new formulations for interface problems in polar coordinates in this paper numerical methods are proposed for some interface problems in polar or cartesian coordinates the new methods are based on a formulation that transforms the interface problem with a nonsmooth or discontinuous solution into a problem with a smooth solution the new formulation leads to a simple second order finite difference scheme for the partial differential equation and a new interpolation scheme for the normal derivative of the solution in conjunction with the fast immersed interface method a fast solver has been developed for the interface problems with a piecewise constant but a discontinuous coefficient using the new formulation in a polar coordinate system

caratheodory toeplitz and nehari problems for matrix valued almost periodic functions in this paper the positive and strictly contractive extension problems for almost periodic matrix functions are treated we present nece ssary and sufficient conditions for the existence of extensions in terms of toeplit z and hankel operators on besicovitch spaces and lebesgue spaces furthermore when a solution exists a special extension the band extension is constructed which enjoys a maximum entropy property a linear fractional parameterization of the set of all extensions is also provided the techniques used xn the proofs include factorizations of matrix valued almost periodic functions and a general algebraic scheme called the band method

a cartesian grid embedded boundary method for solving the poisson and heat equations with discontinuous coefficients in three dimensions we present a method for solving poisson and heat equations with discontinuous coefficients in two and three dimensions it uses a cartesian cut cell embedded boundary method to represent the interface between materials as described in johansen and colella 1998 matching conditions across the interface are enforced using an approximation to fluxes at the boundary overall second order accuracy is achieved as indicated by an array of tests using non trivial interface geometries both the elliptic and heat solvers are shown to remain stable and efficient for material coefficient contrasts up to 10 6 thanks in part to the use of geometric multigrid a test of accuracy when adaptive mesh refinement capabilities are utilized is also performed an example problem relevant to nuclear reactor core simulation is presented demonstrating the ability of the method to solve problems with realistic physical parameters

algebras of almost periodic functions with bohr fourier spectrum in a semigroup hermite property and its applications abstract it is proved that the unital banach algebra of almost periodic functions of several variables with bohr fourier spectrum in a given additive semigroup is an hermite ring the same property holds for the wiener algebra of functions that in addition have absolutely convergent bohr fourier series as applications of the hermite property of these algebras we study factorizations of wiener hopf type of rectangular matrix functions and the toeplitz corona problem in the context of almost periodic functions of several variables

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

solving partial differential equations on irregular domains with moving interfaces with applications to superconformal electrodeposition in semiconductor manufacturing we present a numerical algorithm for solving partial differential equations on irregular domains with moving interfaces instead of the typical approach of solving in a larger rectangular domain our approach performs most calculations only in the desired domain to do so efficiently we have developed a one sided multigrid method to solve the corresponding large sparse linear systems our focus is on the simulation of the electrodeposition process in semiconductor manufacturing in both two and three dimensions our goal is to track the position of the interface between the metal and the electrolyte as the features are filled and to determine which initial configurations and physical parameters lead to superfilling we begin by motivating the set of equations which model the electrodeposition process building on existing models for superconformal electrodeposition we develop a model which naturally arises from a conservation law form of surface additive evolution we then introduce several numerical algorithms including a conservative material transport level set method and our multigrid method for one sided diffusion equations we then analyze the accuracy of our numerical methods finally we compare our result with experiment over a wide range of physical parameters

smoothed hat functions in subdivision following the general idea of smoothing through convolution we discuss schemes for smoothing hat functions on the three directional and the four directional mesh the smoothing masks which in the four directional case are matrix masks have the same support as the refinement mask for the hat functions several smoothing schemes are presented and in some cases the smoothness properties of the smoothed functions are analysed in particular the results for the four directional case follow new and original lines of research

enforcement of constraints and maximum principles in the variational multiscale method abstract we present a new theoretical framework for the enforcement of constraints in variational multiscale vms analysis the theory is first presented in an abstract operator format and subsequently specialized for the steady advection diffusion equation the approach borrows heavily from results in constrained and convex optimization an exact expression for the fine scales is derived in terms of variational derivatives of the constraints lagrange multipliers and a fine scale green s function the methodology described enables the development of numerical methods which satisfy predefined attributes a practical and effective procedure for solving the steady advection diffusion equation is presented based on a vms inspired stabilized method weakly enforced dirichlet boundary conditions and enforcement of a maximum principle and conservation constraint

