a convergence rate theorem for finite difference approximations to delta functions we prove a rate of convergence theorem for approximations to certain integrals over codimension one manifolds in r n the type of manifold involved here is defined by the zero level set of a smooth mapping u r email protected r our approximations are based on the two finite difference methods for discretizing delta functions presented in 16 we included a convergence proof in that paper but only proved rates of convergence in some greatly simplified situations numerical experiments indicated that our two methods were at least first and second order accurate respectively in this note we prove those empirical convergence rates for the two algorithms under fairly general hypotheses

a reconstruction based chebyshev collocation method for the poisson equation an accurate treatment of the gibbs wilbraham phenomenon on irregular interfaces abstract an accurate numerical solution of the laplace equation and poisson equation is desirable in physical problems related to fluid flows and spectral methods are well suited towards this goal however in the presence of an interface the solution is discontinuous owing to the interface jump conditions and thus the numerical approximation is susceptible to the gibbs wilbraham phenomenon leading to a reduced solution accuracy in order to resolve the discontinuity at the interface we propose a reconstruction technique wherein the approximate solution is expressed as the sum of an infinitely differentiable smooth function and a modified heaviside function the smooth function is composed of chebyshev polynomials while the modified heaviside function is expressed in a weak form using the jump conditions across the interface the reconstruction framework allows us to impose the interfacial conditions exactly to motivate further application of the technique we first present the underlying rationale by revisiting the work of wilbraham 1848 13 followed by the numerical implementation we present numerical solution of four equations to illustrate the application of the proposed technique i an ordinary differential equation ode ii the euler bernoulli equation iii the laplace equation and iv the poisson equation the laplace and poisson equation are solved on a square domain embedded with complex interfacial geometries in two dimensions and on a cube with a spherical interface in three dimensions the method achieves error in the maximum norm o 10 14 using approximately 30 modes in each direction and requires a computational time of less than a second for all the problems we also conduct a sensitivity analysis to demonstrate the robustness of the algorithm under sub grid scale perturbations to interface location an alternate derivation for the modified heaviside function is proposed that is distinct from the taylor series expansion used to treat interface conditions in difference methods the pointwise error plots are subsequently presented to demonstrate that the gibbs wilbraham phenomenon is accurately resolved finally we present concluding thoughts on the potential applications of the reconstruction technique

handling solid fluid interfaces for viscous flows explicit jump approximation vs ghost cell approaches the ghost cell approaches gca for handling stationary solid boundaries regular or irregular are first investigated theoretically and numerically for the diffusion equation with dirichlet boundary conditions the main conclusion of this part of investigation is that the approximation for the diffusion term has to be second order accurate everywhere in order for the numerical solution to be rigorously second order accurate violating this principle the linear and quadratic gcas have the following shortcomings 1 restrictive constraints on grid size when the viscosity is small 2 susceptibleness to instability of a time explicit formulation for strongly transient flows 3 convergence deterioration to zeroth or first order for solutions with high frequency modes therefore the widely used linear extrapolation for enforcing no slip boundary conditions should be avoided even for regular solid boundaries as a remedy a simple method based on explicit jump approximation eja is proposed eja hinges on the idea that a velocity derivative jump at the boundary reduces to the value of the velocity derivative at the fluid side because the velocity of the stationary boundary is zero although the time marching linear system of eja is not symmetric it is strictly diagonal dominant with positive diagonal entries numerical results show that over a large range of viscosity and grid sizes eja performs much better than gcas in terms of stability and accuracy furthermore the second order convergence of eja does not depend on viscosity and the spectrum of the solution as those of gcas do this paper is written with enough details so that one can reproduce the numerical results

a new coupled approach high accuracy numerical method for the solution of 3d non linear biharmonic equations in this paper we derive a new fourth order finite difference approximation based on arithmetic average discretization for the solution of three dimensional non linear biharmonic partial differential equations on a 19 point compact stencil using coupled approach the numerical solutions of unknown variable u x y z and its laplacian 2u are obtained at each internal grid point the resulting stencil algorithm is presented which can be used to solve many physical problems the proposed method allows us to use the dirichlet boundary conditions directly and there is no need to discretize the derivative boundary conditions near the boundary we also show that special treatment is required to handle the boundary conditions the new method is tested on three problems and the results are compared with the corresponding second order approximation which we also discuss using coupled approach

optimal reconstruction of material properties in complex multiphysics phenomena we develop an optimization based approach to the problem of reconstructing temperature dependent material properties in complex thermo fluid systems described by the equations for the conservation of mass momentum and energy our goal is to estimate the temperature dependence of the viscosity coefficient in the momentum equation based on some noisy temperature measurements where the temperature is governed by a separate energy equation we show that an elegant and computationally efficient solution of this inverse problem is obtained by formulating it as a pde constrained optimization problem which can be solved with a gradient based descent method a key element of the proposed approach the cost functional gradients are characterized by mathematical structure quite different than in typical problems of pde constrained optimization and are expressed in terms of integrals defined over the level sets of the temperature field advanced techniques of integration on manifolds are required to evaluate numerically such gradients and we systematically compare three different methods as a model system we consider a two dimensional unsteady flow in a lid driven cavity with heat transfer and present a number of computational tests to validate our approach and illustrate its performance

numerical method for solving matrix coefficient elliptic equation with sharp edged interfaces solving elliptic equations with sharp edged interfaces is a challenging problem for most existing methods especially when the solution is highly oscillatory nonetheless it has wide applications in engineering and science an accurate and efficient method is desired we propose an efficient non traditional finite element method with non body fitting grids to solve the matrix coefficient elliptic equations with sharp edged interfaces extensive numerical experiments show that this method is second order accurate in the l norm and that it can handle both sharp edged interface and oscillatory solutions

systematic derivation of jump conditions for the immersed interface method in three dimensional flow simulation in this paper we systematically derive jump conditions for the immersed interface method siam j numer anal 31 1994 pp 1019 1044 siam j sci comput 18 1997 pp 709 735 to simulate three dimensional incompressible viscous flows subject to moving surfaces the surfaces are represented as singular forces in the navier stokes equations which give rise to discontinuities of flow quantities the principal jump conditions across a closed surface of the velocity the pressure and their normal derivatives have been derived by lai and li appl math lett 14 2001 pp 149 154 in this paper we first extend their derivation to generalized surface parametrization starting from the principal jump conditions we then derive the jump conditions of all first second and third order spatial derivatives of the velocity and the pressure we also derive the jump conditions of first and second order temporal derivatives of the velocity using these jump conditions the immersed interface method is applicable to the simulation of three dimensional incompressible viscous flows subject to moving surfaces where near the surfaces the first and second order spatial derivatives of the velocity and the pressure can be discretized with respectively third and second order accuracy and the first order temporal derivatives of the velocity can be discretized with second order accuracy

