constructing spectral schemes of the immersed interface method via a global description of discontinuous functions a global description of discontinuous functions is introduced in this paper by expressing a discontinuous function as the sum of a smooth function and a correction term determined by jump conditions we turn the unknown function from a discontinuous one into a sufficiently smooth one when solving a differential equation spectral schemes are developed based on this concept with the intention of eliminating or reducing the gibbs oscillation finite difference schemes are also constructed as an alternative of the current immersed interface methods both spectral and finite difference schemes are tested on one and two dimensional cases

bilum block versions of multielimination and multilevel ilu preconditioner for general sparse linear systems we introduce block versions of the multielimination incomplete lu ilum factorization preconditioning technique for solving general sparse unstructured linear systems these preconditioners have a multilevel structure and for certain types of problems may exhibit properties that are typically enjoyed by multigrid methods several heuristic strategies for forming blocks of independent sets are introduced and their relative merits are discussed the advantages of block ilum over point ilum include increased robustness and efficiency we compare several versions of the block ilum point ilum and the dual threshold based ilut preconditioners in particular tests with some convection diffusion problems show that it may be possible to obtain convergence that is nearly independent of the reynolds number as well as of the grid size

augmented coupling interface method for solving eigenvalue problems with sign changed coefficients in this paper we propose an augmented coupling interface method on a cartesian grid for solving eigenvalue problems with sign changed coefficients the underlying idea of the method is the correct local construction near the interface which incorporates the jump conditions the method which is very easy to implement is based on finite difference discretization the main ingredients of the proposed method comprise i an adaptive order strategy of using interpolating polynomials of different orders on different sides of interfaces which avoids the singularity of the local linear system and enables us to handle complex interfaces ii when the interface condition involves the eigenvalue the original problem is reduced to a quadratic eigenvalue problem by introducing an auxiliary variable and an interfacial operator on the interface iii the auxiliary variable is discretized uniformly on the interface the rest of variables are discretized on an underlying rectangular grid and a proper interpolation between these two grids are designed to reduce the number of stencil points several examples are tested to show the robustness and accuracy of the schemes

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

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 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 kernel free boundary integral method for elliptic boundary value problems this paper presents a class of kernel free boundary integral kfbi methods for general elliptic boundary value problems bvps the boundary integral equations reformulated from the bvps are solved iteratively with the gmres method during the iteration the boundary and volume integrals involving green s functions are approximated by structured grid based numerical solutions which avoids the need to know the analytical expressions of green s functions the kfbi method assumes that the larger regular domain which embeds the original complex domain can be easily partitioned into a hierarchy of structured grids so that fast elliptic solvers such as the fast fourier transform fft based poisson helmholtz solvers or those based on geometric multigrid iterations are applicable the structured grid based solutions are obtained with standard finite difference method fdm or finite element method fem where the right hand side of the resulting linear system is appropriately modified at irregular grid nodes to recover the formal accuracy of the underlying numerical scheme numerical results demonstrating the efficiency and accuracy of the kfbi methods are presented it is observed that the number of gmres iterations used by the method for solving isotropic and moderately anisotropic bvps is independent of the sizes of the grids that are employed to approximate the boundary and volume integrals with the standard second order fems and fdms the kfbi method shows a second order convergence rate in accuracy for all of the tested dirichlet neumann bvps when the anisotropy of the diffusion tensor is not too strong

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

log determinant relaxation for approximate inference in discrete markov random fields graphical models are well suited to capture the complex and non gaussian statistical dependencies that arise in many real world signals a fundamental problem common to any signal processing application of a graphical model is that of computing approximate marginal probabilities over subsets of nodes this paper proposes a novel method applicable to discrete valued markov random fields mrfs on arbitrary graphs for approximately solving this marginalization problem the foundation of our method is a reformulation of the marginalization problem as the solution of a low dimensional convex optimization problem over the marginal polytope exactly solving this problem for general graphs is intractable for binary markov random fields we describe how to relax it by using a gaussian bound on the discrete entropy and a semidefinite outer bound on the marginal polytope this combination leads to a log determinant maximization problem that can be solved efficiently by interior point methods thereby providing approximations to the exact marginals we show how a slightly weakened log determinant relaxation can be solved even more efficiently by a dual reformulation when applied to denoising problems in a coupled mixture of gaussian model defined on a binary mrf with cycles we find that the performance of this log determinant relaxation is comparable or superior to the widely used sum product algorithm over a range of experimental conditions