parameter identification for semiconductor diodes by lbic imaging laser beam induced current lbic imaging is a nondestructive technique used for the characterization of the electrical structure within a semiconductor in this paper a model is formulated for this technique using the standard drift diffusion model and subsequently an approximate version and its dual are derived for the study of the inverse problem the formulation is then applied to a cross sectional model for n on p devices of finite depth to study in detail the relation between the lbic images and the device parameters numerical methods are developed for the simulation of the lbic image of a diode as well as for the identification of parameters from the lbic image by least squares formulation numerical examples are presented to illustrate the success of identifying parameters such as junction depth diffusion length and equilibrium potential of an abrupt p n junction diode from its lbic image the differentiability of the image with respect to the parameters also is established

fast solvers for 3d poisson equations involving interfaces in a finite or the infinite domain in this paper numerical methods are proposed for poisson equations defined in a finite or infinite domain in three dimensions in the domain there can exists an interface across which the source term the flux and therefore the solution may be discontinuous the existence and uniqueness of the solution are also discussed to deal with the discontinuity in the source term and in the flux the original problem is transformed to a new one with a smooth solution such a transformation can be carried out easily through an extension of the jumps along the normal direction if the interface is expressed as the zero level set of a three dimensional function an auxiliary sphere is used to separate the infinite region into an interior and exterior domain the kelvin s inversion is used to map the exterior domain into an interior domain the two poisson equations defined in the interior and the exterior written in spherical coordinates are solved simultaneously by choosing the mesh size carefully and exploiting the fast fourier transform the resulting finite difference equations can be solved efficiently the approach in dealing with the interface has also been used with the artificial boundary condition technique which truncates the infinite domain numerical results demonstrate second order accuracy of our algorithms

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

the immersed interface method for nonlinear differential equations with discontinuous coefficients and singular sources we extend the immersed interface method of leveque and li to find numerical solutions of one dimensional parabolic partial differential equations of the form u t beta x t u x x lambda x t u x kappa u u x f x where beta u beta u x and f may have known discontinuities at a known location x alpha at each time step a large weakly nonlinear system is set up using a difference scheme which is standard away from x alpha and which is derived for grid points near alpha by solving small linear systems which are determined from the jumps at x alpha the time stepping is done with a crank nicholson scheme and the nonlinear systems are solved with a levenberg marquardt method as an example we consider the flow of cars on a one lane highway with an entrance or exit where traffic is treated as a continuous fluid numerical examples show that we can compute solutions to these equations with second order accuracy

an immersed interface method for solving anisotropic elliptic boundary value problems in three dimensions we have developed a first order stable cartesian grid discretization that uses only interior grid points for inhomogeneous anisotropic elliptic operators subject to neumann boundary conditions on a bounded nonrectangular geometry in three dimensions for this discretization method a necessary and sufficient condition depending on the mesh size h for the existence of this first order stable scheme at a regular i e interior grid point is found in terms of the anisotropy matrix for this discretization method a way to analyze the existence of a first order stable scheme at an irregular i e boundary grid point is also given the arguments are identical to those for the two dimensional case m a dumett and j p keener a numerical method for solving anisotropic elliptic boundary value problems on an irregular domain in 2d manuscript only the details change unlike in m a dumett and j p keener a numerical method for solving anisotropic elliptic boundary value problems on an irregular domain in 2d manuscript a discussion of dirichlet and robin boundary conditions is also included in particular it is shown that the gerschgorin condition does not impose sign restrictions on irregular grid points stencil coefficients as in the neumann case

higher order cartesian grid based finite difference schemes for elliptic equations on irregular domains second and fourth order cartesian grid based finite difference methods are proposed for elliptic and parabolic partial differential equations and associated eigenvalue problems on irregular domains with general boundary conditions our methods are based on the continuation of a solution idea using multivariable taylor s expansion of the solution about selected boundary points and the core ideas of the immersed interface method the methods offer systematic treatment of the general boundary conditions in two and three dimensional domains and are directly applied to semi discretize heat equations on irregular domains convergence analysis and numerical examples are presented the validity and effectiveness of the proposed methods are demonstrated through our numerical results including computations of the eigenvalues of the associated eigenvalue problem