logically rectangular grids and finite volume methods for pdes in circular and spherical domains we describe a class of logically rectangular quadrilateral and hexahedral grids for solving pdes in circular and spherical domains including grid mappings for the circle the surface of the sphere and the three dimensional ball the grids are logically rectangular and the computational domain is a single cartesian grid compared to alternative approaches based on a multiblock data structure or unstructured triangulations this approach simplifies the implementation of numerical methods and the use of adaptive refinement a more general domain with a smooth boundary can be gridded by composing one of the mappings from this paper with another smooth mapping from the circle or sphere to the desired domain although these grids are highly nonorthogonal we show that the high resolution wave propagation algorithm implemented in clawpack can be used effectively to approximate hyperbolic problems on these grids since the ratio between the largest and smallest grids is below 2 for most of our grid mappings explicit finite volume methods such as the wave propagation algorithm do not suffer from the center or pole singularities that arise with polar or latitude longitude grids numerical test calculations illustrate the potential use of these grids for a variety of applications including euler equations shallow water equations and acoustics in a heterogeneous medium pattern formation from a reaction diffusion equation on the sphere is also considered all examples are implemented in the clawpack software package and full source code is available on the web along with matlab routines for the various mappings

a supra convergent finite difference scheme for the variable coefficient poisson equation on non graded grids we introduce a method for solving the variable coefficient poisson equation on non graded cartesian grids that yields second order accuracy for the solutions and their gradients we employ quadtree in 2d and octree in 3d data structures as an efficient means to represent the cartesian grid allowing for constraint free grid generation the schemes take advantage of sampling the solution at the nodes vertices of each cell in particular the discretization at one cell s node only uses nodes of two 2d or three 3d adjacent cells producing schemes that are straightforward to implement numerical results in two and three spatial dimensions demonstrate supra convergence in the l norm

the method of regularized stokeslets a numerical method for computing stokes flows in the presence of immersed boundaries and obstacles is presented the method is based on the smoothing of the forces leading to regularized stokeslets the resulting expressions provide the pressure and velocity field as functions of the forcing the latter expression can also be inverted to find the forces that impose a given velocity boundary condition the numerical examples presented demonstrate the wide applicability of the method and its properties solutions converge with second order accuracy when forces are exerted along smooth boundaries examples of segmented boundaries and forcing at random points are also presented

mib galerkin method for elliptic interface problems material interfaces are omnipresent in the real world structures and devices mathematical modeling of material interfaces often leads to elliptic partial differential equations pdes with discontinuous coefficients and singular sources which are commonly called elliptic interface problems the development of high order numerical schemes for elliptic interface problems has become a well defined field in applied and computational mathematics and attracted much attention in the past decades despite of significant advances challenges remain in the construction of high order schemes for nonsmooth interfaces i e interfaces with geometric singularities such as tips cusps and sharp edges the challenge of geometric singularities is amplified when they are associated with low solution regularities e g tip geometry effects in many fields the present work introduces a matched interface and boundary mib galerkin method for solving two dimensional 2d elliptic pdes with complex interfaces geometric singularities and low solution regularities the cartesian grid based triangular elements are employed to avoid the time consuming mesh generation procedure consequently the interface cuts through elements to ensure the continuity of classic basis functions across the interface two sets of overlapping elements called mib elements are defined near the interface as a result differentiation can be computed near the interface as if there is no interface interpolation functions are constructed on mib element spaces to smoothly extend function values across the interface a set of lowest order interface jump conditions is enforced on the interface which in turn determines the interpolation functions the performance of the proposed mib galerkin finite element method is validated by numerical experiments with a wide range of interface geometries geometric singularities low regularity solutions and grid resolutions extensive numerical studies confirm the designed second order convergence of the mib galerkin method in the l and l 2 errors some of the best results are obtained in the present work when the interface is c 1 or lipschitz continuous and the solution is c 2 continuous

a source term method for poisson problems on irregular domains abstract this paper presents a finite difference method for solving poisson problems on a two dimensional irregular domain an implicit level set representation of the domain boundary is assumed as well as a cartesian grid that is not fitted to the domain the algorithm is based on a scheme for interface problems which captures the jump conditions via singular source terms this paper adapts that method to deal with boundary value problems by employing a simple iterative process that simultaneously enforces the boundary condition and solves for an unknown jump condition the benefit and novelty of this method is that the boundary condition is captured via easily implemented source terms the system of equations that results at each iteration can be solved using a fft based fast poisson solver the scheme can accommodate dirichlet neumann and robin boundary conditions we first address the constant coefficient poisson equation and then extend the scheme to accommodate the variable coefficient equation numerical examples indicate second order accuracy or close to it for the solution the method also produces useful gradient approximations but with generally lower convergence rates

a new particular solution strategy for hyperbolic boundary value problems using hybrid trefftz displacement elements summary a new indirect approach to the problem of approximating the particular solution of non homogeneous hyperbolic boundary value problems is presented unlike the dual reciprocity method which constructs approximate particular solutions using radial basis functions polynomials or trigonometric functions the method reported here uses the homogeneous solutions of the problem obtained by discarding all time derivative terms from the governing equation nevertheless what typifies the present approach from a conceptual standpoint is the option of not using these trial functions exclusively for the approximation of the particular solution but to fully integrate them with the trefftz compliant homogeneous solution basis the particular solution trial basis is capable of significantly improving the trefftz solution even when the original equation is genuinely homogeneous an advantage that is lost if the basis is used exclusively for the recovery of the source terms similarly a sufficiently refined trefftz compliant basis is able to compensate for possible weaknesses of the particular solution approximation the method is implemented using the displacement model of the hybrid trefftz finite element method the functions used in the particular solution basis reduce most terms of the matrix of coefficients to boundary integral expressions and preserve the hermitian sparse and localized structure of the solving system that typifies hybrid trefftz formulations even when domain integrals are present they are generally easy to handle because the integrand presents no singularity copyright 2015 john wiley sons ltd