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

an embedded boundary method for the wave equation with discontinuous coefficients a second order accurate embedded boundary method for the two dimensional wave equation with discontinuous wave propagation speed is described the wave equation is discretized on a cartesian grid with constant grid size and the interface across which the wave speed is discontinuous is allowed to intersect the mesh in an arbitrary fashion by using ghost points on either side of the interface previous embedded boundary techniques for the neumann and dirichlet problems are generalized to satisfy the jump conditions across the interface to second order accuracy the resulting discretization of the jump conditions has the desirable property that each ghost point can be updated independently of all other ghost points resulting in a fully explicit time integration method we prove that the one dimensional restriction of the method is stable without damping for arbitrary locations of the interface relative to the grid for the two dimensional case the previously developed fourth order a t a dissipation is generalized to handle jump conditions we demonstrate that this operator provides sufficient stabilization to enable long time simulations while being weak enough to preserve the accuracy of the solution numerical examples are given where the method is used to study electromagnetic scattering of a plane wave by a dielectric cylinder the numerical solutions are evaluated against the analytical solution due to mie and pointwise second order accuracy is confirmed

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

a level set approach for computing solutions to incompressible two phase flow a level set approach for computing solutions to incompressible two phase flow is presented the interface between the two fluids is considered to be sharp and is described as the zero level set of a smooth function a new treatment of the level set method allows us to efficiently maintain the level set function as the signed distance from the interface we never have to explicitly reconstruct or find the zero level set consequently we are able to handle arbitrarily complex topologies large density and viscosity ratios and surface tension on relatively coarse grids we use a second order projection method along with a second order upwinded procedure for advecting the momentum and level set equations we consider the motion of air bubbles and water drops we also compute flows with multiple fluids such as air oil and water

interpolating stabilized moving least squares mls approximation for 2d elliptic interface problems abstract the main aim of the current paper is to propose a new truly meshless numerical technique to solve the one and two dimensional elliptic interface problems the employed meshless approach is based on a new class of mls approximation the numerical procedure is based on the interpolating stabilized mls ismls approximation the new shape functions that have been made by the ismls technique have the kronecker property thus the dirichlet boundary conditions can be applied directly in the current investigation we propose a meshless collocation method for solving elliptic interface problems based on the shape functions of ismls technique several test problems with sufficient complexity have been studied to check the accuracy and efficiency of the new numerical procedure moreover test problems show the acceptable accuracy and efficiency of the proposed scheme

on the fictitious domain and interpolation formulations of the matched interface and boundary mib method this work overcomes the difficulty of dealing with large curvatures in a high order matched interface and boundary mib method proposed for solving elliptic interface problems the mib method smoothly extends the solution across the interface so that standard high order central finite difference schemes can be used without the loss of accuracy one feature of the mib is that it disassociates the discretization of the elliptic equation from the enforcement of interface jump conditions the other is to make iterative use of only the lowest order jump conditions to determine the fictitious values on extended domains it is of arbitrarily high order in convergence in principle however its applicability was hindered by the lack of sufficiently many grid points to determine all the fictitious values required for high order schemes at the location where the curvature of the interface is relatively large we remove this obstacle by introducing a new concept the disassociation between the discretization and the domain extension we show that the improved mib method is robust for handling general irregular interfaces by extensive numerical experiments on the poisson equation and the helmholtz equation to better understand the mib method and other potential high order interface schemes we propose an alternative interpolation formulation of the mib method and show that the new formulation is essentially equivalent to the improved one

how to obtain an accurate gradient for interface problems abstract it is well known that the immersed interface method iim is second order accurate for interface problems but the accuracy of the first order derivatives or gradients for short is not so clear and is often assumed to be first order accurate in this paper new strategies based on iim are proposed for elliptic interface problems to compute the gradient at grid points both regular and irregular and at the interface from each side of the interface second order in 1d or nearly second order except a factor of log h convergence in 2d of the computed gradient is obtained with almost no extra cost and has been explained in intuition and verified by non trivial numerical tests numerical examples in one two dimensions radial and axis symmetric cases in polar and spherical coordinates are presented to validate the numerical methods and analysis

