solving a linear system on the fine mesh the error analysis shows that the two level stabilized finite element methods provide an approximate solution with the convergence rate of the same order as the usual stabilized finite element solution solving the navier stokes equations on a fine mesh for a related choice of mesh widths therefore the two level methods are of practical importance in scientific computation finally the performance of two kinds of two level stabilized methods are compared in efficiency and precision aspects by a series of numerical experiments the conclusion is that simple two level stabilized method is a viable choice for the lowest equal order approximations of the stationary navier stokes problem

a smoothing technique for discrete delta functions with application to immersed boundary method in moving boundary simulations the effects of complex boundary conditions on flows are represented by a volume force in the immersed boundary methods the problem with this representation is that the volume force exhibits non physical oscillations in moving boundary simulations a smoothing technique for discrete delta functions has been developed in this paper to suppress the non physical oscillations in the volume forces we have found that the non physical oscillations are mainly due to the fact that the derivatives of the regular discrete delta functions do not satisfy certain moment conditions it has been shown that the smoothed discrete delta functions constructed in this paper have one order higher derivative than the regular ones moreover not only the smoothed discrete delta functions satisfy the first two discrete moment conditions but also their derivatives satisfy one order higher moment condition than the regular ones the smoothed discrete delta functions are tested by three test cases a one dimensional heat equation with a moving singular force a two dimensional flow past an oscillating cylinder and the vortex induced vibration of a cylinder the numerical examples in these cases demonstrate that the smoothed discrete delta functions can effectively suppress the non physical oscillations in the volume forces and improve the accuracy of the immersed boundary method with direct forcing in moving boundary simulations

an adaptive fast multipole accelerated poisson solver for complex geometries abstract we present a fast direct and adaptive poisson solver for complex two dimensional geometries based on potential theory and fast multipole acceleration more precisely the solver relies on the standard decomposition of the solution as the sum of a volume integral to account for the source distribution and a layer potential to enforce the desired boundary condition the volume integral is computed by applying the fmm on a square box that encloses the domain of interest for the sake of efficiency and convergence acceleration we first extend the source distribution the right hand side in the poisson equation to the enclosing box as a c 0 function using a fast boundary integral based method we demonstrate on multiply connected domains with irregular boundaries that this continuous extension leads to high accuracy without excessive adaptive refinement near the boundary and as a result to an extremely efficient black box fast solver

a new proof of the voronoi summation formula we present a short alternative proof of the voronoi summation formula which plays an important role in dirichlet s divisor problem and has recently found an application in physics as a trace formula for a schrodinger operator on a noncompact quantum graph

a high order immersed interface method for simulating unsteady incompressible flows on irregular domains immersed boundary methods and immersed interface methods are becoming increasingly popular for the computation of unsteady flows around complex geometries using a cartesian grid while good results both qualitative and quantitative have been obtained most of the methods rely on low order corrections to account for the immersed boundary the objective of the present work is to present as an alternative a high order modified immersed interface method for the 2d unsteady incompressible navier stokes equations in stream function vorticity formulation the method employs an explicit fourth order runge kutta time integration scheme fourth order compact finite differences for computation of spatial derivatives and a nine point fourth order compact discretization of the poisson equation for computation of the stream function corrections to the finite difference schemes are used to maintain high formal accuracy at the immersed boundary as confirmed by analytical tests to validate the method in its application to incompressible flows several physically relevant test cases are computed including uniform flow past a circular cylinder and tollmien schlichting waves in a boundary layer

an arbitrary lagrangian eulerian finite element method for solving three dimensional free surface flows abstract this paper discusses numerical solution of unsteady three dimensional free surface flows the governing equilibrium equations are written in the framework of the arbitrary lagrangian eulerian kinematic description the corresponding variational formulation is established afterwards since the variational problems are nonlinear with respect to the moving coordinates a second order approximate variational problem is derived after a consistent linearization of the referential motion stability of the discrete formulations is ensured with the help of a new stabilization method a robust preconditioned gmres algorithm is then used to solve the resulting set of nonlinear equations finally the computational algorithms are assessed through numerical studies of various problems a large sloshing flow in a three dimensional reservoir a discharge flow from a reservoir simulation of a liquid vortex produced inside a cylindrical container with a disk rotating at the bottom and a three dimensional practical hydraulic problem

