imposing dirichlet boundary conditions in the extended finite element method this paper is devoted to the imposition of dirichlet type conditions within the extended finite element method x fem this method allows one to easily model surfaces of discontinuity or domain boundaries on a mesh not necessarily conforming to these surfaces imposing neumann boundary conditions on boundaries running through the elements is straightforward and does preserve the optimal rate of convergence of the background mesh observed numerically in earlier papers on the contrary much less work has been devoted to dirichlet boundary conditions for the x fem or the limiting case of stiff boundary conditions in this paper we introduce a strategy to impose dirichlet boundary conditions while preserving the optimal rate of convergence the key aspect is the construction of the correct lagrange multiplier space on the boundary as an application we suggest to use this new approach to impose precisely zero pressure on the moving resin front in resin transfer moulding rtm process while avoiding remeshing the case of inner conditions is also discussed as well as two important practical cases material interfaces and phase transformation front capturing copyright 2006 john wiley sons ltd

third order boundary value problems with nonlocal boundary conditions we treat the existence of multiple positive solutions for a third order boundary value problem with nonlocal boundary conditions in our work each of the three boundary conditions is a linear functional on c 0 1 that is given by a stieltjes integral but is not necessarily a positive functional

a new mixed finite element formulation and the mac method for the stokes equations a new mixed finite element method is formulated for the stokes equations in which the two components of the velocity and the pressure are defined on different meshes first order error estimates are obtained for both the velocity and the pressure also the well known mac method is derived from the resulting finite element method

positive solutions to differential inclusions with nonlocal nonlinear boundary conditions we demonstrate the existence of at least one positive solution to the differential inclusion u t f t u t email protected 0 1 equipped with the boundary conditions u 0 h f u and u 1 0 here f is linear functional realized as a lebesgue stieltjes integral and h is a nonlinear function consequently the boundary condition at t 0 can be both nonlocal and nonlinear by imposing an asymptotic condition on h we provide conditions under which the problem will have at least one positive solution

fictitious domain approach with hp finite element approximation for incompressible fluid flow we consider the application of fictitious domain approach combined with least squares spectral elements for the numerical solution of fluid dynamic incompressible equations fictitious domain methods allow problems formulated on a complicated shaped domain to be solved on a simpler domain containing least squares spectral element method has been used to develop the discrete model as this scheme combines the generality of finite element methods with the accuracy of spectral methods moreover the least squares methods have theoretical and computational advantages in the algorithmic design and implementation this paper presents the formulation and validation of the fictitious domain least squares spectral element approach for the steady incompressible navier stokes equations the convergence of the approximated solution is verified solving two dimensional benchmark problems demonstrating the predictive capability of the proposed formulation

existence and multiplicity of non negative solutions for systems of perturbed hammerstein integral equations abstract we study the existence of positive solutions for a system of perturbed hammerstein integral equations our setting is quite general and covers a wide class of systems of boundary value problems the main ingredient in our theory is the classical fixed point index theory for compact maps

iterative process with errors to locally strictly pseudocontractive maps in banach spaces abstract let k be a nonempty subset of a uniformly smooth banach space x and a k x be a locally strickly pseudocontractive map with bounded range the mann and ishikawa iterative sequences with errors which strongly converge to the unique fixed point of a are given under weaker conditions these results answer positively the open problems mentioned by deng and ding 1 p 122 and chidume 2 p 550 the related results deal with the problems that the mann and ishikawa iterative sequences with errors strongly converge to the unique solution of the equation tx f when t x x is locally strongly accretive such that the range of i t is bounded

new harnack inequalities and existence theorems for radially symmetric solutions of elliptic pdes with sign changing or vanishing green s function abstract we consider the perturbed hammerstein integral equation y t 1 t h 1 1 y 2 t h 2 2 y 0 1 g t s f s y s d s where 1 and 2 are linear functionals realized as stieltjes integrals with associated signed stieltjes measures we demonstrate that by utilizing a nonstandard order cone together with an associated nonstandard open set existence of a positive solution to the integral equation can be guaranteed under relatively mild hypotheses on the various constituent functions and functionals in the case in which the green s function t s g t s is allowed to both vanish and change sign as an example we apply our results to radially symmetric elliptic pdes of the form u h x g u x x r 1 r 2 r u x x b r 2 0 u r 1 x u r x x b 1 h u where 0 r 1 r r 2 is a linear functional and h is a continuous function as part of this analysis we demonstrate a new methodology for constructing a suitable harnack like inequality associated to the boundary value problem

existence and nonexistence results for positive solutions of an integral boundary value problem abstract this paper deals with the existence and nonexistence of positive solutions for the integral boundary value problem a u b u g t f t u cos 0 u 0 sin 0 u 0 0 1 u d cos 1 u 1 sin 1 u 1 0 1 u d we provide sufficient conditions for the existence of either none or one or more positive solutions of the above problem the main tool used in the proofs of existence results is a fixed point theorem in a cone due to krasnoselskii and zabreiko the results presented substantially unify and extend the existing results in the literature

a second order virtual node algorithm for nearly incompressible linear elasticity in irregular domains we present a cut cell method in r 2 for enforcing dirichlet and neumann boundary conditions with nearly incompressible linear elastic materials in irregular domains virtual nodes on cut uniform grid cells are used to provide geometric flexibility in the domain boundary shape without sacrificing accuracy we use a mixed formulation utilizing a mac type staggered grid with piecewise bilinear displacements centered at cell faces and piecewise constant pressures at cell centers these discretization choices provide the necessary stability in the incompressible limit and the necessary accuracy in cut cells numerical experiments suggest second order accuracy in l we target high resolution problems and present a class of geometric multigrid methods for solving the discrete equations for displacements and pressures that achieves nearly optimal convergence rates independent of grid resolution

