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

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

solving elliptic problems with discontinuities on irregular domains the voronoi interface method we introduce a simple method dubbed the voronoi interface method to solve elliptic problems with discontinuities across the interface of irregular domains this method produces a linear system that is symmetric positive definite with only its right hand side affected by the jump conditions the solution and the solution s gradients are second order accurate and first order accurate respectively in the l norm even in the case of large ratios in the diffusion coefficient this approach is also applicable to arbitrary meshes additional degrees of freedom are placed close to the interface and a voronoi partition centered at each of these points is used to discretize the equations in a finite volume approach both the locations of the additional degrees of freedom and their voronoi discretizations are straightforward in two and three spatial dimensions

cyclotomic matrices over real quadratic integer rings abstract we classify all cyclotomic matrices over real quadratic integer rings and we show that this classification is the same as classifying cyclotomic matrices over the compositum all real quadratic integer rings r moreover we enumerate a related class of symmetric r matrices those r matrices whose eigenvalues are contained inside the interval 2 2 but whose characteristic polynomials are not in z x

high order matched interface and boundary methods for the helmholtz equation in media with arbitrarily curved interfaces this work overcomes the difficulty of the previous matched interface and boundary mib method in dealing with interfaces with non constant curvatures for optical waveguide analysis this difficulty is essentially bypassed by avoiding the use of local cylindrical coordinates in the improved mib method instead novel jump conditions are derived along global cartesian directions for the transverse magnetic field components effective interface treatments are proposed to rigorously impose jump conditions across arbitrarily curved interfaces based on a simple cartesian grid even though each field component satisfies the scalar helmholtz equation the enforcement of jump conditions couples two transverse magnetic field components so that the resulting mib method is a full vectorial approach for the modal analysis of optical waveguides the numerical performance of the proposed mib method is investigated by considering interface problems with both constant and general curvatures the mib method is shown to be able to deliver a fourth order of accuracy in all cases even when a high frequency solution is involved

piecewise polynomial discretization and krylov accelerated multigrid for elliptic interface problems a new piecewise polynomial interface method pim for discretizing elliptic problems with complex interfaces between high contrast materials is derived analyzed and tested a krylov accelerated interface multigrid approach img solves the discretization efficiently stability and convergence are proved in one dimension while an extensive array of numerical experiments with complex interfaces and large coefficient transitions demonstrate the accuracy efficiency and robustness of the method in two dimensions

a two component matched interface and boundary mib regularization for charge singularity in implicit solvation abstract we present a new matched interface and boundary mib regularization method for treating charge singularity in solvated biomolecules whose electrostatics are described by the poisson boltzmann pb equation in a regularization method by decomposing the potential function into two or three components the singular component can be analytically represented by the green s function while other components possess a higher regularity our new regularization combines the efficiency of two component schemes with the accuracy of the three component schemes based on this regularization a new mib finite difference algorithm is developed for solving both linear and nonlinear pb equations where the nonlinearity is handled by using the inexact newton s method compared with the existing mib pb solver based on a three component regularization the present algorithm is simpler to implement by circumventing the work to solve a boundary value poisson equation inside the molecular interface and to compute related interface jump conditions numerically moreover the new mib algorithm becomes computationally less expensive while maintains the same second order accuracy this is numerically verified by calculating the electrostatic potential and solvation energy on the kirkwood sphere on which the analytical solutions are available and on a series of proteins with various sizes

cyclotomic matrices and graphs over the ring of integers of some imaginary quadratic fields abstract we determine all hermitian o q d matrices for which every eigenvalue is in the interval 2 2 for each d 2 7 11 15 to do so we generalise charged signed graphs to l graphs for appropriate finite sets l and classify all l graphs satisfying the same eigenvalue constraints we find that as in the integer case any such matrix graph is contained in a maximal example with all eigenvalues 2

an unfitted finite element method based on nitsche s method for elliptic interface problems in this paper we propose a method for the finite element solution of elliptic interface problem using an approach due to nitsche the method allows for discontinuities internal to the elements in the approximation across the interface we show that optimal order of convergence holds without restrictions on the location of the interface relative to the mesh further we derive a posteriori error estimates for the purpose of controlling functionals of the error and present some numerical examples