new pathways and connections in number theory and analysis motivated by two incorrect claims of ramanujan abstract the focus of this paper commences with an examination of three not obviously related pages in ramanujan s lost notebook pages 336 335 and 332 in decreasing order of attention on page 336 ramanujan proposes two identities but the formulas are wrong each is vitiated by divergent series we concentrate on only one of the two incorrect identities which may have been devised to attack the extended divisor problem we prove here a corrected version of ramanujan s claim which contains the convergent series appearing in it the convergent series in ramanujan s faulty claim is similar to one used by g f voronoi g h hardy and others in their study of the classical dirichlet divisor problem this now brings us to page 335 which comprises two formulas featuring doubly infinite series of bessel functions the first being conjoined with the classical circle problem initiated by gauss and the second being associated with the dirichlet divisor problem the first and fourth authors along with sun kim have written several papers providing proofs of these two difficult formulas in different interpretations in this monograph we return to these two formulas and examine them in more general settings the aforementioned convergent series in ramanujan s identity is also similar to one that appears in a curious identity found in chapter 15 in ramanujan s second notebook written in a more elegant equivalent formulation on page 332 in the lost notebook this formula may be regarded as a formula for 1 2 and in 1925 s wigert obtained a generalization giving a formula for 1 k for any even integer k 2 we extend the work of ramanujan and wigert in this paper the voronoi summation formula appears prominently in our study in particular we generalize work of j r wilton and derive an analogue involving the sum of divisors function s n the modified bessel functions k s x arise in several contexts as do lommel functions we establish here new series and integral identities involving modified bessel functions and modified lommel functions among other results we establish a modular transformation for an infinite series involving s n and modified lommel functions we also discuss certain obscure related work of n s koshliakov we define and discuss two new related classes of integral transforms which we call koshliakov transforms because he first found elegant special cases of each

an anisotropic gls stabilized finite element method for incompressible flow problems abstract we consider a stabilized finite element approximation of incompressible flow problems in primitive velocity pressure variables which is based on a gls formulation equal order interpolation for the velocity and the pressure can be employed with this formulation the method introduced here is specially developed to be used on anisotropic finite element meshes with large element aspect ratios rather than using a single stabilization parameter within each element the jacobian matrix of the affine transformation from the reference element is employed for stabilization purposes stability and anisotropic error estimates in a suitable mesh dependent norm are proved for the linear stokes problem numerical results confirm the accuracy and robustness of the proposed method compared to similar previous stabilized schemes

exponential sums and lattice points iii the gauss circle problem and the dirichlet divisor problem are special cases of the problem of counting the points of the integer lattice in a planar domain bounded by a piecewise smooth curve in the bombieri iwaniec mozzochi exponential sums method we must count the number of pairs of arcs of the boundary curve which can be brought into coincidence by an automorphism of the integer lattice these coincidences are parametrised by integer points close to certain plane curves the resonance curves this paper sets up an iteration step from a strong hypothesis about integer points close to curves to a bound for the discrepancy the number of integer points minus the area as in the latest work on single exponential sums the bombieri iwaniec mozzochi method itself gives bounds for the number of integer points close to a curve in part of the required range and it can in principle be used iteratively we use a bound obtained by swinnerton dyer s approximation determinant method in the discrepancy estimate o r k log r lambda in terms of the maximum radius of curvature r we reduce k from 2 3 classical and 46 73 paper ii in this series to 131 208 the corresponding exponent in the dirichlet divisor problem becomes k 2 131 416

spectral domain embedding for elliptic pdes in complex domains spectral methods are a class of methods for solving partial differential equations pdes when the solution of the pde is analytic it is known that the spectral solutions converge exponentially as a function of the number of modes used the basic spectral method works only for regular domains such as rectangles or disks domain decomposition methods spectral element methods extend the applicability of spectral methods to more complex geometries an alternative is to embed the irregular domain into a regular one this paper uses the spectral method with domain embedding to solve pdes on complex geometry the running time of the new algorithm has the same order as that for the usual spectral collocation method for pdes on regular geometry the algorithm is extremely simple and can handle dirichlet neumann boundary conditions as well as nonlinear equations

on a two level finite element method for the incompressible navier stokes equations we consider the galerkin finite element method for the incompressible navier stokes equations in two dimensions where the finite dimensional space s employed consist of piecewise polynomials enriched with residual free bubble rfb functions to find the bubble part of the solution a two level finite element method tlfem is described and its application to the navier stokes equation is displayed numerical solutions employing the tlfem are presented for three benchmark problems we compare the numerical solutions using the tlfem with the numerical solutions using a stabilized method

on the order of accuracy of the immersed boundary method higher order convergence rates for sufficiently smooth problems the immersed boundary method is both a mathematical formulation and a numerical scheme for problems involving the interaction of a viscous incompressible fluid and a visco elastic structure in m c lai simulations of the flow past an array of circular cylinders as a test of the immersed boundary method ph d thesis courant institute of mathematical sciences new york university 1998 m c lai c s peskin an immersed boundary method with formal second order accuracy and reduced numerical viscosity j comput phys 160 2000 705 719 lai and peskin introduced a formally second order accurate immersed boundary method but the convergence properties of their algorithm have only been examined computationally for problems with nonsmooth solutions consequently in practice only first order convergence rates have been observed in the present work we describe a new formally second order accurate immersed boundary method and demonstrate its performance for a prototypical fluid structure interaction problem involving an immersed viscoelastic shell of finite thickness studied over a broad range of reynolds numbers we consider two sets of material properties for the viscoelastic structure including a case where the material properties of the coupled system are discontinuous at the fluid structure interface for both sets of material properties the true solutions appear to possess sufficient smoothness for the method to converge at a second order rate for fully resolved computations