a galerkin formulation of the mib method for three dimensional elliptic interface problems we develop a three dimensional 3d galerkin formulation of the matched interface and boundary mib method for solving elliptic partial differential equations pdes with discontinuous coefficients i e the elliptic interface problem the present approach builds up two sets of elements respectively on two extended subdomains which both include the interface as a result two sets of elements overlap each other near the interface fictitious solutions are defined on the overlapping part of the elements so that the differentiation operations of the original pdes can be discretized as if there was no interface the extra coefficients of polynomial basis functions which furnish the overlapping elements and solve the fictitious solutions are determined by interface jump conditions consequently the interface jump conditions are rigorously enforced on the interface the present method utilizes cartesian meshes to avoid the mesh generation in conventional finite element methods fems we implement the proposed mib galerkin method with three different elements namely rectangular prism element five tetrahedron element and six tetrahedron element which tile the cartesian mesh without introducing any new node the accuracy stability and robustness of the proposed 3d mib galerkin are extensively validated over three types of elliptic interface problems in the first type interfaces are analytically defined by level set functions these interfaces are relatively simple but admit geometric singularities in the second type interfaces are defined by protein surfaces which are truly arbitrarily complex the last type of interfaces originates from multiprotein complexes such as molecular motors near second order accuracy has been confirmed for all of these problems to our knowledge it is the first time for an fem to show a near second order convergence in solving the poisson equation with realistic protein surfaces additionally the present work offers the first known near second order accurate method for c 1 continuous or h 2 continuous solutions associated with a lipschitz continuous interface in a 3d setting

a fast direct solver for elliptic partial differential equations on adaptively refined meshes we present a new class of fast direct solvers for elliptic partial differential equations on adaptively refined meshes these solvers rely on a combination of standard fast solvers for uniform grids and potential theory unlike standard iterative approaches they have a well defined operation count they also preserve the order of accuracy of the uniform grid solver despite the presence of coarse fine interfaces

mib method for elliptic equations with multi material interfaces elliptic partial differential equations pdes are widely used to model real world problems due to the heterogeneous characteristics of many naturally occurring materials and man made structures devices and equipments one frequently needs to solve elliptic pdes with discontinuous coefficients and singular sources the development of high order elliptic interface schemes has been an active research field for decades however challenges remain in the construction of high order schemes and particularly for nonsmooth interfaces i e interfaces with geometric singularities the challenge of geometric singularities is amplified when they are originated from two or more material interfaces joining together or crossing each other high order methods for elliptic equations with multi material interfaces have not been reported in the literature to our knowledge the present work develops matched interface and boundary mib method based schemes for solving two dimensional 2d elliptic pdes with geometric singularities of multi material interfaces a number of new mib schemes are constructed to account for all possible topological variations due to two material interfaces the geometric singularities of three material interfaces are significantly more difficult to handle three new mib schemes are designed to handle a variety of geometric situations and topological variations although not all of them the performance of the proposed new mib schemes is validated by numerical experiments with a wide range of coefficient contrasts geometric singularities and solution types extensive numerical studies confirm the designed second order accuracy of the mib method for multi material interfaces including a case where the derivative of the solution diverges

a cartesian grid finite volume method for the advection diffusion equation in irregular geometries we present a fully conservative high resolution finite volume algorithm for advection diffusion equations in irregular geometries the algorithm uses a cartesian grid in which some cells are cut by the embedded boundary a novel feature is the use of a capacity function to model the fact that some cells are only partially available to the fluid the advection portion then uses the explicit wave propagation methods implemented in clawpack and is stable for courant numbers up to 1 diffusion is modelled with an implicit finite volume algorithm results are shown for several geometries convergence is verified and the 1 norm order of accuracy is found to between 1 2 and 2 depending on the geometry and peclet number software is available on the web

a large deviation result for the range of random walk and for the wiener sausage let s n be a random walk on d and let r n be the number of different points among 0 s1 s n 1 we prove here that if d 2 then x lim n 1 n logp r n nx exists for x 0 and establish some convexity and monotonicity properties of x the one dimensional case will be treated in a separate paper

xgfm recovering convergence of fluxes in the ghost fluid method abstract we introduce a simple but effective correction to the ghost fluid method gfm introduced by liu et al 2000 30 for capturing sharp interface conditions when solving poisson equations with discontinuities our method is shown to alleviate the gfm lack of convergence for the flux across the interfaces pointwise convergence of the gradient is recovered by ensuring consistent jump conditions with an iterative process compared to the gfm the discretization stencil and the symmetric positive definiteness of the resulting linear system are not modified only the right hand side of the linear system is affected we first show that the original gfm is consistent with the second order accurate symmetric scheme of gibou et al 2002 17 for dirichlet conditions in separate regions of the computational domain where the parameters and the solution are smooth but with matching conditions at the interface that are not fully consistent in general our proposed method recovers consistency and is shown to produce convergent gradients through several numerical examples in two and three spatial dimensions

an immersed finite element method with integral equation correction summary we propose a robust immersed finite element method in which an integral equation formulation is used to enforce essential boundary conditions the solution of a boundary value problem is expressed as the superposition of a finite element solution and an integral equation solution for computing the finite element solution the physical domain is embedded into a slightly larger cartesian box shaped domain and is discretized using a block structured mesh the defect in the essential boundary conditions which occurs along the physical domain boundaries is subsequently corrected with an integral equation method in order to facilitate the mapping between the finite element and integral equation solutions the physical domain boundary is represented with a signed distance function on the block structured mesh as a result only a boundary mesh of the physical domain is necessary and no domain mesh needs to be generated except for the non boundary conforming block structured mesh the overall approach is first presented for the poisson equation and then generalized to incompressible viscous flow equations as an example of fluid structure coupling the settling of a heavy rigid particle in a closed tank is considered copyright 2010 john wiley sons ltd

branching random walk in random environment phase transitions for local and global growth rates let n be the infinite particle system on whose evolution is as follows at each unit of time each particle independently is replaced by a new generation the size of a new generation descending from a particle at sitex has distributionf x and each of its members independently jumps to sitex 1 with probability 1 h 2 h 0 1 the sequence f x is i i d with uniformly bounded second moment and is kept fixed during the evolution the initial configuration 0 is shift invariant and ergodic two quantities are considered

new cartesian grid methods for interface problems using the finite element formulation new finite element methods based on cartesian triangulations are presented for two dimensional elliptic interface problems involving discontinuities in the coefficients the triangulations in these methods do not need to fit the interfaces the basis functions in these methods are constructed to satisfy the interface jump conditions either exactly or approximately both non conforming and conforming finite element spaces are considered corresponding interpolation functions are proved to be second order accurate in the maximum norm the conforming finite element method has been shown to be convergent with cartesian triangulations these new methods can be used as finite difference methods numerical examples are provided to support the methods and the theoretical analysis