an adaptive formally second order accurate version of the immersed boundary method like many problems in biofluid mechanics cardiac mechanics can be modeled as the dynamic interaction of a viscous incompressible fluid the blood and a visco elastic structure the muscular walls and the valves of the heart the immersed boundary method is a mathematical formulation and numerical approach to such problems that was originally introduced to study blood flow through heart valves and extensions of this work have yielded a three dimensional model of the heart and great vessels in the present work we introduce a new adaptive version of the immersed boundary method this adaptive scheme employs the same hierarchical structured grid approach but a different numerical scheme as the two dimensional adaptive immersed boundary method of roma et al a multilevel self adaptive version of the immersed boundary method ph d thesis courant institute of mathematical sciences new york university 1996 an adaptive version of the immersed boundary method j comput phys 153 2 1999 509 534 and is based on a formally second order accurate i e second order accurate for problems with sufficiently smooth solutions version of the immersed boundary method that we have recently described b e griffith c s peskin on the order of accuracy of the immersed boundary method higher order convergence rates for sufficiently smooth problems j comput phys 208 1 2005 75 105 actual second order convergence rates are obtained for both the uniform and adaptive methods by considering the interaction of a viscous incompressible flow and an anisotropic incompressible viscoelastic shell we also present initial results from the application of this methodology to the three dimensional simulation of blood flow in the heart and great vessels the results obtained by the adaptive method show good qualitative agreement with simulation results obtained by earlier non adaptive versions of the method but the flow in the vicinity of the model heart valves indicates that the new methodology provides enhanced boundary layer resolution differences are also observed in the flow about the mitral valve leaflets

an immersed interface method for incompressible navier stokes equations the method developed in this paper is motivated by peskin s immersed boundary ib method and allows one to model the motion of flexible membranes or other structures immersed in viscous incompressible fluid using a fluid solver on a fixed cartesian grid the ib method uses a set of discrete delta functions to spread the entire singular force exerted by the immersed boundary to the nearby fluid grid points our method instead incorporates part of this force into jump conditions for the pressure avoiding discrete dipole terms that adversely affect the accuracy near the immersed boundary this has been implemented for the two dimensional incompressible navier stokes equations using a high resolution finite volume method for the advective terms and a projection method to enforce incompressibility in the projection step the correct jump in pressure is imposed in the course of solving the poisson problem this gives sharp resolution of the pressure across the interface and also gives better volume conservation than the traditional ib method comparisons between this method and the ib method are presented for several test problems numerical studies of the convergence and order of accuracy are included

a new 3 d numerical approach to the solution of pdes with optimal accuracy on irregular domains and cartesian meshes abstract a new 3 d numerical approach for the time dependent wave and heat equations as well as for the time independent laplace equation on irregular domains with the dirichlet boundary conditions has been developed trivial cartesian meshes and simple 27 point uniform and nonuniform stencil equations are used for 3 d irregular domains the calculation of the coefficients of the stencil equations is based on the minimization of the local truncation error of the stencil equations and yields the optimal order of accuracy of the new technique very small distances 0 1 h 1 0 9 h where h is the grid size between the grid points of a cartesian mesh and the boundary do not worsen the accuracy of the new technique at similar 27 point stencils the accuracy of the new approach is two orders higher than that for the linear finite elements the numerical results for irregular domains show that at the same number of degrees of freedom the new approach is even much more accurate than the high order up to the fifth order tetrahedral finite elements with much wider stencils the wave and heat equations can be uniformly treated with the new approach the order of the time derivative in these equations does not affect the coefficients of the stencil equations of the semi discrete systems the new approach can be directly applied to other partial differential equations

immersed interface methods for stokes flow with elastic boundaries or surface tension a second order accurate interface tracking method for the solution of incompressible stokes flow problems with moving interfaces on a uniform cartesian grid is presented the interface may consist of an elastic boundary immersed in the fluid or an interface between two different fluids the interface is represented by a cubic spline along which the singularly supported elastic or surface tension force can be computed the stokes equations are then discretized using the second order accurate finite difference methods for elliptic equations with singular sources developed in our previous paper siam j numer anal 31 1994 pp 1019 1044 the resulting velocities are interpolated to the interface to determine the motion of the interface an implicit quasi newton method is developed that allows reasonable time steps to be used

positive solutions to a system of second order nonlocal boundary value problems abstract in this paper we study the existence of positive solutions to a system of second order nonlocal boundary value problems by using fixed point index theory in a cone our hypotheses imposed on nonlinearities are those which characterize systems of nonlocal boundary value problems and our boundary value conditions are expressed in terms of possibly nonlinear functions of riemann stieltjes integrals thus generalizing and unifying the boundary value conditions in the literature therefore our results cannot be routinely deduced from the ones for single nonlocal problem in the literature

a three dimensional computational method for blood flow in the heart 1 immersed elastic fibers in a viscous incompressible fluid abstract this paper describes the numerical solution of the 3 dimensional equations of motion of a viscous incompressible fluid that contains an immersed system of elastic fibers implementation details such as vectorization and the efficient use of external memory are discussed the method is applied to the damped vibrations of a fiber wound toroidal tube and empirical evidence of convergence is presented

an immersed interface method for solving incompressible viscous flows with piecewise constant viscosity across a moving elastic membrane this paper presents an implementation of the second order accurate immersed interface method to simulate the motion of the flexible elastic membrane immersed in two viscous incompressible fluids with different viscosities which further develops the work reported in tan et al z j tan d v le k m lim b c khoo an immersed interface method for the incompressible navier stokes equations with discontinuous viscosity across the interface submitted for publication focussing mainly on the fixed interface problems in this work we introduce the velocity components at the membrane as two augmented unknown interface variables to decouple the originally coupled jump conditions for the velocity and pressure three forms of augmented equation are derived to determine the augmented variables to satisfy the continuous condition of the velocity the velocity at the membrane which determine the motion of the membrane is then solved by the gmres iterative method the forces calculated from the configuration of the flexible elastic membrane and the augmented variables are interpolated using cubic splines and applied to the fluid through the jump conditions the position of the flexible elastic membrane is updated implicitly using a quasi newton method bfgs within each time step the navier stokes equations are solved on a staggered cartesian grid using a second order accurate projection method with the incorporation of spatial and temporal jump conditions in addition we also show that the inclusion of the temporal jump contributions has non negligible effect on the simulation results when the grids are crossed by the membrane using the above method we assess the effect of different viscosities on the flow solution and membrane motion