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

formulation of closed loop min max mpc as a quadratically constrained quadratic program in this note we show that min max model predictive control mpc for linearly constrained polytopic systems with quadratic cost can be cast as a quadratically constrained quadratic program qcqp we use the rigorous closed loop formulation of min max mpc and show that any such min max mpc problem with convex costs and constraints can be cast as a finite dimensional convex optimization problem with the qcqp arising from quadratic costs as a special case at the base of the proof is a lemma showing the convexity of the dynamic programming cost to go which implies that the worst case on an infinite polytopic set is assumed on one of its finitely many vertices as the approach is based on a scenario tree formulation the number of variables in this problem grows exponentially with the horizon length fortunately the qcqp is tree structured and can thus be efficiently solved by specially tailored interior point methods whose computational costs are linear in the number of variables the new formulation as a tree sparse qcqp promises to facilitate online solution of the rigorous min max mpc problem with quadratic costs

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

finite element network approximation of conductivity in particle composites a new finite element method computes conductivity in some unstructured particle reinforced composite material the 2 phase material under consideration is composed of a poorly conducting matrix material filled by highly conducting circular inclusions which are randomly dispersed the mathematical model is a poisson type problem with discontinuous coefficients the discontinuities are huge in contrast and quantity the proposed method generalizes classical continuous piecewise affine finite elements to special computational meshes which encode the particles in a network structure important geometric parameters such as the volume fraction are preserved exactly the computational complexity of the method is almost proportional to the number of inclusions this is minimal in the sense that the representation of the underlying geometry via the positions and radii of the inclusions is of the same complexity the discretization error is proportional to the distance of neighboring inclusions and independent of the conductivity contrast in the medium

numerical approximation of elliptic interface problems via isoparametric finite element methods this paper is concerned with the numerical approximation of elliptic interface problems via isoparametric finite element methods first a convergence analysis is carried which asserts that optimal rates of convergence are recovered in both the energy and the l 2 norms subsequently the efficiency of isoparametric finite elements for the problem at hand is also assessed via numerical experiments results both with smooth and piecewise regular interfaces are presented and discussed the numerical predictions corroborate the theoretical results and they also indicate that second order convergence is achieved in the l norm comparisons between isoparametric finite elements lagrangian finite elements and the immersed interface method are also performed

robust dynamic programming for min max model predictive control of constrained uncertain systems we address min max model predictive control mpc for uncertain discrete time systems by a robust dynamic programming approach and develop an algorithm that is suitable for linearly constrained polytopic systems with piecewise affine cost functions the method uses polyhedral representations of the cost to go functions and feasible sets and performs multiparametric programming by a duality based approach in each recursion step we show how to apply the method to robust mpc and give conditions guaranteeing closed loop stability finally we apply the method to a tutorial example a parking car with uncertain mass

robust constrained model predictive control using linear matrix inequalities abstract the primary disadvantage of current design techniques for model predictive control mpc is their inability to deal explicitly with plant model uncertainty in this paper we present a new approach for robust mpc synthesis that allows explicit incorporation of the description of plant uncertainty in the problem formulation the uncertainty is expressed in both the time and frequency domains the goal is to design at each time step a state feedback control law that minimizes a worst case infinite horizon objective function subject to constraints on the control input and plant output using standard techniques the problem of minimizing an upper bound on the worst case objective function subject to input and output constraints is reduced to a convex optimization involving linear matrix inequalities lmis it is shown that the feasible receding horizon state feedback control design robustly stabilizes the set of uncertain plants several extensions such as application to systems with time delays problems involving constant set point tracking trajectory tracking and disturbance rejection which follow naturally from our formulation are discussed the controller design is illustrated with two examples