short note a proof that a discrete delta function is second order accurate it is proved that a discrete delta function introduced by smereka p smereka the numerical approximation of a delta function with application to level set methods j comput phys 211 2006 77 90 gives a second order accurate quadrature rule for surface integrals using values on a regular background grid the delta function is found using a technique of mayo a mayo the fast solution of poisson s and the biharmonic equations on irregular regions siam j numer anal 21 1984 285 299 it can be expressed naturally using a level set function

a compact difference scheme for the biharmonic equation in planar irregular domains we present a finite difference scheme applicable to general irregular planar domains to approximate the biharmonic equation the irregular domain is embedded in a cartesian grid in order to approximate delta 2 phi at a grid point we interpolate the data on the irregular stencil by a polynomial of degree six the finite difference scheme is delta 2 q phi 0 0 where q phi is the interpolation polynomial the interpolation polynomial is not uniquely determined we present a method to construct such an interpolation polynomial and prove that our construction is second order accurate for a regular stencil m ben artzi j p croisille and d fishelov siam j sci comput 31 2008 pp 303 333 shows that the proposed interpolation polynomial is fourth order accurate we present some suitable numerical examples

smoothing properties of implicit finite difference methods for a diffusion equation in maximum norm we prove a regularity property of finite difference schemes for the heat or diffusion equation u t delta u in maximum norm with large time steps for a class of time discretizations including l stable single step methods and the second order backward difference formula with the usual second order laplacian we show that solutions of the scheme gain first spatial differences boundedly and also second differences except for logarithmic factors with respect to nonhomogeneous terms a weaker property is shown for the crank nicolson method as a consequence we show that the numerical solution of a convection diffusion equation with an interface can allow o h truncation error near the interface and still have a solution with uniform o h 2 accuracy and first differences of uniform accuracy almost o h 2

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

an iterative boundary potential method for the infinite domain poisson problem with interior dirichlet boundaries an iterative method is developed for the solution of poisson s problem on an infinite domain in the presence of interior boundaries held at fixed potential in three dimensions the method combines pre existing fast multigrid based poisson solvers for data represented on cartesian grids with the fast multipole method interior boundaries are represented with the embedded boundary formalism the implementation is in parallel and uses adaptive mesh refinement examples are presented for a smooth interior boundary for which an analytical result is known and for an irregular interior boundary problem second order accuracy in l 1 with respect to the grid resolution is demonstrated for both problems

a cartesian grid method for modeling multiple moving objects in 2d incompressible viscous flow we present an efficient method for solving 2d incompressible viscous flows around multiple moving objects our method employs an underlying regular cartesian grid to solve the system using a streamfunction vorticity formulation and with discontinuities representing the embedded objects the no penentration condition for the moving geometry is satisfied by superposing a homogenous solution to the poisson s equation for the streamfunction the no slip condition is satisfied by generating vorticity on the surfaces of the objects both the initial poisson solution and the evaluation of the homogenous solution require embedding irregular discontinuities in a fast poisson solver computation time is dictated by the time required to do a fast poisson solution plus solve an integral form of laplace s equation there is no significant increase in computational cost if the geometry of the embedded objects is variable and moving relative to the underlying grid we test the method against the canonical example of flow past a cylinder and obtained new results on the flow and forces of two cylinders moving relative to each other

mining spatial association rules in image databases in this paper we propose a novel spatial mining algorithm called 9dlt miner to mine the spatial association rules from an image database where every image is represented by the 9dlt representation the proposed method consists of two phases first we find all frequent patterns of length one next we use frequent k patterns k 1 to generate all candidate k 1 patterns for each candidate pattern generated we scan the database to count the pattern s support and check if it is frequent the steps in the second phase are repeated until no more frequent patterns can be found since our proposed algorithm prunes most of impossible candidates it is more efficient than the apriori algorithm the experiment results show that 9dlt miner runs 2 5 times faster than the apriori algorithm

a coupling interface method for elliptic interface problems we propose a coupling interface method cim under cartesian grid for solving elliptic complex interface problems in arbitrary dimensions where the coefficients the source terms and the solutions may be discontinuous or singular across the interfaces it consists of a first order version cim1 and a second order version cim2 in one dimension the cim1 is derived from a linear approximation on both sides of the interface the method is extended to high dimensions through a dimension by dimension approach to connect information from each dimension a coupled equation for the first order derivatives is derived through the jump conditions in each coordinate direction the resulting stencil uses the standard 5 grid points in two dimensions and 7 grid points in three dimensions similarly the cim2 is derived from a quadratic approximation in each dimension in high dimensions a coupled equation for the principal second order derivatives u x k x k is derived through the jump conditions in each coordinate direction the cross derivatives are approximated by one side interpolation this approach reduces the number of grid points needed for one side interpolation the resulting stencil involves 8 grid points in two dimensions and 12 14 grid points in three dimensions a numerical study for the condition number of the resulting linear system of the cim2 in one dimension has been performed it is shown that the condition number has the same behavior as that of the discrete laplacian independent of the relative location of the interface in a grid cell further we also give a proof of the solvability of the coupling equations provided the curvature k of the interface satisfies kh

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 fast vortex method for computing 2d viscous flow abstract we present a fast version of the random vortex method for computing incompressible viscous flow at large reynolds numbers the basis of this method is anderson s method of local corrections and similar ideas for handling the potential and boundary layer flows the goal of these ideas is to reduce the cost involved in computing the velocity field at each time step from being quadratic to linear as a function of the number of vortex elements we present the results of a numerical study of the flow in a closed box due to a vortex fixed at its center our results demonstrate that the addition of the viscous portions of the random vortex method to the method of local corrections does not add appreciably to the cost furthermore the cost of the resulting method is linear when o 104 vortex elements are used in spite of the fact that the majority of these elements lie in a thin band adjacent to the boundary

large deviations and exponential decay for the magnetization in a gaussian random field we consider a continuous model for transverse magnetization of spins diffusing in a homogeneous gaussian random longitudinal field lambda v x x in bbb r d where lambda is the coupling constant giving the intensity of the random field in this setting the transverse magnetization is given by the formula m t bbb e exp lambda 2 int 0 t int 0 t k b r b s ds dr where b t t ge 0 is the standard process of brownian motion and k x is the covariance function of the original random field v x we use large deviation techniques to show that the limit s lambda lim t rightarrow infty frac 1 t ln m t exists we also determine the small lambda behavior of the rate s lambda and show that it is indeed decaying as conjectured in the physics literature