a second order virtual node algorithm for stokes flow problems with interfacial forces discontinuous material properties and irregular domains we present numerical methods for the solution of the stokes equations that handle interfacial discontinuities discontinuous material properties and irregular domains the discretization couples a lagrangian representation of the material interface with eulerian representations of the fluid velocity and pressure the methods are efficient easy to implement and yield discretely divergence free velocities that are second order accurate in l for the special case of continuous fluid viscosity we present a method that decouples the stokes equations into three poisson interface problems which we use the techniques in bedrossian 2010 1 to solve we also solve a fourth poisson equation to enforce a discrete divergence free condition in this case we discretize all equations using an embedded approach on a uniform mac grid employing virtual nodes and duplicated cells at the interfaces these additional degrees of freedom allow for accurate resolution of discontinuities in the fluid stress at the material interface in the case of discontinuous viscosity we require a lagrange multiplier term to enforce continuity of the fluid velocity we provide a novel discretization of this term that accurately resolves constant pressure null modes we show that the accurate resolution of these modes significantly improves performance the discrete coupled equations for the velocity pressure and lagrange multipliers are in the form of a symmetric kkt system however if both fluids have the same viscosity then all four linear systems involved are symmetric positive definite with three of the four having the standard 5 point laplace stencil everywhere numerical results indicate second order accuracy for the velocities and first order accuracy for the pressure in the general case for the continuous viscosity case numerical results indicate second order accuracy for both velocities and pressure

iteration processes for nonlinear lipschitzian strongly accretive mappings in lp spaces abstract suppose x l p or l p 1 p 2 let t x x be a lipschitzian and strongly accretive map with constant k 0 1 and lipschitz constant l satisfying p p 1 l 2 p 2 p 1 1 k define s x x by sx tx x for arbitrary x 0 x the sequence x n n 1 is defined by x n 1 1 n x n n sy n y n 1 n x n n sx n n 0 where n n 1 n n 1 are two real sequences satisfying i 0 n 2 1 p 2 p 1 1 k k n l 2 n p 2 p l 2 1 w n w l 2 1 2 n p 2 p 1 1 k k n l 2 n p 1 2 l 2 1 w n w l 2 1 2 n 1 1 for each n ii 0 n min w w l 2 1 1 w 4 p 1 l 2 k p 1 2 l 2 w 1 for each n iii n n where w p 2 p 1 1 k p 2 p l 2 then x n 1 converges strongly to the unique solution of tx moreover if n 2 1 p 2 p 1 1 k k l 2 p 2 p l 2 1 w w l 2 1 2 p 2 p 1 1 k k l 2 p 1 2 l 2 1 w w l 2 1 2 1 1 and n for each n and some 0 min w w l 2 1 1 w 4 p 1 l 2 k p 1 2 l 2 w 1 then x n 1 q p n 2 x 1 q where q denotes the solution of tx and 1 4 1 p 2 p 1 1 k k l 2 p 2 p l 2 1 w w l 2 1 2 2 p 2 p 1 1 k k l 2 p 1 2 l 2 1 w w l 2 1 2 1 1 0 1 a related result details with the iterative approximation of lipschitz strongly pseudocontractive maps in x

on strategies for enforcing interfacial constraints and evaluating jump conditions with the extended finite element method we consider a problem stemming from recent models of phase transitions in stimulus responsive hydrogels wherein a sharp interface separates swelled and collapsed phases extended finite element methods that approximate the local solution with an enriched basis such that the mesh need not explicitly fit the interface geometry are emphasized attention is focused on the weak enforcement of the normal configurational force balance and various options for evaluating the jump in the normal component of the solute flux at the interface we show that as the reciprocal interfacial mobility vanishes it plays the role of a penalty parameter enforcing a pure dirichlet constraint eventually triggering oscillations in the interfacial velocity we also examine alternative formulations employing a lagrange multiplier to enforce this constraint it is shown that the most convenient choice of basis for the lagrange multiplier results in oscillations in the multiplier field and a decrease in accuracy and rate of convergence in local error norms suggesting a lack of stability in the discrete formulation under such conditions neither the direct evaluation of the gradient of the approximation at the phase interface nor the interpretation of the lagrange multiplier field provide a robust means to obtain the jump in the normal component of solute flux fortunately the adaptation and use of local domain integral methodologies considerably improves the flux evaluations several example problems are presented to compare and contrast the various techniques and methods

enrichment of enhanced assumed strain approximations for representing strong discontinuities addressing volumetric incompressibility and the discontinuous patch test summary we present a geometrically nonlinear assumed strain method that allows for the presence of arbitrary intra finite element discontinuities in the deformation map special attention is placed on the coarsemesh accuracy of these methods and their ability to avoid mesh locking in the incompressible limit given an underlying mesh and an arbitrary failure surface we first construct an enriched approximation for the deformation map with the nonlinear analogue of the extended finite element method x fem with regard to the richer space of functions spanned by the gradient of the enriched approximation we then adopt a broader interpretation of variational consistency for the construction of the enhanced strain in particular in those elements intersected by the failure surface we construct enhanced strain approximations which are orthogonal to piecewise constant stress fields contrast is drawn with existing strong discontinuity approaches where the enhanced strain variations in localized elements were constructed to be orthogonal to constant nominal stress fields importantly the present formulation gives rise to a symmetric tangent stiffness matrix even in localized elements the present modification also allows for the satisfaction of a discontinuous patch test wherein two different constant stress fields on each side of the failure surface lie in the solution space we demonstrate how the proposed modifications eliminate spurious stress oscillations along the failure surface particularly for nearly incompressible material response additional numerical examples are provided to illustrate the efficacy of the modified method for problems in hyperelastic fracture mechanics copyright c 2000 john wiley sons ltd