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

worst case formulations of model predictive control for systems with bounded parameters two different predictive control formulations are developed based on minimization of the worst case quadratic cost for systems with bounded parameters the two formulations differ on the assumptions made about the future inputs in optimizing the current input one assumes open loop control while the other considers closed loop control their closed loop properties such as asymptotic stability are examined we then focus on a moving average model with an integrator and derive computationally simpler suboptimal algorithms

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

approximate robust dynamic programming and robustly stable mpc we present a technique for approximate robust dynamic programming that is suitable for linearly constrained polytopic systems with piecewise affine cost functions the approximation method uses polyhedral representations of the cost to go function and feasible set and can considerably reduce the computational burden compared to recently proposed methods for exact robust dynamic programming bemporad a borrelli f morari m 2003 min max control of constrained uncertain discrete time linear systems ieee transactions on automatic control 48 9 1600 1606 diehl m bjornberg j 2004 robust dynamic programming for min max model predictive control of constrained uncertain systems ieee transactions on automatic control 49 12 2253 2257 we show how to apply the method to robust mpc and give conditions that guarantee closed loop stability we finish by applying the method to a state constrained tutorial example a parking car with uncertain mass

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

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

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

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

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

lower eigenvalue bounds for singular pencils of matrices abstract beauwens procedure for obtaining lower eigenvalue bounds for regular pencils of matrices a b is simplified and extended to the singular case the theory is then compared through a particular perturbed modified incomplete factorization with notay s generalization of another approach initiated by gustafsson and developed by axelsson and barker and by wilmet

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 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

differential dynamic programming and newton s method a modified version of the original differential dynamic programming ddp algorithm for unconstrained discrete optimal control problems is described this version which differs from the original b

extrinsic meshfree approximation using asymptotic expansion for interfacial discontinuity of derivative a sharp meshfree approximation for derivative discontinuities across arbitrary interfaces is proposed the interface can be arbitrarily located in a domain in which nodes are distributed uniformly or irregularly the proposed meshfree approximations consist of two parts singular and regular the moving least square meshfree approximation is used together with the local wedge function as basis functions the approximations for discontinuities are applied in a meshfree point collocation method to obtain solutions of the poisson problem with a layer delta source on the interface and second order elliptic problems with discontinuous coefficients and or the singular layer sources along the interface the numerical calculations show that this method has good performance even on irregular node models

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

finite volume formulation of the mib method for elliptic interface problems the matched interface and boundary mib method has a proven ability for delivering the second order accuracy in handling elliptic interface problems with arbitrarily complex interface geometries however its collocation formulation requires relatively high solution regularity finite volume method fvm has its merit in dealing with conservation law problems and its integral formulation works well with relatively low solution regularity we propose an mibfvm to take the advantages of both mib and fvm for solving elliptic interface problems we construct the proposed method on cartesian meshes with vertex centered control volumes a large number of numerical experiments are designed to validate the present method in both two dimensional 2d and three dimensional 3d domains it is found that the proposed mibfvm achieves the second order convergence for elliptic interface problems with complex interface geometries in both l and l2 norms

a level set formulation of eulerian interface capturing methods for incompressible fluid flows a level set formulation is derived for incompressible immiscible navier stokes equations separated by a free surface the interface is identified as the zero level set of a smooth function eulerian finite difference methods based on this level set formulation are proposed these methods are robust and efficient and are capable of computing interface singularities such as merging and reconnection numerical experiments are presented to demonstrate the effectiveness of the methods

comparison of taylor finite difference and window finite difference and their application in fdtd the finite difference time domain fdtd method is an important tool in numerical electromagnetic simulation there are many ways to construct a finite difference approximation such as the taylor series expansion theorem the filtering theory etc this paper aims to provide the comparison between the taylor finite difference tfd scheme based on the taylor series expansion theorem and the window finite difference wfd scheme based on the filtering theory their properties have been examined in detail separately in addition the formula of the generalized finite difference gfd scheme is presented which can include both the tfd scheme and the wfd scheme furthermore their application in the numerical solution of maxwell s equations is presented the formulas for the stability criterion and the numerical dispersion relation are derived and analyzed in order to evaluate their performance more accurately a new definition of error is presented upon it the effect of several factors including the grid resolution the courant number and the aspect ratio of the cell on the performance of the numerical dispersion is examined