rapid solution of integral equations of classical potential theory an algorithm is described for rapid solution of classical boundary value problems dirichlet an neumann for the laplace equation based on iteratively solving integral equations of potential theory cpu time requirements for previously published algorithms of this type are proportional to n2 where n is the number of nodes in the discretization of the boundary of the region the cpu time requirements for the algorithm of the present paper are proportional to n making it considerably more practical for large scale problems

fast integral equation methods for the modified helmholtz equation we present integral equation methods for the solution to the two dimensional modified helmholtz equation u x email protected email protected x 0 in bounded or unbounded multiply connected domains we consider both dirichlet and neumann problems we derive well conditioned fredholm integral equations of the second kind which are discretized using high order hybrid gauss trapezoid rules our fast multipole based iterative solution procedure requires only o n operations where n is the number of nodes in the discretization of the boundary we demonstrate the performance of our methods on several numerical examples and we show that they have both the ability to handle highly complex geometry and the potential to solve large scale problems

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

critical large deviations of one dimensional annealed brownian motion in a poissonian potential we derive a large deviation principle for the position at large times t of a one dimensional annealed brownian motion in a poissonian potential in the critical spatial scale t 1 3 here annealed means that averages are taken with respect to both the path and environment measures in contrast to the d dimensional case for d 2 in the critical scale t d d 2 as treated by sznitman the rate function which measures the large deviations exhibits three different regimes these regimes depend on the position of the path at time t our large deviation principle has a natural application to the study of a one dimensional annealed brownian motion with a constant drift in a poissonian potential

robust second order accurate discretizations of the multi dimensional heaviside and dirac delta functions we present a robust second order accurate method for discretizing the multi dimensional heaviside and the dirac delta functions on irregular domains the method is robust in the following ways 1 integrations of source terms on a co dimension one surface are independent of the underlying grid and therefore stable under perturbations of the domain s boundary 2 the method depends only on the function value of a level function not on its derivatives we present the discretizations in tabulated form to make their implementations straightforward we present numerical results in two and three spatial dimensions to demonstrate the second order accuracy in the l 1 norm in the case of the solution of pdes with singular source terms in the case of evaluating the contribution of singular source terms on interfaces the method is also second order accurate in the l norm

a pressure boundary integral method for direct fluid particle simulations on cartesian grids we consider a new cartesian grid method for direct numerical simulations of fully coupled interaction of incompressible flow and spherical particles based on a discontinuous extension of the pressure poisson equation ppe across particle boundaries we give a complete mathematical description of the boundary integral treatment of the discontinuous ppe that includes the derivation of a new pressure boundary condition for accelerating boundaries and the solution of the system of boundary integral equations using spherical harmonics expansions the model was validated with the standard test for finite reynolds number flow around a sphere and with a novel test using the analytical solution for the stokes flow past two adjacent spheres moving with the same velocity the model capability and numerical efficiency was demonstrated with simulations for the collective settling of groups of 64 512 particles

accurate gradient approximation for complex interface problems in 3d by an improved coupling interface method most elliptic interface solvers become complicated for complex interface problems at those exceptional points where there are not enough neighboring interior points for high order interpolation such complication increases especially in three dimensions usually the solvers are thus reduced to low order accuracy in this paper we classify these exceptional points and propose two recipes to maintain order of accuracy there aiming at improving the previous coupling interface method 26 yet the idea is also applicable to other interface solvers the main idea is to have at least first order approximations for second order derivatives at those exceptional points recipe 1 is to use the finite difference approximation for the second order derivatives at a nearby interior grid point whenever this is possible recipe 2 is to flip domain signatures and introduce a ghost state so that a second order method can be applied this ghost state is a smooth extension of the solution at the exceptional point from the other side of the interface the original state is recovered by a post processing using nearby states and jump conditions the choice of recipes is determined by a classification scheme of the exceptional points the method renders the solution and its gradient uniformly second order accurate in the entire computed domain numerical examples are provided to illustrate the second order accuracy of the presently proposed method in approximating the gradients of the original states for some complex interfaces which we had tested previous in two and three dimensions and a real molecule 1d63 which is double helix shape and composed of hundreds of atoms

biased random walk conditioned on survival among bernoulli obstacles subcritical phase we consider a discrete time biased random walk conditioned to avoid bernoulli obstacles on mathbb z d d ge 2 up to time n this model is known to undergo a phase transition for a large bias the walk is ballistic whereas for a small bias it is sub ballistic we prove that in the sub ballistic phase the random walk is contained in a ball of radius o n 1 d 2 which is the same scale as for the unbiased case as an intermediate step we also prove large deviation principles for the endpoint distribution for the unbiased random walk at scales between n 1 d 2 and o n d d 2 these results improve and complement earlier work by sznitman ann sci ecole norm sup 4 28 3 345 370 371 390 1995

asymptotics for the wiener sausage with drift a particle is considered which moves in d according to a brownian motion with drifth 0 the space is assumed to contain random traps the probability of survival of the particle up to timet decays exponentially ast with a positive decay rate is shown to be a non analytic function of h for small h the decay rate is given by h 1 2 h 2 but if h exceeds a certain critical value h depends also on the parameters describing trapping upper and lower bounds for h are given which imply the asymptotic linearity of h for large h the critical point marks a transition from localized to delocalized behavior a variational formula for the decay rate is given on the level of generalized processes which elucidates the mathematical mechanism behind observations made earlier by grassberger and procaccia on the basis of computer simulations

a second order cartesian method for the simulation of electropermeabilization cell models in this paper we present a new finite differences method to simulate electropermeabilization models like the model of neu and krassowska or the recent model of kavian et al these models are based on the evolution of the electric potential in a cell embedded in a conducting medium the main feature lies in the transmission of the voltage potential across the cell membrane the jump of the potential is proportional to the normal flux thanks to the well known kirchoff law an adapted scheme is thus necessary to accurately simulate the voltage potential in the whole cell notably at the membrane separating the cell from the outer medium we present a second order finite differences scheme in the spirit of the method introduced by cisternino and weynans for elliptic problems with immersed interfaces this is a cartesian grid method based on the accurate discretization of the fluxes at the interface through the use of additional interface unknowns the main novelty of our present work lies in the fact that the jump of the potential is proportional to the flux and therefore is not explicitly known the original use of interface unknowns makes it possible to discretize the transmission conditions with enough accuracy to obtain a second order spatial convergence we prove the second order spatial convergence in the stationary linear one dimensional case and the first order temporal convergence for the dynamical non linear model in one dimension we then perform numerical experiments in two dimensions that corroborate these results