fourier embedded domain methods extending a function defined on an irregular region to a rectangle so that the extension is spatially periodic and c a simple way to solve a partial differential equation in a non rectangular domain w is to embed the domain in a rectangle b and solve the problem more easily in the rectangle to apply a fourier spectral method on the rectangle it is necessary to extend the inhomogeneous term in the pde f x y to b in such a way that the extended function g x y is periodic and infinitely differentiable and yet is equal to f x y everywhere in the irregular domain w if the boundary of w email protected is defined as the zero isoline of a function f x y then a suitable extension is g x y f x y h email protected x y constant where h is a smoothed approximation to the step function which is defined in the main text

an immersed interface method for simulating the interaction of a fluid with moving boundaries in the immersed interface method boundaries are represented as singular force in the navier stokes equations which enters a numerical scheme as jump conditions recently we systematically derived all the necessary spatial and temporal jump conditions for simulating incompressible viscous flows subject to moving boundaries in 3d with second order spatial and temporal accuracy near the boundaries sheng xu z jane wang systematic derivation of jump conditions for the immersed interface method in three dimensional flow simulation siam j sci comput 2006 in press in this paper we implement the immersed interface method to incorporate these jump conditions in a 2d numerical scheme we study the accuracy efficiency and robustness of our method by simulating taylor couette flow flow induced by a relaxing balloon flow past single and multiple cylinders and flow around a flapping wing our results show that 1 our code has second order accuracy in the infinity norm for both the velocity and the pressure 2 the addition of an object introduces relatively insignificant computational cost 3 the method is equally effective in computing flow subject to boundaries with prescribed force or boundaries with prescribed motion

matched interface and boundary mib method for elliptic problems with sharp edged interfaces elliptic problems with sharp edged interfaces thin layered interfaces and interfaces that intersect with geometric boundary are notoriously challenging to existing numerical methods particularly when the solution is highly oscillatory this work generalizes the matched interface and boundary mib method previously designed for solving elliptic problems with curved interfaces to the aforementioned problems we classify these problems into five distinct topological relations involving the interfaces and the cartesian mesh lines flexible strategies are developed to systematically extends the computational domains near the interface so that the standard central finite difference scheme can be applied without the loss of accuracy fictitious values on the extended domains are determined by enforcing the physical jump conditions on the interface according to the local topology of the irregular point the concepts of primary and secondary fictitious values are introduced to deal with sharp edged interfaces for corner singularity or tip singularity an appropriate polynomial is multiplied to the solution to remove the singularity extensive numerical experiments confirm the designed second order convergence of the proposed method

fast immersed interface poisson solver for 3d unbounded problems around arbitrary geometries abstract we present a fast and efficient fourier based solver for the poisson problem around an arbitrary geometry in an unbounded 3d domain this solver merges two rewarding approaches the lattice green s function method and the immersed interface method using the sherman morrison woodbury decomposition formula the method is intended to be second order up to the boundary this is verified on two potential flow benchmarks we also further analyse the iterative process and the convergence behavior of the proposed algorithm the method is applicable to a wide range of problems involving a poisson equation around inner bodies which goes well beyond the present validation on potential flows

first order system least squares for coupled stokes darcy flow the coupled problem with stokes flow in one subdomain and a darcy flow model in a second subdomain is studied in this paper both flow problems are treated as first order systems involving pseudostress and velocity in the stokes case and using a flux pressure formulation in the darcy subdomain as process variables respectively the beavers joseph saffman interface conditions are treated by an appropriate interface functional which is added to the least squares functional associated with the subdomain problems a combination of h mathrm div conforming raviart thomas and standard h 1 conforming elements is used for the stokes as well as for the darcy subsystem the homogeneous least squares functional is shown to be equivalent to an appropriate norm allowing the use of standard finite element approximation estimates it also establishes the fact that the local evaluation of the least squares functional itself constitutes an a posteriori error estimator to be used for adaptive refinement strategies

pattern transitions and complexity for a nonlocal logistic map abstract we examine the pattern solutions in a generalized nonlocal logistic map that includes spatial kernels in both growth and competition terms we show that this map includes as a particular case the nonlocal fisher kolmogorov equation and we demonstrate the existence of three kinds of stationary nonlinear solutions one uniform one cosine type that we refer to as wavelike solution and another in the form of gaussian we also obtain analytical expressions that describe the nonlinear pattern behavior in the system and we establish the stability criterion we define thermodynamics quantities such as entropy and the order parameter based on this the pattern no pattern and pattern pattern transitions are properly analyzed we show that these pattern solutions may be related to the recently observed peak adding phenomenon in nonlinear optics

impossibility of fast stable approximation of analytic functions from equispaced samples it is shown that no stable procedure for approximating functions from equally spaced samples can converge exponentially for analytic functions to avoid instability one must settle for root exponential convergence the proof combines a bernstein inequality of 1912 with an estimate due to coppersmith and rivlin in 1992