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

a dimension by dimension splitting immersed interface method for heat conduction equation with interfaces the numerical method proposed in this paper is an improvement of the adi method by li and mayo 1994 the proposed method is unconditionally stable for both two and three dimensional heat conduction interface problems while li s adi method is only stable for two dimensional problems the method is a modification of a locally one dimensional lod difference scheme with correction term added to the right hand side of the standard lod difference scheme at irregular points the correction term is determined so that the local truncation error is of order o h at irregular points then the method is two order convergent in both time and space directions numerical examples show good agreement with exact solutions and confirm the order of convergence and stability

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

min max control of constrained uncertain discrete time linear systems for discrete time uncertain linear systems with constraints on inputs and states we develop an approach to determine state feedback controllers based on a min max control formulation robustness is achieved against additive norm bounded input disturbances and or polyhedral parametric uncertainties in the state space matrices we show that the finite horizon robust optimal control law is a continuous piecewise affine function of the state vector and can be calculated by solving a sequence of multiparametric linear programs when the optimal control law is implemented in a receding horizon scheme only a piecewise affine function needs to be evaluated on line at each time step the technique computes the robust optimal feedback controller for a rather general class of systems with modest computational effort without needing to resort to gridding of the state space

min max feedback model predictive control for constrained linear systems min max feedback formulations of model predictive control are discussed both in the fixed and variable horizon contexts the control schemes the authors discuss introduce in the control optimization the notion that feedback is present in the receding horizon implementation of the control this leads to improved performance compared to standard model predictive control and resolves the feasibility difficulties that arise with the min max techniques that are documented in the literature the stabilizing properties of the methods are discussed as well as some practical implementation details

lu decomposition and numerical structure for solving large sparse nonsymmetric linear systems abstract in this work the solution of a large sparse linear system of equations with an arbitrary sparsity pattern is obtained by using lu decomposition method as well as numerical structure approach the lu decomposition method is based on doolittle s method while the numerical structure approach is based on cramer s rule the numerical structure approach produces direct solution without facing fill in problems as encountered in lu decomposition in order to reduce the fill ins in the decomposition the powers of a boolean matrix obtained from the coefficient matrix a are taken so that the fill ins in the structure of a can be known in advance the position of fill ins in a are thus determined in the best choice manner that is it is very effective and memory wise cheap we also outline a method by using numerical structure with reduced computation efforts finally experiments are performed on eight examples to compare the efficiency of the proposed methods the results obtained are reported in a table it is found that the lu decomposition method is much better than numerical structure the usefulness of numerical structure approach is also discussed

calculation of incompressible viscous flows by an unconditionally stable projection fem this paper investigates the numerical performance of a finite element implementation of a new incremental fractional step method to compute steady and unsteady incompressible viscous flows under general boundary conditions and using unstructured meshes a variational framework is adopted which accommodates two different spaces for representing and approximating the velocity fields calculated respectively in the viscous and inviscid phases of the method but which leads to a very simple numerical scheme in terms of only one discrete velocity field an unconditionally stable semi implicit approximation of the nonlinear term is used to eliminate any time step restriction as far as the numerical stability is concerned numerical results for five test problems in two dimensions are reported to illustrate the flexibility of the proposed method

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

a 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

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

immersed interface method for elliptic equations based on a piecewise second order polynomial in this paper we develop finite difference methods for elliptic equations in a domain w r d d 1 2 within the region w we suppose there is an irregular surface c of codimension 1 hereafter called an interface across which the function u or some of its derivatives are known to be discontinuous we use uniform grid and a piecewise second order polynomial to approximate u then we get a second order method for these problems at last we give several examples to show the correctness and efficiency of the scheme

a preconditioning technique for indefinite systems resulting from mixed approximations of elliptic problems this paper provides a preconditioned iterative technique for the solution of saddle point problems these problems typically arise in the numerical approximation of partial differential equations by lagrange multiplier techniques and or mixed methods the saddle point problem is reformulated as a symmetric positive definite system which is then solved by conjugate gradient iteration applications to the equations of elasticity and stokes are discussed and the results of numerical experiments are given