the extended finite element method for rigid particles in stokes flow a new method for the simulation of particulate flows based on the extended finite element method x fem is described in this method the particle surfaces need not conform to the finite element boundaries so that moving particles can be simulated without remeshing the near field form of the fluid flow about each particle is built into the finite element basis using a partition of unity enrichment allowing the simple enforcement of boundary conditions and improved accuracy over other methods on a coarse mesh we present a weak form of the equations of motion useful for the simulation of freely moving particles and solve example problems for particles with prescribed and unknown velocities copyright 2001 john wiley sons ltd

a second order virtual node method for elliptic problems with interfaces and irregular domains in three dimensions we present a numerical method for the variable coefficient poisson equation in three dimensional irregular domains and with interfacial discontinuities the discretization embeds the domain and interface into a uniform cartesian grid augmented with virtual degrees of freedom to provide accurate treatment of jump and boundary conditions the matrix associated with the discretization is symmetric positive definite and equal to the standard 7 point finite difference poisson stencil away from embedded interfaces and boundaries numerical evidence suggests second order accuracy in the l norm our approach improves the treatment of dirichlet and jump constraints in the recent work of bedrossian et al 1 and introduces innovations necessary in three dimensions specifically we construct new constraint based lagrange multiplier spaces that significantly improve the conditioning of the associated linear system of equations we provide a method for cell local polyhedral approximation to the zero isocontour surface of a level set needed for three dimensional embedding and we show that the new lagrange multiplier spaces naturally lead to a class of easy to implement multigrid methods that achieve near optimal efficiency as shown by numerical examples for the specific case of a continuous poisson coefficient in interface problems we provide an expansive treatment of the construction of a particular solution that satisfies the value jump and flux jump constraints as in 1 this is used in a discontinuity removal technique that yields the standard 7 point stencil across the interface and only requires a modification to the right hand side of the linear system

modeling holes and inclusions by level sets in the extended finite element method a methodology to model arbitrary holes and material interfaces inclusions without meshing the internal boundaries is proposed the numerical method couples the level set method s osher j a sethian j comput phys 79 1 1988 12 to the extended finite element method x fem n moes j dolbow t belytschko int j numer methods engrg 46 1 1999 131 in the x fem the finite element approximation is enriched by additional functions through the notion of partition of unity the level set method is used for representing the location of holes and material interfaces and in addition the level set function is used to develop the local enrichment for material interfaces numerical examples in two dimensional linear elastostatics are presented to demonstrate the accuracy and potential of the new technique

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

a fictitious domain approach to the direct numerical simulation of incompressible viscous flow past moving rigid bodies application to particulate flow abstract in this article we discuss a methodology that allows the direct numerical simulation of incompressible viscous fluid flow past moving rigid bodies the simulation methods rest essentially on the combination of a lagrange multiplier based fictitious domain methods which allow the fluid flow computations to be done in a fixed flow region b finite element approximations of the navier stokes equations occurring in the global model c time discretizations by operator splitting schemes in order to treat optimally the various operators present in the model the above methodology is particularly well suited to the direct numerical simulation of particulate flow such as the flow of mixtures of rigid solid particles and incompressible viscous fluids possibly non newtonian we conclude this article with the presentation of the results of various numerical experiments including the simulation of store separation for rigid airfoils and of sedimentation and fluidization phenomena in two and three dimensions

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

immersed finite element method abstract in this paper the immersed finite element method ifem is proposed for the solution of complex fluid and deformable structure interaction problems encountered in many physical models in ifem a lagrangian solid mesh moves on top of a background eulerian fluid mesh which spans over the entire computational domain hence the mesh generation is greatly simplified moreover both fluid and solid domains are modeled with the finite element methods and the continuity between the fluid and solid sub domains are enforced via the interpolation of the velocities and the distribution of the forces with the reproducing kernel particle method rkpm delta function in comparison with the immersed boundary ib method the higher ordered rkpm delta function enables the fluid domain to have nonuniform spatial meshes with arbitrary geometries and boundary conditions the use of such kernel functions may eventually open doors to multi scale and multi resolution modelings of complex fluid structure interaction problems rigid and deformable spheres dropping in channels are simulated to demonstrate the unique capabilities of the proposed method the results compare well with the experimental data to the authors knowledge these are the first solutions that deal with particulate flows with very flexible solids

arbitrary branched and intersecting cracks with the extended finite element method summary extensions of a new technique for the nite element modelling of cracks with multiple branches multiple holes and cracks emanating from holes are presented this extended nite element method x fem allows the representation of crack discontinuities and voids independently of the mesh a standard displacementbased approximation is enriched by incorporating discontinuous elds through a partition of unity method a methodology that constructs the enriched approximation based on the interaction of the discontinuous geometric features with the mesh is developed computation of the stress intensity factors sif in dierent examples involving branched and intersecting cracks as well as cracks emanating from holes are presented to demonstrate the accuracy and the robustness of the proposed technique copyright 2000 john wiley sons ltd

constructive solutions for nonlinear multiparameter eigenvalue problems abstract based on the three well known banach mann and ishikawa iteration processes constructive solutions to a general nonlinear nonhomogenious two parameter eigenvalue problem of the form ax bx cx f are established the allowed range of the complex parameters and is discussed error estimates are studied the obtained results are natural extensions to the two parameter case of the corresponding one parameter problem discussed in the author s paper 1

an adaptive version of the immersed boundary method a computational setting for the immersed boundary method employing an adaptive mesh refinement is presented enhanced accuracy for the method is attained locally by covering an immersed boundary vicinity with a sequence of nested progressively finer rectangular grid patches which dynamically follow the immersed boundary motion the set of equations describing the interaction between a non stationary viscous incompressible fluid and an immersed elastic boundary is solved by coupling a projection method specially designed for locally refined meshes to an implicit formulation of the immersed boundary method the main contributions of this work concern the formulation and the implementation of a multilevel self adaptive version of the immersed boundary method on locally refined meshes this approach is tested for a particular two dimensional model problem for which no significant difference is found between the solutions obtained on a mesh refined locally around the immersed boundary and on the associated uniform mesh built with the resolution of the finest level