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

a weak formulation for solving elliptic interface problems without body fitted grid a typical elliptic interface problem is casted as piecewise defined elliptic partial differential equations pde in different regions which are coupled together with interface conditions such as jumps in solution and flux across the interface in many situations such as the interface is moving the challenge is how to solve such a problem accurately robustly and efficiently without generating a body fitted mesh the key issue is how to capture complex geometry of the interface and jump conditions across the interface effectively on a fixed mesh while the interface is not aligned with the mesh and the pde is not valid across the interface in this work we present a systematic formulation and further study of a second order accurate numerical method proposed in hou and liu 2005 16 for elliptic interface problem the key idea is to decompose the solution into two parts a singular part and a regular part the singular part captures the interface conditions while the regular part belongs to an appropriate space in the whole domain which can be solved by a standard finite element formulation in a general setup the two parts are coupled together we give an explicit study of the construction of the singular part and the discretized system for the regular part one key advantage of using weak formulation is that one can avoid assuming using more regularity than necessary of the solution and the interface we present the numerical algorithm and numerical tests in 3d to demonstrate the accuracy and other properties of our method

a kernel free boundary integral method for implicitly defined surfaces the kernel free boundary integral kfbi method is a structured grid method for general elliptic partial differential equations unlike the standard boundary integral method it avoids direct evaluation of volume and boundary integrals which needs to know analytical expressions for the integral kernels to evaluate a boundary or volume integral the kfbi method first solves a corrected interface problem on a structured grid and then the numerical solution on the structured grid is interpolated to get approximate values of the integral at points on the boundary selection of control points of the boundary plays a key role in the kfbi method since both the correction for the interface equations and the interpolation with the structured grid based solution involve calculation of tangential derivatives of boundary data while stability and efficiency of the numerical differentiation critically depend on the distribution of control points this work proposes a new point selection method based on an overlapping surface decomposition of the boundary which is implicitly defined by a level set function the points selected are intersection points of the boundary with the grid lines of an underlying cartesian grid by the method the interpolation stencils can be easily chosen to be locally uniform along a coordinate axis in two space dimensions and locally uniform on a coordinate plane in three space dimensions which allows efficient numerical differentiation and boundary reconstruction representation an additional equilibrating process of boundary data further guarantees stable numerical differentiation numerical results demonstrating the method with examples in both two and three space dimensions are presented

an association based case reduction technique for case based reasoning case based reasoning cbr is a type of problem solving technique which uses previous cases to solve new unseen and different problems although a larger number of cases in the memory can improve the coverage of the problem space the retrieval efficiency will be downgraded if the size of the case base grows to an unacceptable level in cbr systems the tradeoff between the number of cases stored in the case base and the retrieval efficiency is a critical issue this paper addresses the problem of case base maintenance by developing a new technique the association based case reduction technique acrt to reduce the size of the case base in order to enhance the efficiency while maintaining or even improving the accuracy of the cbr the experiments on 12 uci datasets and an actual case from taiwan s hospital have shown superior generalization accuracy for cbr with acrt cbr acrt as well as a greater solving efficiency

immersed interface methods for neumann and related problems in two and three dimensions we develop and apply a finite difference method to discretize the laplacian operator with neumann boundary conditions on an irregular domain using a regular cartesian grid the method is an extension of the immersed interface method developed by leveque and li siam j numer anal 31 1994 1019 1044 with careful selection of stencils the method is second order accurate and produces a matrix that is stable diagonally semidominant the method is illustrated on several two dimensional problems and one three dimensional problem

computation of interface reflection and regular or diffuse transmission of the planar symmetric radiative transfer equation with isotropic scattering and its diffusion limit in this paper we develop a numerical scheme for the interface problem in the planar symmetric radiative transfer equation with isotropic scattering such problems arise in the modeling of the propagation of energy density for waves in heterogeneous media with weak random fluctuation in the high frequency regime the idea following the earlier work of jin and wen for regular transmission and reflection is to build the interface condition which characterizes the reflection and transmission into the numerical flux the new contribution of this article is to deal with the diffuse transmission at the interface the positivity of the scheme is proven moreover we show via numerical examples that this new scheme is able to capture the correct regular or diffuse transmission and reflection through the interface and as the mean free path goes to zero capture the diffusion limit with the correct interface conditions

a note on immersed interface method for three dimensional elliptic equations abstract the immersed interface method proposed by leveque and li 1 is extended to three dimensional elliptic equations of the form x u x x u x f x we study the situation in which there is an irregular interface surface s contained in the solution domain across which and f may be discontinuous or even singular as a result the solution u will usually be nonsmooth or even discontinuous a finite difference approach with a uniform cartesian grid is used in the discretization local truncation error analysis is performed to estimate the accuracy of the numerical solution

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

the fast solution of poisson s and the biharmonic equations on irregular regions we present fast methods for solving laplace s and the biharmonic equations on irregular regions with smooth boundaries the methods used for solving both equations make use of fast poisson solvers on a rectangular region in which the irregular region is embedded they also both use an integral equation formulation of the problem where the integral equations are fredholm integral equations of the second kind the main idea is to use the integral equation formulation to define a discontinuous extension of the solution to the rest of the rectangular region fast solvers are then used to compute the extended solution aside from solving the equations we have also been able to compute derivatives of the solutions with little loss of accuracy when the data was sufficiently smooth

short note controlling self force errors at refinement boundaries for amr pic we analyze the source of the self force errors in the node centered adaptive mesh refinement particle in cell amr pic algorithm and propose a method for reducing those self forces our approach is based on a method of charge deposition due to mayo a mayo the fast solution of poisson s and the biharmonic equations on irregular regions siam journal of numerical analysis 21 2 1984 285 299 that can reduce the self force error to any specified degree of accuracy

a multi level boundary element method for stokes flows in irregular two dimensional domains recently we have developed multi level boundary element methods mlbem for the solution of the laplace and helmholtz equations that involve asymptotically decaying non oscillatory and oscillatory singular kernels respectively the accuracy and efficiency of the fast boundary element methods for steady state heat diffusion and acoustics problems have been investigated for square domains the current work extends the mlbem methodology to the solution of stokes equation in more complex two dimensional domains the performance of the fast boundary element method for the stokes flows is first investigated for a model problem in a unit square then we consider an example problem possessing an analytical solution in a rectangular domain with 5 1 aspect ratio and finally we study the performance of the mlbem algorithm in a c shaped domain