a parallel multithreaded sparse triangular linear system solver abstract we propose a parallel sparse triangular linear system solver based on the spike algorithm sparse triangular systems are required to be solved in many applications often they are a bottleneck due to their inherently sequential nature furthermore typically many successive systems with the same coefficient matrix and with different right hand side vectors are required to be solved the proposed solver decouples the problem at the cost of extra arithmetic operations as in the banded case compared to the banded case there are extra savings due to the sparsity of the triangular coefficient matrix we show the parallel performance of the proposed solver against the state of the art parallel sparse triangular solver in intel s math kernel library mkl on a multicore architecture we also show the effect of various sparse matrix reordering schemes numerical results show that the proposed solver outperforms mkl s solver in 80 of cases by a factor of 2 47 on average

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 continuum method for modeling surface tension abstract a new method for modeling surface tension effects on fluid motion has been developed interfaces between fluids of different properties or colors are represented as transition regions of finite thickness across which the color variable varies continuously at each point in the transition region a force density is defined which is proportional to the curvature of the surface of constant color at that point it is normalized so that the conventional description of surface tension on an interface is recovered when the ratio of local transition region thickness to local radius of curvature approaches zero the continuum method eliminates the need for interface reconstruction simplifies the calculation of surface tension enables accurate modeling of two and three dimensional fluid flows driven by surface forces and does not impose any modeling restrictions on the number complexity or dynamic evolution of fluid interfaces having surface tension computational results for two dimensional flows are given to illustrate the properties of the method

solving elliptic interface problems with jump conditions on cartesian grids abstract we present a simple numerical algorithm for solving elliptic equations where the diffusion coefficient the source term the solution and its flux are discontinuous across an irregular interface the algorithm produces second order accurate solutions and first order accurate gradients in the l norm on cartesian grids the condition number is bounded regardless of the ratio of the diffusion constant and scales like that of the standard 5 point stencil approximation on a rectangular grid with no interface numerical examples are given in two and three spatial dimensions

sparse matrix methods for use in electrical impedance tomography the objective of electrical impedance tomography is to reconstruct images representing the electrical impedance properties within a region from measurements on its surface the region of interest is usually first discretized into finite elements and its impedance distribution updated using an iterative process this iterative process comprises two problems the forward problem and the inverse problem the inverse problem is the term given to the procedure to find the internal impedance distribution from a set of boundary measurements and the forward problem is the determination of the internal voltages given the impedance distribution and boundary conditions in this paper several finite element labelling algorithms implemented by the authors in c are investigated and their impact on the forward problem solver efficiency analysed the algorithms investigated are nested dissection nd minimum degree mdg minimum deficiency mdf and simulated annealing for fill in safr reduction during cholesky factorization these renumbering strategies were applied to a collection of representative two dimensional meshes used in electrical impedance tomography and a number of sparse symmetric matrices from the harwell boeing sparse matrix collection for comparison purposes 1997 by john wiley sons ltd

a front tracking method for viscous incompressible multi fluid flows abstract a method to simulate unsteady multi fluid flows in which a sharp interface or a front separates incompressible fluids of different density and viscosity is described the flow field is discretized by a conservative finite difference approximation on a stationary grid and the interface is explicitly represented by a separate unstructured grid that moves through the stationary grid since the interface deforms continuously it is necessary to restructure its grid as the calculations proceed in addition to keeping the density and viscosity stratification sharp the tracked interface provides a natural way to include surface tension effects both two and three dimensional full numerical simulations of bubble motion are presented

discontinuous bubble scheme for elliptic problems with jumps in the solution abstract we propose a new numerical method to solve an elliptic problem with jumps both in the solution and derivative along an interface by considering a suitable function which has the same jumps as the solution we transform the problem into one without jumps then we apply the immersed finite element method in which we allow uniform meshes so that the interface may cut through elements to discretize the problem as introduced in 1 2 3 some convenient way of approximating the jumps of the solution by piecewise linear functions is suggested our method can also handle the case when the interface passes through grid points we believe this paper presents the first resolution of such cases numerical experiments for various problems show second order convergence in l2 and first order in h1 norms moreover the convergence order is very robust for all problems tested

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

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