a 3d immersed interface method for fluid solid interaction abstract in immersed interface methods solids in a fluid are represented by singular forces in the navier stokes equations and flow jump conditions induced by the singular forces directly enter into numerical schemes this paper focuses on the implementation of an immersed interface method for simulating fluid solid interaction in 3d the method employs the mac scheme for the spatial discretization the rk4 scheme for the time integration and an fft based poisson solver for the pressure poisson equation a fluid solid interface is tracked by lagrangian markers intersections of the interface with mac grid lines identify finite difference stencils on which jump contributions to finite difference schemes are needed to find the intersections and to interpolate jump conditions from the lagrangian markers to the intersections parametric triangulation of the interface is used the velocity of the lagrangian markers is interpolated directly from surrounding mac grid nodes with interpolation schemes accounting for jump conditions numerical examples demonstrate that 1 the method has near second order accuracy in the infinity norm for velocity and the accuracy for pressure is between first and second order 2 the method conserves the volume enclosed by a no penetration boundary and 3 the method can efficiently handle multiple moving solids with ease

an immersed interface method for viscous incompressible flows involving rigid and flexible boundaries we present an immersed interface method for the incompressible navier stokes equations capable of handling both rigid and flexible boundaries the immersed boundaries are represented by a number of lagrangian control points in order to ensure that the no slip condition on the rigid boundary is satisfied singular forces are applied on the fluid the forces are related to the jumps in pressure and the jumps in the derivatives of both pressure and velocity and are interpolated using cubic splines the strength of the singular forces at the rigid boundary is determined by solving a small system of equations at each timestep for flexible boundaries the forces that the boundary exerts on the fluid are computed from the constitutive relation of the flexible boundary and are applied to the fluid through the jump conditions the position of the flexible boundary is updated implicitly using a quasi newton method bfgs within each timestep the navier stokes equations are discretized on a staggered cartesian grid by a second order accurate projection method for pressure and velocity and the overall scheme is second order accurate

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

ishikawa iteration process for nonlinear lipschitz strongly accretive mappings abstract let e lp p 2 and let t e e be a lipschitzian and strongly accretive mapping let s e e be defined by sx tx x it is proved that under suitable conditions on the real sequences n n 0 and n n 0 the iteration process x0 e xn 1 1 n xn ns 1 n xn nsxn n 0 converges strongly to the unique solution of tx a related result deals with the iterative approximation of fixed points for lipschitz strongly pseudocontractive mappings in e a consequence of our result gives an affirmative answer to a problem posed by c e chidume j math anal appl 151 no 2 1990 453 461

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

the approximation of fixed points of compositions of nonexpansive mappings in hilbert space determining fixed points of nonexpansive mappings is a frequent problem in mathematics and physical sciences an algorithm for finding common fixed points of nonexpansive mappings in hilbert space essentially due to halpern is analyzed the main theorem extends wittmann s recent work and partially generalizes a result by lions algorithms of this kind have been applied to the convex feasibility problem

a cartesian grid projection method for the incompressible euler equations in complex geometries many problems in fluid dynamics require the representation of complicated internal or external boundaries of the flow here we present a method for calculating time dependent incompressible inviscid flow which combines a projection method with a cartesian grid approach for representing geometry in this approach the body is represented as an interface embedded in a regular cartesian mesh the advection step is based on a cartesian grid algorithm for compressible flow in which the discretization of the body near the flow uses a volume of fluid representation a redistribution procedure is used to eliminate time step restrictions due to small cells where the boundary intersects the mesh the projection step uses an approximate projection based on a cartesian grid method for potential flow the method incorporates knowledge of the body through volume and area fractions along with certain other integrals over the mixed cells convergence results are given for the projection itself and for the time dependent algorithm in two dimensions the method is also demonstrated on flow past a half cylinder with vortex shedding

the immersed interface method for elliptic equations with discontinuous coefficients and singular sources the authors develop finite difference methods for elliptic equations of the form nabla cdot beta x nabla u x kappa x u x f x in a region omega in one or two space dimension

existence and uniqueness of positive solutions for an integral boundary value problem abstract this paper deals with the existence and uniqueness of positive solutions for the second order integral boundary value problem u f u 0 u 0 0 1 u d u 1 0 1 u d where f c r r is sign changing the main tools used in the proofs are the a priori estimate method and the leray schauder fixed point theorem

a nitsche extended finite element method for incompressible elasticity with discontinuous modulus of elasticity in this note we propose a finite element method for incompressible or compressible elasticity problems with discontinuous modulus of elasticity or if compressible poisson s ratio the problem is written on mixed form using p1 continuous displacements and elementwise p0 pressures leading to the possibility of eliminating the pressure beforehand in the compressible case in the incompressible case the method is augmented by a stabilization term penalizing the pressure jumps we show a priori error estimates under certain regularity hypothesis in particular we prove that if the exact solution is sufficiently smooth in each subdomain then the convergence order is optimal

an iterative process for nonlinear lipschitzian strongly accretive mappings in lp spaces abstract suppose x l p or l p p 2 let t x x be a lipschitzian and strongly accretive map with constant k 0 1 and lipschitz constant l define s x x by sx f tx x let c n n 1 be a real sequence satisfying 1 i 0 c n k p 1 l 2 2 k 1 1 for each n 2 ii n c n then for arbitrary x 0 x the sequence x n 1 1 c n x n c n sx n n 0 converges strongly to the unique solution of tx f moreover if c n k p 1 l 2 2 k 1 1 for each n then x n 1 q n 2 x 1 q where q denotes the solution of tx f and 1 k p 1 l 2 2 k 1 1 0 1 a related result deals with the iterative approximation of lipschitz strongly pseudocontractive maps in x

holder continuity of weak solutions of p laplacian pdes with vmo coefficients abstract we consider solutions u w 1 p r n of the p laplacian pde a x d u p 2 d u 0 for x r n where is open and bounded more generally we consider solutions of the elliptic system a x g a x d u d u d u 0 x as well as minimizers of the functional g a x d u d x in each case the coefficient map a r is only assumed to be of class v m o l which means that it may be discontinuous without assuming that x a x has any weak differentiability we show that u c loc 0 for each 0 1 the preceding results are in fact a corollary of a much more general result which applies to the functional f x u d u d x in case f is only asymptotically convex