fourth order accurate evaluation of integrals in potential theory on exterior 3d regions we present two methods for the rapid high order accurate evaluation of integrals in potential theory on general unbounded 3d regions our methods allow for direct calculation of derivatives of the integrals as well one of the methods uses a fourth order compact stencil and the other uses a nonstandard variant of richardson extrapolation both methods involve calculation of discontinuities in high order derivatives of the integrals across the boundary of the integration region the extrapolation method in addition involves correction for the discontinuities in truncation error the number of operations required for the methods is essentially equal to twice the number of operations needed to solve poisson s equation on a regular grid both methods avoid problems associated with using quadrature methods to evaluate integrals with singular kernels numerical results are presented for experiments on a variety of geometries in free space

spectral element method for three dimensional elliptic problems with smooth interfaces abstract in this paper we propose a least squares spectral element method for three dimensional elliptic interface problems the differentiability estimates and the main stability theorem using non conforming spectral element functions are proven the proposed method is free from any kind of first order reformulation a suitable preconditioner is constructed with help of the regularity estimate and proposed stability estimates which is used to control the condition number we show that these preconditioners are spectrally equivalent to the quadratic forms by which we approximate them we obtain the error estimates which show the exponential accuracy of the method numerical results are obtained for both straight and curved interfaces to show the efficiency of the proposed method

rapid methods for the conformal mapping of multiply connected regions abstract we present fast methods for the conformal mapping of simply doubly and multiply connected regions onto certain canonical regions in the plane our mapping procedure consists of two parts first we solve an integral equation on the boundary of the region we wish to map the solution of this integral equation is needed to determine the boundary correspondence we have chosen to use the integral equation formulation of mikhlin although it is not widely used this formulation has the advantage that it leads to integral equations of the second kind with unique solutions and bounded kernels the solutions are also periodic allowing for effective use of the trapezoid rule once we have solved the integral equation we use a rapid method we have previously developed to determine the mapping function in the interior of the region this method makes use of fast poisson solvers and thereby circumvents the difficulties associated with computing integrals at points near the boundary of the region and avoids the expense of computing many integrals we also provide results of numerical experiments

looking into the seeds of time discovering temporal patterns in large transaction sets this paper studies the problem of mining frequent itemsets along with their temporal patterns from large transaction sets a model is proposed in which users define a large set of temporal patterns that are interesting or meaningful to them a temporal pattern defines the set of time points where the user expects a discovered itemset to be frequent the model is general in that i no constraints are placed on the interesting patterns given by the users and ii two measures inclusiveness and exclusiveness are used to capture how well the temporal patterns match the time points given by the discovered itemsets intuitively these measures indicate to what extent a discovered itemset is frequent at time points included in a temporal pattern p but not at time points not in p using these two measures one is able to model many temporal data mining problems appeared in the literature as well as those that have not been studied by exploiting the relationship within and between itemset space and pattern space simultaneously a series of pruning techniques are developed to speed up the mining process experiments show that these pruning techniques allow one to obtain performance benefits up to 100 times over a direct extension of non temporal data mining algorithms

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

loss and gain functions for cbr retrieval the method described in this article evaluates case similarity in the retrieval stage of case based reasoning cbr it thus plays a key role in deciding which case to select and therefore in deciding which solution will be eventually applied in cbr there are many retrieval techniques one feature shared by most is that case retrieval is based on attribute similarity and importance however there are other crucial factors that should be considered such as the possible consequences of a given solution in other words its potential loss and gain as their name clearly implies these concepts are defined as functions measuring loss and gain when a given retrieval case solution is applied moreover these functions help the user to choose the best solution so that when a mistake is made the resulting loss is minimal in this way the highest benefit is always obtained

extending fields in a level set method by solving a biharmonic equation we present an approach for computing extensions of velocities or other fields in level set methods by solving a biharmonic equation the approach differs from other commonly used approaches to velocity extension because it deals with the interface fully implicitly through the level set function no explicit properties of the interface such as its location or the velocity on the interface are required in computing the extension these features lead to a particularly simple implementation using either a sparse direct solver or a matrix free conjugate gradient solver furthermore we propose a fast poisson preconditioner that can be used to accelerate the convergence of the latter we demonstrate the biharmonic extension on a number of test problems that serve to illustrate its effectiveness at producing smooth and accurate extensions near interfaces a further feature of the method is the natural way in which it deals with symmetry and periodicity ensuring through its construction that the extension field also respects these symmetries

a low numerical dissipation immersed interface method for the compressible navier stokes equations a numerical method to solve the compressible navier stokes equations around objects of arbitrary shape using cartesian grids is described the approach considered here uses an embedded geometry representation of the objects and approximate the governing equations with a low numerical dissipation centered finite difference discretization the method is suitable for compressible flows without shocks and can be classified as an immersed interface method the objects are sharply captured by the cartesian mesh by appropriately adapting the discretization stencils around the irregular grid nodes located around the boundary in contrast with available methods no jump conditions are used or explicitly derived from the boundary conditions although a number of elements are adopted from previous immersed interface approaches a new element in the present approach is the use of the summation by parts formalism to develop stable non stiff first order derivative approximations at the irregular grid points second order derivative approximations as those appearing in the transport terms can be stiff when irregular grid points are located too close to the boundary this is addressed using a semi implicit time integration method moreover it is shown that the resulting implicit equations can be solved explicitly in the case of constant transport properties convergence studies are performed for a rotating cylinder and vortex shedding behind objects of varying shapes at different mach and reynolds numbers

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

boundary integral solutions of coupled stokes and darcy flows an accurate computational method based on the boundary integral formulation is presented for solving boundary value problems for stokes and darcy flows the method also applies to problems where the equations are coupled across an interface through appropriate boundary conditions the adopted technique consists of first reformulating the singular integrals for the fluid quantities as single and double layer potentials then the layer potentials are regularized and discretized using standard quadratures as a final step the leading term in the regularization error is eliminated in order to gain one more order of accuracy the numerical examples demonstrate the increase of the convergence rate from first to second order and show a decrease in magnitude of the error the coupled problems require the computation of the gradient of the stokes velocity at the common interface this boundary condition is also written as a combination of single and double layer potentials so that the same approach can be used to compute it accurately extensive numerical examples show the increased accuracy gained by the correction terms