fourier embedded domain methods periodic and c extension of a function defined on an irregular region to a rectangle via convolution with gaussian kernels abstract one possible way to solve a partial differential equation in an irregular region is the use of the so called domain embedding methods in where the domain of interest is embedded within a rectangle in order to apply a fourier spectral method on the rectangle the inhomogeneous term f x y has to be extended to a new function g x y that is periodic and infinitely differentiable and equal to f x y everywhere within some authors have given explicit methods to compute extensions with infinite order convergence for the cases in where the boundary of can be defined as the zero isoline of a function x y for the cases in where this is not possible we suggest a new method to build these extensions via convolution with gaussian kernels

high order matched interface and boundary method for elliptic equations with discontinuous coefficients and singular sources this paper introduces a novel high order interface scheme the matched interface and boundary mib method for solving elliptic equations with discontinuous coefficients and singular sources on cartesian grids by appropriate use of auxiliary line and or fictitious points physical jump conditions are enforced at the interface unlike other existing interface schemes the proposed method disassociates the enforcement of physical jump conditions from the discretization of the differential equation under study to construct higher order interface schemes the proposed mib method bypasses the major challenge of implementing high order jump conditions by repeatedly enforcing the lowest order jump conditions the proposed mib method is of arbitrarily high order in principle in treating straight regular interfaces we construct mib schemes up to 16th order for more general elliptic problems with curved irregular interfaces and boundary up to 6th order mib schemes have been demonstrated by employing the standard high order finite difference schemes to discretize the laplacian the present mib method automatically reduces to the standard central difference scheme when the interface is absent the immersed interface method iim is regenerated for a comparison study of the proposed method the robustness of the mib method is verified against the large magnitude of the jump discontinuity across the interface the nature of high efficiency and low memory requirement of the mib method is extensively validated via solving various elliptic immersed interface problems in two and three dimensions ee dimensions

parametric raviart thomas elements for mixed methods on domains with curved surfaces the finite element approximation on curved boundaries using parametric raviart thomas spaces is studied in the context of the mixed formulation of poisson s equation as a saddle point system it is shown that optimal order convergence is retained on domains with piecewise c k 2 boundary for the parametric raviart thomas space of degree k geq 0 under the usual regularity assumptions this extends the analysis in f bertrand s munzenmaier and g starke siam j numer anal 52 2014 pp 3165 3180 from the first order system least squares formulation to mixed approaches of saddle point type in addition a detailed proof of the crucial estimate in three dimensions is given which handles some complications not present in the two dimensional case moreover the appropriate treatment of inhomogeneous flux boundary conditions is discussed the results are confirmed by computational results which also demonstrate that optimal order convergence is not achieved in general if standard raviart th

immersed boundary smooth extension the immersed boundary method is a simple efficient and robust numerical scheme for solving pde in general domains yet it only achieves first order spatial accuracy near embedded boundaries in this paper we introduce a new high order numerical method which we call the immersed boundary smooth extension ibse method the ibse method achieves high order accuracy by smoothly extending the unknown solution of the pde from a given smooth domain to a larger computational domain enabling the use of simple cartesian grid discretizations e g fourier spectral methods the method preserves much of the flexibility and robustness of the original ib method in particular it requires minimal geometric information to describe the boundary and relies only on convolution with regularized delta functions to communicate information between the computational grid and the boundary we present a fast algorithm for solving elliptic equations which forms the basis for simple high order implicit time methods for parabolic pde and implicit explicit methods for related nonlinear pde we apply the ibse method to solve the poisson heat burgers and fitzhugh nagumo equations and demonstrate fourth order pointwise convergence for dirichlet problems and third order pointwise convergence for neumann problems

tether force constraints in stokes flow by the immersed boundary method on a periodic domain the immersed boundary method is an algorithm for simulating the interaction of immersed elastic bodies or boundaries with a viscous incompressible fluid the immersed elastic material is represented in the fluid equations by a system or field of applied forces the particular case of stokes flow with applied forces on a periodic domain involves two related mathematical complications one of these is that an arbitrary constant vector may be added to the fluid velocity and the other is the constraint that the integral of the applied force must be zero typically forces defined on a freely floating elastic immersed boundary or body satisfy this constraint but there are many important classes of forces that do not for example the so called tether forces that are used to prescribe the simulated configuration of an immersed boundary possibly in a time dependent manner typically do not sum to zero another type of force that does not have zero integral is a uniform force density that may be used to simulate an overall pressure gradient driving flow through a system we present a method for periodic stokes flow that when used with tether points admits the use of all forces irrespective of their integral over the domain a byproduct of this method is that the additive constant velocity associated with periodic stokes flow is uniquely determined indeed the additive constant is chosen at each time step so that the sum of the tether forces balances the sum of any other forces that may be applied

