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An Efficient Neural-Network and Finite-Difference Hybrid Method for elliptic interface problems with Applications

Communications in Computational Physics 2023年第4期33卷 1090-1105页

作者： Wei-Fan Hu Te-Sheng Lin Yu-Hau Tseng Ming-Chih Lai Department of Mathematics National Central UniversityTaoyuan 32001Taiwan Department of Applied Mathematics National Yang Ming Chiao Tung UniversityHsinchu 30010Taiwan Department of Applied Mathematics National University of KaohsiungKaohsiung 81148Taiwan National Center for Theoretical Sciences National Taiwan UniversityTaipei 10617Taiwan

A new and efficient neural-network and finite-difference hybrid method is developed for solving Poisson equation in a regular domain with jump discontinuities on embedded irregular interfaces.Since the solution has low regularity across the interface,when applying finite difference discretization to this problem,an additional treatment accounting for the jump discontinuities must be employed.Here,we aim to elevate such an extra effort to ease our implementation by machine learning methodology.The key idea is to decompose the solution into singular and regular parts.The neural network learning machinery incorporating the given jump conditions finds the singular solution,while the standard five-point Laplacian discretization is used to obtain the regular solution with associated boundary conditions.Regardless of the interface geometry,these two tasks only require supervised learning for function approximation and a fast direct solver for Poisson equation,making the hybrid method easy to implement and efficient.The two-and three-dimensional numerical results show that the present hybrid method preserves second-order accuracy for the solution and its derivatives,and it is comparable with the traditional immersed interface method in the literature.As an application,we solve the Stokes equations with singular forces to demonstrate the robustness of the present method.

关键词： Neural networks sharp interface method fast direct solver elliptic interface problem Stokes equations

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A Fourth-Order Kernel-Free Boundary Integral Method for interface problems

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Communications in Computational Physics 2023年第3期33卷 764-794页

作者： Yaning Xie Shuwang Li Wenjun Ying College of Science Zhejiang University of TechnologyXihuHangzhou 310023P.R.China Department of Applied Mathematics Illinois Institute of TechnologyChicagoUnited States School of Mathematical Sciences MOE-LSC and Institute of Natural SciencesShanghai Jiao Tong UniversityMinhangShanghai 200240P.R.China.

This paper presents a fourth-order Cartesian grid based boundary integral method(BIM)for heterogeneous interface problems in two and three dimensional space,where the problem interfaces are irregular and can be explicitly given by parametric curves or implicitly defined by level set functions.The method reformulates the governing equation with interface conditions into boundary integral equations(BIEs)and reinterprets the involved integrals as solutions to some simple interface problems in an extended regular region.Solution of the simple equivalent interface problems for integral evaluation relies on a fourth-order finite difference method with an FFT-based fast elliptic solver.The structure of the coefficient matrix is preserved even with the existence of the interface.In the whole calculation process,analytical expressions of Green’s functions are never determined,formulated or computed.This is the novelty of the proposed kernel-free boundary integral(KFBI)method.Numerical experiments in both two and three dimensions are shown to demonstrate the algorithm efficiency and solution accuracy even for problems with a large diffusion coefficient ratio.

关键词： elliptic interface problem compact scheme finite difference method Cartesian grid method kernel-free boundary integral method boundary integral equation

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An Adaptive Mesh Refinement Strategy for Immersed Boundary/interface Methods

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Communications in Computational Physics 2012年第7期12卷 515-527页

作者： Zhilin Li Peng Song Center for Research in Scientific Computation&Department of Mathematics North Carolina State UniversityRaleighNC 27695USA Nanjing Normal University China Operations Research Program North Carolina StateUniversityRaleighNC 27695USA

An adaptive mesh refinement strategy is proposed in this paper for the Immersed Boundary and Immersed interface methods for two-dimensional elliptic interface problems involving singular sources.The interface is represented by the zero level set of a Lipschitz functionϕ(x,y).Our adaptive mesh refinement is done within a small tube of|ϕ(x,y)|δwith finer Cartesian meshes.The discrete linear system of equations is solved by a multigrid solver.The AMR methods could obtain solutions with accuracy that is similar to those on a uniform fine grid by distributing the mesh more economically,therefore,reduce the size of the linear system of the equations.Numerical examples presented show the efficiency of the grid refinement strategy.

关键词： Adaptive mesh refinement immersed boundary method immersed interface method elliptic interface problem Cartesian grid method level set representation singular sources

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AN EMBEDDED BOUNDARY METHOD FOR elliptic AND PARABOLIC problemS WITH interfaceS AND APPLICATION TO MULTI-MATERIAL SYSTEMS WITH PHASE TRANSITIONS

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Acta Mathematica Scientia 2010年第2期30卷 499-521页

作者： Shuqiang Wang Roman Samulyak Tongfei Guo Department of Applied Mathematics and Statistics Stony Brook University Stony Brook NY 11794 USA Computational Science Center Brookhaven National Laboratory Upton NY 11973 USA

The embedded boundary method for solving elliptic and parabolic problems in geometrically complex domains using Cartesian meshes by Johansen and Colella （1998, J. Comput. Phys. 147, 60） has been extended for elliptic and parabolic problems with interior boundaries or interfaces of discontinuities of material properties or solutions. Second order accuracy is achieved in space and time for both stationary and moving interface problems. The method is conservative for elliptic and parabolic problems with fixed interfaces. Based on this method, a front tracking algorithm for the Stefan problem has been developed. The accuracy of the method is measured through comparison with exact solution to a two-dimensional Stefan problem. The algorithm has been used for the study of melting and solidification problems.