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

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

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

matrix dependent prolongations and restrictions in a blackbox multigrid solver multigrid methods are studied for the solution of linear systems resulting from the 9 point discretization of a general linear second order elliptic partial differential equation in two dimensions the rate of convergence of standard multigrid methods often deteriorates when the coefficients in the differential equation are discontinuous or when dominating first order terms are present these difficulties may be overcome by choosing the prolongation and restriction operators in a special way a novel way to do this is proposed as a result a blackbox solver written in standard fortran 77 has been developed numerical experiments for several hard test problems are described and comparison is made with other algorithms the standard mg method and a method introduced by kettler a significant improvement of robustness and efficiency is found

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

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

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

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

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

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

invertibility of toeplitz operators and corona conditions in a strip abstract a toeplitz operator with symbol g such that det g 1 is invertible if there is a non trivial solution to a riemann hilbert problem g with and satisfying the corona conditions in c and c respectively however determining such a solution and verifying that the corona conditions are satisfied are in general difficult problems in this paper on one hand we establish conditions on which are equivalent to the corona conditions but easier to verify if g 1 are analytic and bounded in a strip this happens in particular with almost periodic symbols on the other hand we identify new classes of symbols g for which a non trivial solution to g can be explicitly determined and the corona conditions can be verified by the above mentioned approach thus obtaining invertibility criteria for the associated toeplitz operators

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

a multigrid ghost point level set method for incompressible navier stokes equations on moving domains with curved boundaries abstract in this paper we present a numerical approach to solve the navier stokes equations on moving domains with second order accuracy the space discretization is based on the ghost point method which falls under the category of unfitted boundary methods since the mesh does not adapt to the moving boundary the equations are advanced in time by using crank nicholson the momentum and continuity equations are solved simultaneously for the velocity and the pressure by adopting a proper multigrid approach to avoid the checkerboard instability for the pressure a staggered grid is adopted where velocities are defined at the sides of the cell and the pressure is defined at the centre the lack of uniqueness for the pressure is circumvented by the inclusion of an additional scalar unknown representing the average divergence of the velocity and an additional equation to set the average pressure to zero several tests are performed to simulate the motion of an incompressible fluid around a moving object as well as the lid driven cavity tests around steady objects the object is implicitly defined by a level set approach that allows a natural computation of geometrical properties such as distance from the boundary normal directions and curvature different shapes are tested circle ellipse and flower numerical results show the second order accuracy for the velocity and the divergence that decays to zero with second order and the efficiency of the multigrid that is comparable with the tests available in literature for rectangular domains without objects showing that the presence of a complex shaped object does not degrade the performance

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

convergence analysis of the q1 finite element method for elliptic problems with non boundary fitted meshes summary the aim of this paper is to derive a priori error estimates when the mesh does not fit the original domain s boundary this problematic of the last century e g the finite difference methodology returns to topical studies with the huge development of domain embedding fictitious domain or cartesiangrid methods these methods use regular structured meshes most often cartesian for non aligned domains although non boundary fitted approaches become more and more applied very few studies are devoted to theoretical error estimates in this paper the convergence of a q1 nonconforming finite element method is analyzed for secondorder elliptic problems with dirichlet robin or neumann boundary conditions the finite element method uses standard q1 rectangular finite elements as the finite element approximate space is not contained in the original solution space this method is referred to as nonconforming a stairstep boundary defined from the cartesian mesh approximates the original domain s boundary the convergence analysis of the finite element method for such a kind of non boundary fitted stairstepped approximation is no treated in the literature the study of dirichlet problems is based on similar techniques as those classically used with boundary fitted linear triangular finite elements the estimates obtained for robin problems are novel and use some more technical arguments the rate of convergence is proved to be in o h 1 2 for the h 1 norm for all general boundary conditions and classical duality arguments allow to obtain an o h error estimate in the l 2 norm for dirichlet problems numerical results obtained with fictitious domain techniques that impose original boundary conditions on a non boundary fitted approximate immersed interface are presented these results confirm the theoretical rates of convergence copyright c 2007 john wiley sons ltd

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

lagrange multiplier approach with optimized finite difference stencils for pricing american options under stochastic volatility the deterministic numerical valuation of american options under heston s stochastic volatility model is considered the prices are given by a linear complementarity problem with a two dimensional parabolic partial differential operator a new truncation of the domain is described for small asset values while for large asset values and variance a standard truncation is used the finite difference discretization is constructed by numerically solving a quadratic optimization problem aiming to minimize the truncation error at each grid point a lagrange approach is used to treat the linear complementarity problems numerical examples demonstrate the accuracy and effectiveness of the proposed approach