a spectral fc solver for the compressible navier stokes equations in general domains i explicit time stepping we present a fourier continuation fc algorithm for the solution of the fully nonlinear compressible navier stokes equations in general spatial domains the new scheme is based on the recently introduced accelerated fc method which enables use of highly accurate fourier expansions as the main building block of general domain pde solvers previous fc based pde solvers are restricted to linear scalar equations with constant coefficients the fc methodology presented in this text thus constitutes a significant generalization of the previous fc schemes as it yields general domain fc solvers for nonlinear systems of pdes while not restricted to periodic boundary conditions and therefore applicable to general boundary value problems on arbitrary domains the proposed algorithm inherits many of the highly desirable properties arising from rapidly convergent fourier expansions including high order convergence essentially spectrally accurate dispersion relations and much milder cfl constraints than those imposed by polynomial based spectral methods since for example the spectral radius of the fc first derivative grows linearly with the number of spatial discretization points we demonstrate the accuracy and optimal parallel efficiency of the algorithm in a variety of scientific and engineering contexts relevant to fluid dynamics and nonlinear acoustics

first order system least squares on curved boundaries higher order raviart thomas elements with this paper our investigation of the finite element approximation on curved boundaries using raviart thomas spaces in the context of first order system least squares methods is continued and extended to the higher order case it is shown that the optimal order of convergence is retained from the lowest order case if a parametric version of raviart thomas elements is used this is illustrated numerically for an elliptic boundary value problem involving a circular boundary curve

immersed boundary smooth extension ibse a high order method for solving incompressible flows in arbitrary smooth domains abstract the immersed boundary method is a simple efficient and robust numerical scheme for solving pde in general domains yet for fluid problems it only achieves first order spatial accuracy near embedded boundaries for the velocity field and fails to converge pointwise for elements of the stress tensor in a previous work we introduced the immersed boundary smooth extension ibse method a variation of the ib method that achieves high order accuracy for elliptic pde by smoothly extending the unknown solution of the pde from a given smooth domain to a larger computational domain enabling the use of simple cartesian grid discretizations in this work we extend the ibse method to allow for the imposition of a divergence constraint and demonstrate high order convergence for the stokes and incompressible navier stokes equations up to third order pointwise convergence for the velocity field and second order pointwise convergence for all elements of the stress tensor the method is flexible to the underlying discretization we demonstrate solutions produced using both a fourier spectral discretization and a standard second order finite difference discretization

local projection stabilization of equal order interpolation applied to the stokes problem the local projection stabilization allows us to circumvent the babuska brezzi condition and to use equal order interpolation for discretizing the stokes problem the projection is usually done in a two level approach by projecting the pressure gradient onto a discontinuous finite element space living on a patch of elements we propose a new local projection stabilization method based on possibly enriched finite element spaces and discontinuous projection spaces defined on the same mesh optimal order of convergence is shown for pairs of approximation and projection spaces satisfying a certain inf sup condition examples are enriched simplicial finite elements and standard quadrilateral hexahedral elements the new approach overcomes the problem of an increasing discretization stencil and thus is simple to implement in existing computer codes numerical tests confirm the theoretical convergence results which are robust with respect to the user chosen stabilization parameter

high order solution of poisson problems with piecewise constant coefficients and interface jumps we present a fast and accurate algorithm to solve poisson problems in complex geometries using regular cartesian grids we consider a variety of configurations including poisson problems with interfaces across which the solution is discontinuous of the type arising in multi fluid flows the algorithm is based on a combination of the correction function method cfm and boundary integral methods bim interface and boundary conditions can be treated in a fast and accurate manner using boundary integral equations and the associated bim unfortunately bim can be costly when the solution is needed everywhere in a grid e g fluid flow problems we use the cfm to circumvent this issue the solution from the bim is used to rewrite the problem as a series of poisson problems in rectangular domainswhich requires the bim solution at interfaces boundaries only these poisson problems involve discontinuities at interfaces of the type that the cfm can handle hence we use the cfm to solve them to high order of accuracy with finite differences and a fast fourier transform based fast poisson solver we present 2 d examples of the algorithm applied to poisson problems involving complex geometries including cases in which the solution is discontinuous we show that the algorithm produces solutions that converge with either 3rd or 4th order of accuracy depending on the type of boundary condition and solution discontinuity

analysis of the small viscosity and large reaction coefficient in the computation of the generalized stokes problem by a novel stabilized finite element method abstract in this paper we propose and analyze a novel stabilized finite element method fem for the system of generalized stokes equations arising from the time discretization of transient stokes problem the system involves a small viscosity which is proportional to the inverse of large reynolds number and a large reaction coefficient which is the inverse of small time step the proposed stabilized fem employs the c 0 piecewise linear elements for both velocity field and pressure on the same mesh and uses the residuals of the momentum equation and the divergence free equation to define the stabilization terms the stabilization parameters are fixed and element independent without a comparison of the viscosity the reaction coefficient and the mesh size using the finite element solution of an auxiliary boundary value problem as the interpolating function for velocity and the h 1 seminorm projection for pressure instead of the usual nodal interpolants we derive error estimates for the stabilized finite element approximations to velocity and pressure in the l 2 and h 1 norms and most importantly we explicitly establish the dependence of error bounds on the viscosity the reaction coefficient and the mesh size our analysis reveals that this stabilized fem is particularly suitable for the generalized stokes system with a small viscosity and a large reaction coefficient which has never been achieved before in the error analysis of other stabilization methods in the literature we then numerically confirm the effectiveness of the proposed stabilized fem comparisons made with other existing stabilization methods show that the newly proposed method can attain better accuracy and stability