sharp interface cartesian grid method i an easily implemented technique for 3d moving boundary computations a cartesian grid method is developed for the simulation of incompressible flows around stationary and moving three dimensional immersed boundaries the embedded boundaries are represented using level sets and treated in a sharp manner without the use of source terms to represent boundary effects the narrow band distance function field in the level set boundary representation facilitates implementation of the finite difference flow solver the resulting algorithm is implemented in a straightforward manner in three dimensions and retains global second order accuracy the accuracy of the finite difference scheme is established and shown to be comparable to finite volume schemes that are considerably more difficult to implement moving boundaries are handled naturally the pressure solver is accelerated using an algebraic multigrid technique adapted to be effective in the presence of moving embedded boundaries benchmarking of the method is performed against available numerical as well as experimental results

a fast solver for the stokes equations with distributed forces in complex geometries we present a new method for the solution of the stokes equations the main features of our method are 1 it can be applied to arbitrary geometries in a black box fashion 2 it is second order accurate and 3 it has optimal algorithmic complexity our approach to which we refer as the embedded boundary integral method ebi is based on anita mayo s work for the poisson s equation the fast solution of poisson s and the biharmonic equations on irregular regions siam journal on numerical analysis 21 1984 285 299 we embed the domain in a rectangular domain for which fast solvers are available and we impose the boundary conditions as interface jump conditions on the velocities and tractions we use an indirect boundary integral formulation for the homogeneous stokes equations to compute the jumps the resulting equations are discretized by nystrom s method the rectangular domain problem is discretized by finite elements for a velocity pressure formulation with equal order interpolation bilinear elements q1 q1 stabilization is used to circumvent the inf sup condition for the pressure space for the integral equations fast matrix vector multiplications are achieved via an n logn algorithm based on a block representation of the discrete integral operator combined with kernel independent singular value decomposition to sparsify low rank blocks the regular grid solver is a krylov method conjugate residuals combined with an optimal two level schwartz preconditioner for the integral equation we use gmres we have tested our algorithm on several numerical examples and we have observed optimal convergence rates

positive solutions to boundary value problems with nonlinear boundary conditions abstract in this paper we consider the boundary value problem y t f t y t subject to the boundary conditions y a y and y 2 b 0 in this setting c rd a 2 b t r r is a continuous functional which represents a nonlinear nonlocal boundary condition by imposing sufficient structure on and the nonlinearity f we deduce the existence of at least one positive solution to this problem the novelty in our setting lies in the fact that may be strictly nonpositive for some y 0 our results are achieved by appealing to the krasnosel ski fixed point theorem we conclude with several examples illustrating our results and the generalizations that they afford

fixed point theorems for nonlinear operators some new fixed point theorems are presented for operators of accretive nonlinear contractive or nonexpansive type these results are then used to establish a new existence principle for second order boundary value problems in hilbert spaces

an immersed interface method for the incompressible navier stokes equations with discontinuous viscosity across the interface we present an immersed interface method for solving the incompressible navier stokes equations with discontinuous viscosity across the interface and singular forces the method is based on the augmented strategy proposed by li ito and lai comput fluids 36 2007 pp 622 635 to decouple the jump conditions of the fluid variables through the introduction of two augmented variables in the proposed method the immersed interface is represented by a number of lagrangian control points and the augmented interface variables applicable along the interface are solved by the lu method or the gmres iterative method in addition we also derive a new second order accurate bilinear interpolating scheme for the discontinuous velocity and a new jump condition for the normal derivative of the pressure the jumps in both pressure and velocity and the jumps in their derivatives are related to the augmented interface variables and or the forces which are interpolated using cubic splines for a flexible interface the forces that the interface exerts on the fluid are computed from the constitutive relation of the flexible interface and are applied to the fluid through the jump conditions the position of the flexible interface is updated implicitly within each time step the navier stokes equations are discretized on a staggered cartesian grid by a second order accurate projection method the numerical results show that the overall scheme is second order accurate

mimetic finite difference method for the stokes problem on polygonal meshes various approaches to extend finite element methods to non traditional elements general polygons pyramids polyhedra etc have been developed over the last decade the construction of basis functions for such elements is a challenging task and may require extensive geometrical analysis the mimetic finite difference mfd method works on general polygonal meshes and has many similarities with low order finite element methods both schemes try to preserve the fundamental properties of the underlying physical and mathematical models the essential difference between the two schemes is that the mfd method uses only the surface representation of discrete unknowns to build the stiffness and mass matrices since no extension of basis functions inside the mesh elements is required practical implementation of the mfd method is simple for polygonal meshes that may include degenerate and non convex elements in this article we present a new mfd method for the stokes problem on arbitrary polygonal meshes and analyze its stability the method is developed for the general case of tensor coefficients which allows us to apply it to a linear elasticity problem as well numerical experiments show for the velocity variable second order convergence in a discrete l 2 norm and first order convergence in a discrete h 1 norm for the pressure variable first order convergence is shown in the l 2 norm

two phase flows on interface refined grids modeled with vof staggered finite volumes and spline interpolants abstract a two phase 2d model that combines the volume of fluid vof method with implicit staggered finite volumes discretization of the navier stokes equation is presented staggered finite volumes are developed on the basis of nonconforming crouzeix raviart finite elements where all components of the velocity lie in the middle of the element edges and the pressure degrees of freedom are found in the centers of mass of the elements staggered finite volumes extend marker and cell mac regular staggered grids to unstructured mesh a linear saddle point problem resulting from either the discretization or the newton method is solved for all unknown pressures and velocities interface is represented with spline interpolants which follow the vof distribution adaptive mesh refinement is used to obtain a high level of uniform refining at the domain of dependence of the interface the aligned grid is obtained by irregular refining of the cells which are intersected by a curve the boundaries of its elements coincide with the slope segments going through the intersections of the curve with the underlying regular elements boundary the deformable computational grids are used only to discretize the navier stokes equation the advection of volume fractions is done on the advection mesh which corresponds to highest regular refining on the computational grid approximation of the surface tension on spline interpolants offers a straightforward way to describe correctly the pressure jumps on interface fitted staggered grids this allows deletion of the anomalous currents around a statical bubble and their effective reduction in real simulations on the aligned grid the continuity of the viscous stress is modeled exactly due to the finite volume approach using the proposed numerical techniques single bubble rise is analyzed