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

a fourth order kernel free boundary integral method for implicitly defined surfaces in three space dimensions abstract the kernel free boundary integral kfbi method is a finite difference version of the traditional boundary integral method for elliptic and parabolic partial differential equations on complex domains it evaluates boundary or volume integrals involved in the solution of boundary integral equations bies by solving equivalent but simple interface problems on regular grids so that the integral kernel or green function is never needed or computed this is the essential difference of the kfbi method from the traditional ones it takes advantage of the well conditioning property of discrete bies so that the number of krylov subspace iterations is essentially independent of discretization parameter or system dimension this paper presents a fourth order kernel free boundary integral method for second order elliptic partial differential equations on complex domains in three space dimensions whose boundaries are given by implicitly defined surfaces it represents the domain boundary and discretizes data on it by intersection points of the surface with cartesian grid lines the approach has a variety of advantages the current work solves simple interface problems with corrected 27 point compact finite difference schemes and calculates the discrete equations with a fast fourier transform based elliptic solver numerical examples show that the proposed method is efficient as well as accurate

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 embedded finite difference method for the poisson equation in a domain with an irregular boundary and dirichlet boundary conditions the poisson equation subject to dirichlet boundary conditions on an irregular domain can be treated by embedding the region in a rectangular domain and solving using finite differences over the rectangle the crucial issue is the discretization of the boundaries of the irregular domain in the past both linear and quadratic boundary treatments have been used and error bounds have been derived in both cases showing that the linear case gives uniform second order accuracy whereas the quadratic case gives third order accuracy at the boundaries and second order accuracy internally thus it has been recommended that the linear boundary treatment be used as it is simpler gives rise to a symmetric matrix formulation and has uniform accuracy the present work shows that this argument is inadequate because the coefficients of the error terms also play an important role we demonstrate this in the 1 d case by determining explicit expressions for the error for both the linear and quadratic boundary treatments it is shown that for the linear case the coefficient of error is in general large enough to dominate the calculation and that therefore it is necessary to use a quadratic boundary treatment in order to obtain errors comparable with those obtained for a regular domain we go on to show that the 1 d expressions for error can be used to approximate the boundary error for 2 d problems and that for the linear treatment the boundary error again dominates

the explicit jump immersed interface method finite difference methods for pdes with piecewise smooth solutions many boundary value problems bvps or initial bvps have nonsmooth solutions with jumps along lower dimensional interfaces the explicit jump immersed interface method ejiim was developed following li s fast iterative immersed interface method fiiim recognizing that the foundation for the efficient solution of many such problems is a good solver for elliptic bvps ejiim generalizes the class of problems for which fiiim is applicable it handles interfaces between constant and variable coefficients and extends the immersed interface method iim to bvps on irregular domains with neumann and dirichlet boundary conditions proofs of second order convergence for a one dimensional 1d problem with piecewise constant coefficients and for two dimensional 2d problems with singular sources are given other problems are reduced to the singular sources case with additional equations determining the source strengths the advantages of ejiim are high quality of solutions even on coarse grids and easy adaptation to many problems with complicated geometries while still maintaining the efficiency of the fiiim

three dimensional matched interface and boundary mib method for treating geometric singularities this paper reports the three dimensional 3d generalization of our previous 2d higher order matched interface and boundary mib method for solving elliptic equations with discontinuous coefficients and non smooth interfaces new mib algorithms that make use of two sets of interface jump conditions are proposed to remove the critical acute angle constraint of our earlier mib scheme for treating interfaces with sharp geometric singularities such as sharp edges sharp wedges and sharp tips the resulting 3d mib schemes are of second order accuracy for arbitrarily complex interfaces with sharp geometric singularities of fourth order accuracy for complex interfaces with moderate geometric singularities and of sixth order accuracy for curved smooth interfaces a systematical procedure is introduced to make the mib matrix optimally symmetric and banded by appropriately choosing auxiliary grid points consequently the new mib linear algebraic equations can be solved with fewer number of iterations the proposed mib method makes use of cartesian grids standard finite difference schemes lowest order interface jump conditions and fictitious values the interface jump conditions are enforced at each intersecting point of the interface and mesh lines to overcome the staircase phenomena in finite difference approximation while a pair of fictitious values are determined along a mesh at a time an iterative procedure is proposed to determine all the required fictitious values for higher order schemes by repeatedly using the lowest order jump conditions a variety of mib techniques are developed to overcome geometric constraints the essential strategy of the mib method is to locally reduce a 2d or a 3d interface problem into 1d like ones the proposed mib method is extensively validated in terms of the order of accuracy the speed of convergence the number of iterations and cpu time numerical experiments are carried out to complex interfaces including the molecular surfaces of a protein a missile interface and van der waals surfaces of intersecting spheres

a numerical method for solving variable coefficient elliptic equation with interfaces a new 2nd order accurate numerical method on non body fitting grids is proposed for solving the variable coefficient elliptic equation in disjoint subdomains w separated by interfaces c the variable coefficients the source term and hence the solution itself and its derivatives may be discontinuous across the interfaces jump conditions in solution and its co normal derivative at interface are prescribed instead of smooth the interfaces are only required to be lipschitz continuous as submanifold a weak formulation is developed the existence uniqueness and regularity of the solutions are studied the numerical method is derived by discretizing the weak formulation the method is different from traditional finite element methods extensive numerical experiments are presented and show that the method is 2nd order accurate in solution and 1st order accurate in its gradient in l norm if the interface is c 2 and solutions are c 2 on the closures of the subdomains the method can handle the problems when the solutions and or the interfaces are weaker than c 2 for example email protected h 2 w c is lipschitz continuous and their singularities coincide see example 18 in section 4 the accuracies of the method under various circumstances are listed in table 19

a simplified reproducing kernel method for 1 d elliptic type interface problems abstract in this paper an efficient numerical method is proposed for the 1d elliptic type interface problems we first construct a broken reproducing kernel space and then apply the simplified reproducing kernel method to the interface problems the convergence of the proposed method is discussed finally some representative numerical examples are presented to illustrate the accuracy and robustness of the proposed method

the rapid evaluation of volume integrals of potential theory on general regions abstract we present a new method for the rapid high order accurate evaluation of certain volume integrals in potential theory on general irregular regions the kernels of the integrals are either a fundamental solution or a linear combination of the derivatives of a fundamental solution of a second order linear elliptic differential equation instead of using a standard quadrature formula or the exact evaluation of any integral the methods rely on rapid methods of solving the differential equation which the kernel is the solution of therefore the number of operations needed to evaluate the volume integral is essentially equal to the number of operations needed to solve the differential equation on a rectangular region with a regular grid and the method requires no evaluation of the kernel