a gaussian like immersed boundary kernel with three continuous derivatives and improved translational invariance the immersed boundary ib method is a general mathematical framework for studying problems involving fluid structure interactions in which an elastic structure is immersed in a viscous incompressible fluid in the ib formulation the fluid described by eulerian variables is coupled with the immersed structure described by lagrangian variables via the use of the dirac delta function from a numerical standpoint the lagrangian force spreading and the eulerian velocity interpolation are carried out by a regularized compactly supported discrete delta function which is assumed to be a tensor product of a single variable immersed boundary kernel ib kernels are derived from a set of postulates designed to achieve approximate grid translational invariance interpolation accuracy and computational efficiency in this note we present a new 6 point immersed boundary kernel that is c 3 and yields a substantially improved translational invariance compared to other common ib kernels

population persistence in weakly coupled sinks abstract we consider a single species population obeying a saturated growth model with spatial diffusion taken into account explicitly strong spatial heterogeneity is considered represented by a position dependent reproduction rate the geometry of the problem is that of two patches where the reproductive rate is positive surrounded by unfavorable patches where it is negative we focus on the particular case where the population would not persist in the single patches sinks we find by means of an analytical derivation supplemented by a numerical calculation the conditions for the persistence of the population in the compound system of weakly connected patches we show that persistence is possible even if each individual patch is a sink where the population would go extinct the results are of particular relevance for ecological management at the landscape level showing that small patches may harbor populations as long as the connectivity with adjacent patches is maintained microcosmos experiences with bacteria could be performed for experimental verification of the predictions

edge stabilization for the generalized stokes problem a continuous interior penalty method in this note we introduce and analyze a stabilized finite element method for the generalized stokes equation stability is obtained by adding a least squares penalization of the gradient jumps acro

isolation effects in a system of two mutually communicating identical patches abstract starting from the fisher kolmogorov petrovskii piskunov equation fkpp we model the dynamic of a diffusive system with two mutually communicating identical patches and isolated of the remaining matrix for this system we find the minimal size of each fragment in the explicit form and compare with the explicit results for similar problems found in the literature from this comparison emerges an unexpected result that for a same set of the parameters the isolated system studied in this work with size l can be better or worst than the non isolated systems with the same size l uniquely depending on the parameter a 0 internal conditions of the patches due to the fact that this result is unexpected we propose an experimental verification

optimal spectral approximation of 2n order differential operators by mixed isogeometric analysis abstract we approximate the spectra of a class of 2 n order differential operators using isogeometric analysis in mixed formulations this class includes a wide range of differential operators such as those arising in elliptic biharmonic cahn hilliard swift hohenberg and phase field crystal equations the spectra of the differential operators are approximated by solving differential eigenvalue problems in mixed formulations which require auxiliary parameters the mixed isogeometric formulation when applying classical quadrature rules leads to an eigenvalue error convergence of order 2 p where p is the order of the underlying b spline space we improve this order to be 2 p 2 by applying optimally blended quadrature rules developed in puzyrev et al 2017 caloet al 0000 and this order is an optimum in the view of dispersion error we also compare these results with the mixed finite elements and show numerically that the mixed isogeometric analysis leads to significantly better spectral approximations

stochastic skellam model abstract we consider the dynamics of a biological population described by the fisher kolmogorov petrovskii piskunov fkpp equation in the case where the spatial domain consists of alternating favorable and adverse patches whose sizes are distributed randomly for the one dimensional case we define a stochastic analogue of the classical critical patch size we address the issue of persistence of a population and we show that the minimum fraction of the length of favorable segments to the total length is always smaller in the stochastic case than in a periodic arrangement in this sense spatial stochasticity favors viability of a population

the riemann zeros and eigenvalue asymptotics comparison between formulae for the counting functions of the heights tn of the riemann zeros and of semiclassical quantum eigenvalues en suggests that the tn are eigenvalues of an unknown hermitean operator h obtained by quantizing a classical dynamical system with hamiltonian hcl many features of hcl are provided by the analogy for example the riemann dynamics should be chaotic and have periodic orbits whose periods are multiples of logarithms of prime numbers statistics of the tn have a similar structure to those of the semiclassical en in particular they display random matrix universality at short range and nonuniversal behaviour over longer ranges very refined features of the statistics of the tn can be computed accurately from formulae with quantum analogues the riemann siegel formula for the zeta function is described in detail its interpretation as a relation between long and short periodic orbits gives further insights into the quantum spectral fluctuations we speculate that the riemann dynamics is related to the trajectories generated by the classical hamiltonian hcl xp