an extended finite element method lagrange multiplier based approach for fluid structure interaction abstract this paper presents a new fixed grid fluid structure interaction scheme that can be applied to the interaction of most general structures with incompressible flow it is based on an extended finite element method xfem based strategy the extended eulerian fluid field and the lagrangian structural field are partitioned and iteratively coupled using lagrange multiplier techniques for non matching grids the approach allows the simulation of the interaction of thin and bulky structures exhibiting large deformations finally qualitative examples and a benchmark computation demonstrate key features and accuracy of the method

iterative and approximate solutions of general parameter dependent nonlinear operator equations in hilbert spaces iterative solutions of a general nonlinear operator equation of the form ax tx f where a and t are in general nonlinear operators in an appropriate space have not been developed so far in this paper the three well known banach mann and ishikawa iteration processes are used to find solutions of this equation in the hilbert space case an error estimate in each case is given the established results generalize many of the known results in literature

a fictitious domain method for external incompressible viscous flow modeled by navier stokes equations abstract in this article we discuss a fictitious domain method for the numerical solutions of three dimensional elliptic problems with dirichlet boundary conditions and also of the navier stokes equations modeling incompressible viscous flow the methodology for the navier stokes equations described here takes a systematic advantage of time discretization by operator splitting in order to treat separately advection imbedding and incompressibility due to the decoupling fast elliptic solvers can be used to treat the incompressibility condition even if the original problem is taking place on a nonregular geometry the resulting methodology is applied to two dimensional unsteady external incompressible viscous flow problems and three dimensional stokes problems

positive solutions of a second order integral boundary value problem in this paper we study the existence of positive solutions of a second order integral boundary value problems for ordinary differential equations our results presented here unify generalize and substantially improve the existing results in the literature moreover it is worthwhile to point out that our method will dispense with constructing a new green function

the immersed interface method for the navier stokes equations with singular forces abstract peskin s immersed boundary method has been widely used for simulating many fluid mechanics and biology problems one of the essential components of the method is the usage of certain discrete delta functions to deal with singular forces along one or several interfaces in the fluid domain however the immersed boundary method is known to be first order accurate and usually smears out the solutions in this paper we propose an immersed interface method for the incompressible navier stokes equations with singular forces along one or several interfaces in the solution domain the new method is based on a second order projection method with modifications only at grid points near or on the interface from the derivation of the new method we expect fully second order accuracy for the velocity and nearly second order accuracy for the pressure in the maximum norm including those grid points near or on the interface this has been confirmed in our numerical experiments furthermore the computed solutions are sharp across the interface nontrivial numerical results are provided and compared with the immersed boundary method meanwhile a new version of the immersed boundary method using the level set representation of the interface is also proposed in this paper

a boundary condition capturing method for poisson s equation on irregular domains interfaces have a variety of boundary conditions or jump conditions that need to be enforced the ghost fluid method gfm was developed to capture the boundary conditions at a contact discontinuity in the inviscid euler equations and has been extended to treat more general discontinuities such as shocks detonations and deflagrations and compressible viscous flows in this paper a similar boundary condition capturing approach is used to develop a new numerical method for the variable coefficient poisson equation in the presence of interfaces where both the variable coefficients and the solution itself may be discontinuous this new method is robust and easy to implement even in three spatial dimensions furthermore the coefficient matrix of the associated linear system is the standard symmetric matrix for the variable coefficient poisson equation in the absence of interfaces allowing for straightforward application of standard black box solvers

a second order virtual node method for elliptic problems with interfaces and irregular domains we present a second order accurate geometrically flexible and easy to implement method for solving the variable coefficient poisson equation with interfacial discontinuities or on irregular domains handling both cases with the same approach we discretize the equations using an embedded approach on a uniform cartesian grid employing virtual nodes at interfaces and boundaries a variational method is used to define numerical stencils near these special virtual nodes and a lagrange multiplier approach is used to enforce jump conditions and dirichlet boundary conditions our combination of these two aspects yields a symmetric positive definite discretization in the general case we obtain the standard 5 point stencil away from the interface for the specific case of interface problems with continuous coefficients we present a discontinuity removal technique that admits use of the standard 5 point finite difference stencil everywhere in the domain numerical experiments indicate second order accuracy in l

existence of nontrivial solutions for a nonlinear sturm liouville problem with integral boundary conditions abstract this paper deals with the existence of nontrivial solutions for a nonlinear singular sturm liouville problem with integral boundary conditions

the numerical approximation of a delta function with application to level set methods it is shown that a discrete delta function can be constructed using a technique developed by anita mayo the fast solution of poisson s and the biharmonic equations on irregular regions siam j sci comput 21 1984 285 299 for the numerical solution of elliptic equations with discontinuous source terms this delta function is concentrated on the zero level set of a continuous function in two space dimensions this corresponds to a line and a surface in three space dimensions delta functions that are first and second order accurate are formulated in both two and three dimensions in terms of a level set function the numerical implementation of these delta functions achieves the expected order of accuracy

a finite volume method for incompressible linear elasticity abstract this paper describes development of a finite volume based method for modelling of incompressible linear elastic body problems the method is based on the solution of the integral form of conservation equations governing momentum balance and the introduction of pressure as an additional variable the incompressibility is enforced by employing the volumetric constraint equation a collocated variable arrangement is used and the spatial domain is discretised using finite volumes of arbitrary polyhedral shapes a segregated approach is employed to solve resulting set of coupled linear algebraic equations embedding a simple based algorithm for displacement pressure coupling numerical experiments showed that the developed method appears to be locking free

on nonlinear boundary conditions satisfying certain asymptotic behavior abstract in this paper we consider a second order boundary value problem of the form y t a t g y t y 0 y y 1 0 we demonstrate that if the nonlinear functional y satisfies certain asymptotic behavior then the problem will possess at least one positive solution these results generalize and improve on some recent results in the literature on boundary value problems with nonlocal nonlinear boundary conditions and we provide some examples which illustrate these generalizations and improvements