a new weak galerkin finite element method for elliptic interface problems a new weak galerkin wg finite element method is introduced and analyzed in this paper for solving second order elliptic equations with discontinuous coefficients and interfaces comparing with the existing wg algorithm for solving the same type problems the present wg method has a simpler variational formulation and fewer unknowns moreover the new wg algorithm allows the use of finite element partitions consisting of general polytopal meshes and can be easily generalized to high orders optimal order error estimates in both h 1 and l 2 norms are established for the present wg finite element solutions extensive numerical experiments have been conducted to examine the accuracy flexibility and robustness of the proposed wg interface approach in solving regular elliptic interface problems high order convergences are numerically confirmed by using piecewise polynomial basis functions of high degrees moreover the wg method is shown to be able to accommodate very complicated interfaces due to its flexibility in choosing finite element partitions finally in dealing with challenging problems with low regularities the piecewise linear wg method is capable of delivering a second order of accuracy in l norm for both c 1 and h 2 continuous solutions

a decomposed immersed interface method for variable coefficient elliptic equations with non smooth and discontinuous solutions a second order accurate finite difference method is presented for solving two dimensional variable coefficient elliptic equations on cartesian grids in which the coefficients the source term the solution and its derivatives may be nonsmooth or discontinuous across an interface a correction term is introduced to the standard central difference stencil so that the numerical discretization is well defined across the interface we also propose a new method to approximate the correction term as part of the iterative procedure the method is easy to implement since the correction term only needs to be added to the right hand side of the system therefore the coefficient matrix remains symmetric and diagonally dominant allowing for most standard solvers to be used numerical examples show good agreements with exact solutions and the order of accuracy is comparable with other immersed interface methods

adaptive mesh refinement and superconvergence for two dimensional interface problems adaptive mesh refinement and the borgers algorithm are combined to generate a body fitted mesh which can resolve the interface with fine geometric details standard linear finite element method bas

a spatially second order alternating direction implicit adi method for solving three dimensional parabolic interface problems abstract a new matched alternating direction implicit adi method is proposed in this paper for solving three dimensional 3d parabolic interface problems with discontinuous jumps and complex interfaces this scheme inherits the merits of its ancestor for two dimensional problems while possesses several novel features such as a non orthogonal local coordinate system for decoupling the jump conditions two side estimation of tangential derivatives at an interface point and a new douglas rachford adi formulation that minimizes the number of perturbation terms to attack more challenging 3d problems in time discretization this new adi method is found to be of first order and stable in various experiments in space discretization the matched adi method achieves the second order accuracy based on simple cartesian grids for various irregularly shaped surfaces and spatial temporal dependent jumps computationally the matched adi method is as efficient as the fastest implicit scheme based on the geometrical multigrid for solving 3d parabolic equations in the sense that its complexity in each time step scales linearly with respect to the spatial degree of freedom n i e o n furthermore unlike iterative methods the adi method is an exact or non iterative algebraic solver which guarantees to stop after a certain number of computations for a fixed n therefore the proposed matched adi method provides a very promising tool for solving 3d parabolic interface problems

the immersed interface multigrid methods for interface problems new multigrid methods are developed for the maximum principle preserving immersed interface method applied to second order linear elliptic and parabolic pdes that involve interfaces and discontinuities for elliptic interface problems the multigrid solver developed in this paper works while some other multigrid solvers do not for linear parabolic equations we have developed the second order maximum principle preserving finite difference scheme in this paper we use the crank nicolson scheme to deal with the diffusion part and an explicit scheme for the first order derivatives numerical examples are also presented