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

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

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

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

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 sharp interface finite volume method for elliptic equations on cartesian grids we present a second order sharp interface finite volume method for the solution of the three dimensional elliptic equation x u x f x with variable coefficients on cartesian grids in particular we focus on interface problems with discontinuities in the coefficient the source term the solution and the fluxes across the interface the method uses standard piecewise trilinear finite elements for normal cells and a double piecewise trilinear ansatz for the solution on cells intersected by the interface resulting always in a compact 27 point stencil singularities associated with vanishing partial volumes of intersected grid cells are removed by a two term asymptotic approach in contrast to the 2d method presented by two of the authors in m oevermann r klein a cartesian grid finite volume method for elliptic equations with variable coefficients and embedded interfaces journal of computational physics 219 2006 749 769 we use a minimization technique to determine the unknown coefficients of the double trilinear ansatz this simplifies the treatment of the different cut cell types and avoids additional special operations for degenerated interface topologies the resulting set of linear equations has been solved with a bicgstab solver preconditioned with an algebraic multigrid in various testcases including large ratios and non smooth interfaces the method achieves second order of accuracy in the l and l 2 norm

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

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

an augmented cartesian grid method for stokes darcy fluid structure interactions summary a new finite difference method based on cartesian meshes is proposed for solving the fluid structure interaction between a fluid flow modeled by the stokes equations and a porous media modeled by the darcy s law the idea is to introduce several augmented variables along the interface between the fluid flow and the porous media so that the problem can be decoupled as several poisson equations the augmented variables should be chosen so that the beavers joseph saffman and other interface conditions are satisfied in the discretization the augmented variables have co dimension one compared with that of the primitive variables and are solved through the schur complement system a non trivial analytic solution with a circular interface is constructed to check second order convergency of the proposed method numerical examples with various interfaces and parameters are also presented some simulations show interesting behaviors of the fluid structure interaction between the fluid flow and the porous media the computational framework can be applied to other multi phase and multi physics problems copyright 2015 john wiley sons ltd

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 new multiscale finite element method for high contrast elliptic interface problems we introduce a new multiscale finite element method which is able to accurately capture solutions of elliptic interface problems with high contrast coefficients by using only coarse quasiuniform meshes and without resolving the interfaces a typical application would be the modelling of flow in a porous medium containing a number of inclusions of low or high permeability embedded in a matrix of high respectively low permeability our method is h 1 conforming with degrees of freedom at the nodes of a triangular mesh and requiring the solution of subgrid problems for the basis functions on elements which straddle the coefficient interface but which use standard linear approximation otherwise a key point is the introduction of novel coefficientdependent boundary conditions for the subgrid problems under moderate assumptions we prove that our methods have optimal convergence rate of o h in the energy norm and o h 2 in the l 2 norm where h is the coarse mesh diameter and the hidden constants in these estimates are independent of the contrast i e ratio of largest to smallest value of the pde coefficient for standard elements the best estimate in the energy norm would be o h 1 2 e with a hidden constant which in general depends on the contrast the new interior boundary conditions depend not only on the contrast of the coefficients but also on the angles of intersection of the interface with the element edges

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

factorization of matrix functions with subgroup supported fourier coefficients abstract it is proved that if a wiener class matrix function on a connected compact abelian group admits a factorization with respect to a connected compact abelian supergroup then it admits a factorization with respect to the original group in particular all factorization indices are characters of the original group an illustrative example is considered

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

multivariate matrix refinable functions with arbitrary matrix dilation characterizations of the stability and orthonormality of a multivariate matrix refinable function d with arbitrary matrix dilation m are provided in terms of the eigenvalue and 1 eigenvector properties of the restricted transition operator under mild conditions it is shown that the approximation order of d is equivalent to the order of the vanishing moment conditions of the matrix refinement mask pa the restricted transition operator associated with the matrix refinement mask pa is represented by a finite matrix ami j i j with a idet m i 1 e p j q3 p e and p j q3 p being the kronecker product of matrices p j and p the spectral properties of the transition operator are studied the sobolev regularity estimate of a matrix refinable function 4 is given in terms of the spectral radius of the restricted transition operator to an invariant subspace this estimate is analyzed in an example