an extension of the immersed boundary method based on the distributed lagrange multiplier approach an extended formulation of the immersed boundary method which facilitates simulation of incompressible isothermal and natural convection flows around immersed bodies and which may be applied for linear stability analysis of the flows is presented the lagrangian forces and heat sources are distributed on the fluid structure interface the method treats pressure the lagrangian forces and heat sources as distributed lagrange multipliers thereby implicitly providing the kinematic constraints of no slip and the corresponding thermal boundary conditions for immersed surfaces extensive verification of the developed method for both isothermal and natural convection 2d flows is provided strategies for adapting the developed approach to realistic 3d configurations are discussed

analysis of a one dimensional model for the immersed boundary method numerical methods are studied for the one dimensional heat equation with a singular forcing term u t u xx c t delta x alpha t the delta function delta x is replaced by a discrete approximation d h x and the resulting equation is solved by a crank nicolson method on a uniform grid the accuracy of this method is analyzed for various choices of d h the case where c t is specified and also the case where c is determined implicitly by a constraint on the solution at the point a are studied these problems serve as a model for the immersed boundary method of peskin for incompressible flow problems in irregular regions some insight is gained into the accuracy that can be achieved and the importance of choosing appropriate discrete delta functions

first order system least squares for second order partial differential equations part i this paper develops a least squares functional that arises from recasting general second order uniformly elliptic partial differential equations in n 2 or 3 dimensions as a system of first order equations in part i z cai r d lazarov t manteuffel and s mccormick siam j numer anal 31 1994 pp 1785 1799 a similar functional was developed and shown to be elliptic in the h divv times h 1 norm and to yield optimal convergence for finite element subspaces of h divv times h 1 in this paper the functional is modified by adding a compatible constraint and imposing additional boundary conditions on the first order system the resulting functional is proved to be elliptic in the h 1 n 1 norm this immediately implies optimal error estimates for finite element approximation by standard subspaces of h 1 n 1 another direct consequence of this ellipticity is that multiplicative and additive multigrid algorithms applied to the resulting discrete functionals are optimally convergent as an alternative to perturbation based approaches the least squares approach developed here applies directly to convection diffusion reaction equations in a unified way and also admits a fast multigrid solver historically a missing ingredient in least squares methodology

high order unconditionally stable fc ad solvers for general smooth domains i basic elements we introduce a new methodology for the numerical solution of partial differential equations in general spatial domains our algorithms are based on the use of the well known alternating direction implicit adi approach in conjunction with a certain fourier continuation fc method for the resolution of the gibbs phenomenon unlike previous alternating direction methods of order higher than one which can only deliver unconditional stability for rectangular domains the present high order algorithms possess the desirable property of unconditional stability for general domains the computational time required by our algorithms to advance a solution by one time step in turn grows in an essentially linear manner with the number of spatial discretization points used in this paper we demonstrate the fc ad methodology through a variety of examples concerning the heat and laplace equations in two and three dimensional domains with smooth boundaries applications of the fc ad methodology to hyperbolic pdes together with a theoretical discussion of the method will be put forth in a subsequent contribution the numerical examples presented in this text demonstrate the unconditional stability and high order convergence of the proposed algorithms as well the very significant improvements they can provide in one of our examples we demonstrate a one thousand improvement factor over the computing times required by some of the most efficient alternative general domain solvers

a new high order immersed interface method for solving elliptic equations with imbedded interface of discontinuity this paper presents a new high order immersed interface method for elliptic equations with imbedded interface of discontinuity compared with the original second order immersed interface method of r j leveque z li the immersed interface method for elliptic equations with discontinuous coefficients and singular sources siam j numer anal 31 1994 1001 25 the new method achieves arbitrarily high order accuracy for derivatives at an irregular grid point by imposing only two physical jump conditions together with a wider set of grid stencils the new interface difference formulas are expressed in a general explicit form so that they can be applied to different multi dimensional problems without any modification the new interface algorithms of up to o h 4 accuracy have been derived and tested on several one and two dimensional elliptic equations with imbedded interface compared to the standard second order immersed interface method the test results show that the new fourth order immersed interface method leads to a significant improvement in accuracy of the numerical solutions the proposed method has potential advantages in the application to two phase flow because of its high order accuracy and simplicity in applications

the immersed boundary method a projection approach a new formulation of the immersed boundary method with a structure algebraically identical to the traditional fractional step method is presented for incompressible flow over bodies with prescribed surface motion like previous methods a boundary force is applied at the immersed surface to satisfy the no slip constraint this extra constraint can be added to the incompressible navier stokes equations by introducing regularization and interpolation operators the current method gives prominence to the role of the boundary force acting as a lagrange multiplier to satisfy the no slip condition this role is analogous to the effect of pressure on the momentum equation to satisfy the divergence free constraint the current immersed boundary method removes slip and non divergence free components of the velocity field through a projection the boundary force is determined implicitly without any constitutive relations allowing the present formulation to use larger cfl numbers compared to some past methods symmetry and positive definiteness of the system are preserved such that the conjugate gradient method can be used to solve for the flow field examples show that the current formulation achieves second order temporal accuracy and better than first order spatial accuracy in l 2 norms for one and two dimensional test problems results from two dimensional simulations of flows over stationary and moving cylinders are in good agreement with those from previous experimental and numerical studies