weak galerkin methods for second order elliptic interface problems weak galerkin methods refer to general finite element methods for partial differential equations pdes in which differential operators are approximated by their weak forms as distributions such weak forms give rise to desirable flexibilities in enforcing boundary and interface conditions a weak galerkin finite element method wg fem is developed in this paper for solving elliptic pdes with discontinuous coefficients and interfaces theoretically it is proved that high order numerical schemes can be designed by using the wg fem with polynomials of high order on each element extensive numerical experiments have been carried out to validate the wg fem for solving second order elliptic interface problems high order of convergence is numerically confirmed in both l 2 and l norms for the piecewise linear wg fem special attention is paid to solve many interface problems in which the solution possesses a certain singularity due to the nonsmoothness of the interface a challenge in research is to design nearly second order numerical methods that work well for problems with low regularity in the solution the best known numerical scheme in the literature is of order o h to o h 1 5 for the solution itself in l norm it is demonstrated that the wg fem of the lowest order i e the piecewise constant wg fem is capable of delivering numerical approximations that are of order o h 1 75 to o h 2 in the l norm for c 1 or lipschitz continuous interfaces associated with a c 1 or h 2 continuous solution

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 interface fitted mesh generator and virtual element methods for elliptic interface problems abstract a simple and efficient interface fitted mesh generation algorithm which can produce a semi structured interface fitted mesh in two and three dimensions quickly is developed in this paper elements in such interface fitted meshes are not restricted to simplices but can be polygons or polyhedra especially in 3d the polyhedra instead of tetrahedra can avoid slivers virtual element methods are applied to solve elliptic interface problems with solutions and flux jump conditions algebraic multigrid solvers are used to solve the resulting linear algebraic system numerical results are presented to illustrate the effectiveness of our method

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

accurate solution and gradient computation for elliptic interface problems with variable coefficients a new augmented method is proposed for elliptic interface problems with a piecewise variable coefficient that has a finite jump across a smooth interface the main motivation is to get not only a second order accurate solution but also a second order accurate gradient from each side of the interface key to the new method is introducing the jump in the normal derivative of the solution as an augmented variable and rewriting the interface problem as a new pde that consists of a leading laplacian operator plus lower order derivative terms near the interface in this way the leading second order derivative jump relations are independent of the jump in the coefficient that appears only in the lower order terms after the scaling an upwind type discretization is used for the finite difference discretization at the irregular grid points on or near the interface so that the resulting coefficient matrix is an m matrix a multigrid solver is used to solve the linear system of equations and the gmres iterative me

minimal polynomials of integral symmetric matrices abstract we investigate what the possible minimal polynomials are for integral symmetric matrices we show that the obvious necessary conditions are sufficient for polynomials of degree at most 4 we show that necessary conditions are sufficient for minimal polynomials of self adjoint operators on positive definite unimodular lattices we also give a relatively elementary proof of the result of estes that any total real algebraic integer is the eigenvalue of an integral symmetric matrix this question was asked by alan hoffman who also showed that this result implies that any totally real algebraic integer is the eigenvalue of the adjacency matrix of some graph

maximum principle preserving schemes for interface problems with discontinuous coefficients new finite difference methods using cartesian grids are developed for elliptic interface problems with variable discontinuous coefficients singular sources and nonsmooth or even discontinuous solutions the new finite difference schemes are constructed to satisfy the sign property of the discrete maximum principle using quadratic optimization techniques the methods are shown to converge under certain conditions using comparison functions the coefficient matrix of the resulting linear system of equations is an m matrix and is coupled with a multigrid solver numerical examples are also provided to show the efficiency of the proposed methods

classroom note calculation of weights in finite difference formulas the classical techniques for determining weights in finite difference formulas were either computationally slow or very limited in their scope e g specialized recursions for centered and staggered approximations for adams bashforth adams moulton and bdf formulas for odes etc two recent algorithms overcome these problems for equispaced grids such weights can be found very conveniently with a two line algorithm when using a symbolic language such as mathematica reducing to one line in the case of explicit approximations for arbitrarily spaced grids we describe a computationally very inexpensive numerical algorithm