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

symmetric interior penalty galerkin approaches for two dimensional parabolic interface problems with low regularity solutions abstract this work presents novel finite element approaches for solving a parabolic partial differential equation with discontinuous coefficients and low regularity solutions in a bounded convex polyhedral domain a spatial semi discretization based on symmetric interior penalty galerkin sipg approximations is constructed and analyzed by using discontinuous piecewise linear functions for smooth initial data spatial errors in the broken l 2 h 1 and l 2 h 1 norms are proven to be optimal with respect to low regularity solutions which are only piecewise h 1 s smooth with 0 s 1 furthermore we present the full sipg discretizations based on an euler backward finite difference time discretization the proposed approximations are shown to be unconditionally stable and have nearly the optimal l 2 l 2 and l 2 h 1 error estimates even when the regularities of the solutions are low on the whole domain the error estimates are optimal with respect to time sharply depending on the indexes in the global and local regularity numerical experiments for two dimensional parabolic interface problems verify the theoretical convergence rates

three dimensional elliptic solvers for interface problems and applications second order accurate elliptic solvers using cartesian grids are presented for three dimensional interface problems in which the coefficients the source term the solution and its normal flux may be discontinuous across an interface one of our methods is designed for general interface problems with variable but discontinuous coefficient the scheme preserves the discrete maximum principle using constrained optimization techniques an algebraic multigrid solver is applied to solve the discrete system the second method is designed for interface problems with piecewise constant coefficient the method is based on the fast immersed interface method and a fast 3d poisson solver the second method has been modified to solve helmholtz poisson equations on irregular domains an application of our method to an inverse interface problem of shape identification is also presented in this application the level set method is applied to find the unknown surface iteratively

a novel concept for the design of immersed interface methods the objective of this paper is to present a novel robust high order accurate immersed interface method iim for advection diffusion type equations in contrast to other immersed methods that were designed for consistency and accuracy with a posteriori check of the numerical stability we combine local taylor series expansion at irregular grid points with a local stability constraint as part of the design process stability investigations of the iim are employed to demonstrate that the local stability constraint is sufficient for obtaining a globally stable method as long as the neumann number is less than its limiting value one of the key aspects of this iim is that the irregular finite difference stencils can be isolated from the rest of the computational domain to validate our novel immersed interface approach two dimensional and three dimensional test cases for model equations are presented in addition this method is applied to the incompressible navier stokes equations to conduct stability investigations of a boundary layer flow over a rough surface and for investigations of pulsatile stenotic flows stability investigations of wall bounded flows are challenging for immersed methods because the near wall accuracy is important for correctly capturing the characteristics of the hydrodynamic instability mechanisms in particular regarding the wave relation between the wave velocity components close to the wall

a new subdivision method for bivariate splines on the four directional mesh we present a new bivariate subdivision scheme based on two generators of a four directional spline space in particular we are dealing with piecewise linear continuous splines and with piecewise cubic continuously differentiable splines the subdivision schemes rely on a matrix mask of small support in addition we present some results on related quasi interpolating operators and on approximation order of the underlying shift invariant spaces

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

the corona theorem and the canonical factorization of triangular ap matrix functions effective criteria and explicit formulas abstract using results of the companion paper the authors obtain effective criteria for the existence of canonical factorization for new classes of triangular almost periodic matrix functions of the form formula the important role in the proof of necessity in these criteria is played by a result on the stability of canonical ap factorizability obtained in the present paper by finding corona solutions for a pair of binomial almost periodic functions it is possible to calculate explicitly the factors of a canonical factorization for a class of considered matrix functions with trinomial f

an interpolation matched interface and boundary method for elliptic interface problems an interpolation matched interface and boundary imib method with second order accuracy is developed for elliptic interface problems on cartesian grids based on original mib method proposed by zhou et al y zhou g wei on the fictious domain and interpolation formulations of the matched interface and boundary method j comput phys 219 2006 228 246 explicit and symmetric finite difference formulas at irregular grid points are derived by virtue of the level set function the difference scheme using imib method is shown to satisfy the discrete maximum principle for a certain class of problems rigorous error analyses are given for the imib method applied to one dimensional 1d problems with piecewise constant coefficients and two dimensional 2d problems with singular sources comparison functions are constructed to obtain a sharp error bound for 1d approximate solutions furthermore we compare the ghost fluid method gfm immersed interface method iim mib and imib methods for 1d problems finally numerical examples are provided to show the efficiency and robustness of the proposed method

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