an exact trace formula and zeta functions for an infinite quantum graph with a non standard weyl asymptotics we study a quantum hamiltonian that is given by the negative laplacian and an infinite chain of like potentials with strength 0 on the half line and which is equivalent to a one parameter family of laplacians on an infinite metric graph this graph consists of an infinite chain of edges with the metric structure defined by assigning an interval in 0 ln to each edge with length we show that the one parameter family of quantum graphs possesses a purely discrete and strictly positive spectrum for each 0 and prove that the dirichlet laplacian is the limit of the one parameter family in the strong resolvent sense the spectrum of the resulting dirichlet quantum graph is also purely discrete the eigenvalues are given by n n2 with multiplicities d n where d n denotes the divisor function we can thus relate the spectral problem of this infinite quantum graph to dirichlet s famous divisor problem and infer the non standard weyl asymptotics for the eigenvalue counting function based on an exact trace formula the voronoi summation formula we derive explicit formulae for the trace of the wave group the heat kernel the resolvent and for various spectral zeta functions these results enable us to establish a well defined renormalized secular equation and a selberg like zeta function defined in terms of the classical periodic orbits of the graph for which we derive an exact functional equation and prove that the analogue of the riemann hypothesis is true

multilevel boundary functionals for least squares mixed finite element methods for least squares mixed finite element methods for the first order system formulation of second order elliptic problems a technique for the weak enforcement of boundary conditions is presented this approach is based on least squares boundary functionals which are equivalent to the h 1 2 and h1 2 norms on the trace spaces of lowest order raviart thomas elements for the flux and standard continuous piecewise linear elements for the pressure respectively continuity and coercivity of the resulting bilinear form is proved implying optimal order convergence of the resulting galerkin approximation the boundary least squares functional is implemented using multilevel principles and the technique is tested numerically for a model problem

applicability of the fisher equation to bacterial population dynamics the applicability of the fisher equation which combines diffusion with logistic nonlinearity to population dynamics of bacterial colonies is studied with the help of explicit analytic solutions for the spatial distribution of a stationary bacterial population under a static mask the mask protects bacteria from ultraviolet light the solution which is in terms of jacobian elliptic functions is used to provide a practical prescription to extract fisher equation parameters from observations and to decide on the validity of the fisher equation

an immersed boundary method with divergence free velocity interpolation and force spreading abstract the immersed boundary ib method is a mathematical framework for constructing robust numerical methods to study fluid structure interaction in problems involving an elastic structure immersed in a viscous fluid the ib formulation uses an eulerian representation of the fluid and a lagrangian representation of the structure the lagrangian and eulerian frames are coupled by integral transforms with delta function kernels the discretized ib equations use approximations to these transforms with regularized delta function kernels to interpolate the fluid velocity to the structure and to spread structural forces to the fluid it is well known that the conventional ib method can suffer from poor volume conservation since the interpolated lagrangian velocity field is not generally divergence free and so this can cause spurious volume changes in practice the lack of volume conservation is especially pronounced for cases where there are large pressure differences across thin structural boundaries the aim of this paper is to greatly reduce the volume error of the ib method by introducing velocity interpolation and force spreading schemes with the properties that the interpolated velocity field in which the structure moves is at least c 1 and satisfies a continuous divergence free condition and that the force spreading operator is the adjoint of the velocity interpolation operator we confirm through numerical experiments in two and three spatial dimensions that this new ib method is able to achieve substantial improvement in volume conservation compared to other existing ib methods at the expense of a modest increase in the computational cost further the new method provides smoother lagrangian forces tractions than traditional ib methods the method presented here is restricted to periodic computational domains its generalization to non periodic domains is important future work

stabilized low order finite elements for stokes equations with damping abstract in this paper we consider the stationary stokes equations with damping low order mixed finite element spaces are used to approximate the velocity and the pressure and a local pressure projection stabilization method is used for the pairs to overcome the lack of the inf sup condition the stability of this method is proved and the optimal order error estimates are derived by some nonlinear analysis techniques at last two numerical examples are implemented to test the stability and effectiveness of the proposed method

an adaptive stabilized finite element scheme for a water quality model abstract residual type a posteriori error estimators are introduced in this paper for an advection diffusion reaction problem with a dirac delta source term the error is measured in an adequately weighted w 1 p norm these estimators are proved to yield global upper and local lower bounds for the corresponding norms of the error they are used to guide adaptive procedures which are experimentally shown to lead to optimal orders of convergence