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

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

a fast iterative algorithm for elliptic interface problems a fast second order accurate iterative method is proposed for the elliptic equation grad cdot beta x y grad u f x y in a rectangular region omega in two space dimensions we assume that there is an irregular interface across which the coefficient beta the solution u and its derivatives and or the source term f may have jumps we are especially interested in the cases where the coefficients beta are piecewise constant and the jump in beta is large the interface may or may not align with an underlying cartesian grid the idea in our approach is to precondition the differential equation before applying the immersed interface method proposed by leveque and li siam j numer anal 4 1994 pp 1019 1044 in order to take advantage of fast poisson solvers on a rectangular region an intermediate unknown function the jump in the normal derivative across the interface is introduced our discretization is equivalent to using a second order difference scheme for a corresponding poisson equation in the region and a second order discretization for a neumann like interface condition thus second order accuracy is guaranteed a gmres iteration is employed to solve the schur complement system derived from the discretization a new weighted least squares method is also proposed to approximate interface quantities from a grid function numerical experiments are provided and analyzed the number of iterations in solving the schur complement system appears to be independent of both the jump in the coefficient and the mesh size

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

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

finite element methods and their convergence for elliptic and parabolic interface problems in this paper we consider the finite element methods for solving second order elliptic and parabolic interface problems in two dimensional convex polygonal domains nearly the same optimal l 2 norm and energy norm error estimates as for regular problems are obtained when the interfaces are of arbitrary shape but are smooth though the regularities of the solutions are low on the whole domain the assumptions on the finite element triangulation are reasonable and practical

integer symmetric matrices having all their eigenvalues in the interval 2 2 abstract we completely describe all integer symmetric matrices that have all their eigenvalues in the interval 2 2 along the way we classify all signed graphs and then all charged signed graphs having all their eigenvalues in this same interval we then classify subsets of the above for which the integer symmetric matrices signed graphs and charged signed graphs have all their eigenvalues in the open interval 2 2

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

a high order hybridizable discontinuous galerkin method for elliptic interface problems we present a high order hybridizable discontinuous galerkin method for solving elliptic interface problems in which the solution and gradient are nonsmooth because of jump conditions across the interface the hybridizable discontinuous galerkin method is endowed with several distinct characteristics first they reduce the globally coupled unknowns to the approximate trace of the solution on element boundaries thereby leading to a significant reduction in the global degrees of freedom second they provide for elliptic problems with polygonal interfaces approximations of all the variables that converge with the optimal order of kc1 in the l norm where k denotes the polynomial order of the approximation spaces third they possess some superconvergence properties that allow the use of an inexpensive element by element postprocessing to compute a new approximate solution that converges with order kc2 however for elliptic problems with finite jumps in the solution across the curvilinear interface the approximate solution and gradient do not converge optimally if the elements at the interface are isoparametric the discrepancy between the exact geometry and the approximate triangulation near the curved interfaces results in lower order convergence to recover the optimal convergence for the approximate solution and gradient we propose to use superparametric elements at the interface copyright 2012 john wiley sons ltd

an augmented matched interface and boundary mib method for solving elliptic interface problem abstract in this paper a second order accurate augmented matched interface and boundary mib is introduced for solving two dimensional 2d elliptic interface problems with piecewise constant coefficients the augmented mib seamlessly combines several key ingredients of the standard mib augmented immersed interface method iim and explicit jump iim to produce a new fast interface algorithm based on the mib zeroth and first order jump conditions are enforced across an arbitrarily curved interface which yields fictitious values on cartesian nodes near the interface by using such fictitious values a simple procedure is proposed to reconstruct cartesian derivative jumps as auxiliary variables and couple them with the jump corrected taylor series expansions which allow us to restore the order of the central difference across the interface to two moreover by using the schur complement to disassociate the algebraic computation of auxiliary variables and function values the discrete laplacian can be efficiently inverted by using the fast fourier transform fft it is found in our numerical experiments that the iteration number in solving the auxiliary system weakly depends on the mesh size as a consequence the total computational cost of the augmented mib is about o n 2 log n for a cartesian grid with dimension n n in 2d therefore the augmented mib outperforms the classical mib in all cases by significantly reducing the cpu time while keeping the same second order of accuracy in dealing with complicated interfaces

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