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

three positive solutions of singular nonlocal boundary value problems for systems of nonlinear second order ordinary differential equations abstract in this paper we consider the existence of at least three positive solutions of singular nonlocal boundary value problems for systems of nonlinear second order ordinary differential equations the associated green s function for the boundary value problems is first given the proofs of our main results are based upon the leggett williams fixed point theorem finally we give an example to demonstrate our result

analysis of stiffness in the immersed boundary method and implications for time stepping schemes the immersed boundary method is known to exhibit a high degree of numerical stiffness associated with the interaction of immersed elastic fibres with the surrounding fluid we perform a linear analysis of the underlying equations of motion for immersed fibres and identify a discrete set of fibre modes which are associated solely with the presence of the fibre this work generalises our results in a previous paper 1995 siam j appl math 55 1577 by incorporating the effect of spreading the singular fibre force over a finite smoothing radius corresponding to the approximate delta function used in the immersed boundary method we investigate the stability of the fibre modes their stiffness and their dependence on the problem parameters focusing on the influence of smoothing we then extend the analytical results to include the effect of time discretisation and draw conclusions about the time step restrictions on various explicit schemes as well as the convergence rates for an iterative semi implicit method we draw comparisons with computations and show how the results can be applied to help in choosing alternate time stepping schemes that are specially tailored to handle the stiffness in immersed fibres in particular we present numerical results that show how fully explicit runge kutta schemes perform in comparison with the best of the semi implicit schemes currently in use

improved volume conservation in the computation of flows with immersed elastic boundaries this paper introduces a recipe for the construction of a finite difference divergence operator the coefficients of which are completely determined once an interpolation scheme has been chosen substitution of this divergence operator for the previously used divergence based on central differences makes a dramatic improvement in the overall volume conservation that is observed in an immersed boundary computation this improvement is particularly important for computations in which an elastic boundary separates chambers containing fluid at substantially different pressures a situation which is prominent in cardiac fluid dynamics during the contraction of the ventricles

remarks on a non local boundary value problem we remark that results on existence of symmetric positive solutions obtained in some recent papers can in fact be deduced as special cases of some known results we also correct the values of the principal eigenvalue previously given in some examples

an efficient fluid solid coupling algorithm for single phase flows we present a simple and efficient fluid solid coupling method in two and three spatial dimensions in particular we consider the numerical approximation of the navier stokes equations on irregular domains and propose a novel approach for solving the hodge projection step on arbitrary shaped domains this method is straightforward to implement and leads to a symmetric positive definite linear system for both the projection step and for the implicit treatment of the viscosity we demonstrate the accuracy of our method in the l 1 and l norms and present its removing the errors associated with the conventional rasterization type discretizations we apply this method to the simulation of a flow past a cylinder in two spatial dimensions and show that our method can reproduce the known stable and unstable regimes as well as correct lift and drag forces we also apply this method to the simulation of a flow past a sphere in three spatial dimensions at low and moderate reynolds number to reproduce the known steady axisymmetric and non axisymmetric flow regimes we further apply this algorithm to the coupling of flows with moving rigid bodies

partial regularity of solutions to p x laplacian pdes with discontinuous coefficients abstract for r n an open and bounded region we consider solutions u w loc 1 p x r n with n 1 of the p x laplacian system a x d u p x 2 d u 0 a e x where concerning the coefficient function x a x we assume only that a w 1 q l where q 1 is essentially arbitrary this implies that the coefficient in the pde can be highly irregular and yet in spite of this we still recover that u c loc 0 0 for each 0 1 where 0 is a set of full measure due to the variational methodology that we employ our results apply to the more general question of the regularity of the integral functional a x d u p x d x

a discontinuous galerkin based immersed boundary method a numerical method to approximate partial differential equations on meshes that do not conform to the domain boundaries is introduced the proposed method is conceptually simple and free of user defined parameters starting with a conforming finite element mesh the key ingredient is to switch those elements intersected by the dirichlet boundary to a discontinuous galerkin approximation and impose the dirichlet boundary conditions strongly by virtue of relaxing the continuity constraint at those elements boundary locking is avoided and optimal order convergence is achieved this is shown through numerical experiments in reaction diffusion problems copyright 2008 john wiley sons ltd

a high order numerical approach with cartesian meshes for modeling of wave propagation and heat transfer on irregular domains with inhomogeneous materials abstract recently we have developed a new numerical approach for pdes with constant coefficients on irregular domains and cartesian meshes in this paper we extend it to a much more general case of pdes with variable coefficients that have a lot of applications e g the modeling of functionally graded materials the inhomogeneous materials obtained by 3 d printing and many others here we consider the 2 d wave and heat equations for isotropic and anisotropic inhomogeneous materials the idea of the extension to the case of pdes with variable coefficients is based on the representation of the stencil coefficients as functions of the mesh size this leads to the increase in the size of the local system of algebraic equations solved for each grid point of the new approach however this does not change the size of the global system of semidiscrete equations and practically does not increase the computational costs of the proposed technique similar to our previous technique the new 2 d approach with compact 9 point stencils uses trivial cartesian meshes for complex irregular domains and provides the fourth order of accuracy for the wave and heat equations with variable coefficients the calculation of the coefficients of the stencil equations is based on the minimization of the local truncation error of the stencil equations and yields the optimal order of accuracy of the new technique at similar 9 point stencils the accuracy of the new approach is much higher than that for the linear finite elements the numerical results for irregular domains show that at the same number of degrees of freedom the new approach is even much more accurate than the high order up to the third order finite elements with much wider stencils the wave and heat equations are uniformly treated with the new approach