关键词： embedded boundary method elliptic interface problem front tracking Ste-fan problem

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LOCAL DISCONTINUOUS GALERKIN METHOD FOR elliptic interface problemS

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Acta Mathematica Scientia 2017年第5期37卷 1519-1535页

作者：张志娟蔚喜军常延贞 Department of Mathematics Nanchang University Laboratory of Computational Physics Institute of Applied Physics and Computational Mathematics Department of Mathematics Beijing University of Chemical Technology

In this paper,the minimal dissipation local discontinuous Galerkin method is studied to solve the elliptic interface problems in two-dimensional domains.The interface may be arbitrary smooth curves.It is shown that the error estimates in L;-norm for the solution and the flux are O（h;|log h|）and O(h|log h|;),respectively.In numerical experiments,the successive substitution iterative methods are used to solve the LDG schemes.Numerical results verify the efficiency and accuracy of the method.

关键词： elliptic interface problem minimal dissipation local discontinuous Galerkin method error estimates

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An Augmented Lagrangian Uzawa IterativeMethod for Solving Double Saddle-Point Systems with Semidefinite(2,2)Block and its Application to DLM/FDMethod for elliptic interface problems

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Communications in Computational Physics 2021年第6期30卷 124-143页

作者： Cheng Wang Pengtao Sun School of Mathematical Sciences Tongji UniversityShanghai 200092China Department of Mathematical Sciences University of Nevada Las Vegas4505Maryland ParkwayLas VegasNV 89154USA

.In this paper,an augmented Lagrangian Uzawa iterative method is developed and analyzed for solving a class of double saddle-point systems with semidefinite(2,2)block.Convergence of the iterativemethod is proved under the assumption that the double saddle-point problem exists a unique solution.An application of the iterative method to the double saddle-point systems arising from the distributed Lagrange multiplier/fictitious domain(DLM/FD)finite element method for solving elliptic interface problems is also presented,in which the existence and uniqueness of the double saddle-point system is guaranteed by the analysis of the DLM/FD finite element method.Numerical experiments are conducted to validate the theoretical results and to study the performance of the proposed iterative method.

关键词： Double saddle-point problem augmented Lagrangian Uzawa method elliptic interface problem distributed Lagrange multiplier/fictitious domain(DLM/FD)method

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An Edge-Based Anisotropic Mesh Refinement Algorithm and its Application to interface problems

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Communications in Computational Physics 2010年第8期8卷 511-540页

作者： Duan Wang Ruo Li Ningning Yan LMAM&School of Mathematical Sciences Peking University100871BeijingChina CAPT LMAM&School of Mathematical SciencesPeking University100871BeijingChina LSEC Institute of Systems ScienceAcademy of Mathematics and Systems ScienceChinese Academy of Sciences100080BeijingChina

Based on an error estimate in terms of element edge vectors on arbitrary unstructured simplex meshes,we propose a new edge-based anisotropic mesh refinement algorithm.As the mesh adaptation indicator,the error estimate involves only the gradient of error rather than higher order derivatives.The preferred refinement edge is chosen to reduce the maximal term in the error estimate.The algorithm is implemented in both two-and three-dimensional cases,and applied to the singular function interpolation and the elliptic interface problem.The numerical results demonstrate that the convergence order obtained by using the proposed anisotropic mesh refinement algorithm can be higher than that given by the isotropic one.

关键词： Adaptive finite element method anisotropic mesh refinement elliptic interface problem non-homogeneous jump a posteriori error estimate

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A Parallel Second Order Cartesian Method for elliptic interface problems

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Communications in Computational Physics 2012年第10期12卷 1562-1587页

作者： Marco Cisternino Lisl Weynans Dipartimento di Ingegneria Aeronautica e Spaziale Politecnico di Torino C.so Duca degli Abruzzi 2410129 TorinoItaly Univ.Bordeaux IMBUMR 5251F-33400 TalenceFrance.CNRSIMBUMR 5251F-33400 TalenceFrance.INRIAF-33400 TalenceFrance

We present a parallel Cartesianmethod to solve elliptic problems with complex immersed interfaces.This method is based on a finite-difference scheme and is second-order accurate in the whole domain.The originality of the method lies in the use of additional unknowns located on the interface,allowing to express straightforwardly the interface transmission conditions.We describe the method and the details of its parallelization performed with the PETSc library.Then we present numerical validations in two dimensions,assorted with comparisons to other related methods,and a numerical study of the parallelized method.

关键词： elliptic interface problem Cartesian method second-order scheme interface unknowns

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