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LOCAL DISCONTINUOUS GALERKIN METHOD FOR ELLIPTIC INTERFACE PROBLEMS 认领 引用
1
作者 张志娟 蔚喜军 常延贞 《Acta Mathematica Scientia》 SCIE CSCD 2017年第5期1519-1535,共17页
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 th... 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. 展开更多
关键词 elliptic interface problem minimal dissipation local discontinuous Galerkin method error estimates
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A CELL-CENTERED MULTIGRID SOLVER FOR THE FINITE VOLUME DISCRETIZATION OF ANISOTROPIC ELLIPTIC INTERFACE PROBLEMS ON IRREGULAR DOMAINS 认领 引用
2
作者 Kejia Pan Xiaoxin Wu +1 位作者 Hongling Hu Zhilin Li 《Journal of Computational Mathematics》 SCIE CSCD 2025年第1期18-42,共25页
The aim of this paper is to develop a fast multigrid solver for interpolation-free finite volume (FV) discretization of anisotropic elliptic interface problems on general bounded domains that can be described as a uni... The aim of this paper is to develop a fast multigrid solver for interpolation-free finite volume (FV) discretization of anisotropic elliptic interface problems on general bounded domains that can be described as a union of blocks. We assume that the curved interface falls exactly on the boundaries of blocks. The transfinite interpolation technique is applied to generate block-wise distorted quadrilateral meshes, which can resolve the interface with fine geometric details. By an extensive study of the harmonic average point method, an interpolation-free nine-point FV scheme is then derived on such multi-block grids for anisotropic elliptic interface problems with non-homogeneous jump conditions. Moreover, for the resulting linear algebraic systems from cell-centered FV discretization, a high-order prolongation operator based fast cascadic multigrid solver is developed and shown to be robust with respect to both the problem size and the jump of the diffusion coefficients. Various non-trivial examples including four interface problems and an elliptic problem in complex domain without interface, all with tens of millions of unknowns, are provided to show that the proposed multigrid solver is dozens of times faster than the classical algebraic multigrid method as implemented in the code AMG1R5 by Stüben. 展开更多
关键词 Elliptic interface problem Discontinuous coefficients Anisotropic coefficients Cascadic multigrid method Richardson extrapolation
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Dirichlet-Neumann Learning Algorithm for Solving Elliptic Interface Problems 认领 引用
3
作者 Qi Sun Xuejun Xu Haotian Yi 《Communications in Computational Physics》 SCIE CSCD 2025年第6期248-284,共37页
Non-overlapping domain decomposition methods are well-suited for addressing interface problems across various disciplines,where traditional numerical simulations often require the use of interface-fitted meshes or tec... Non-overlapping domain decomposition methods are well-suited for addressing interface problems across various disciplines,where traditional numerical simulations often require the use of interface-fitted meshes or technically designed basis functions.To remove the burden of mesh generation and to effectively tackle with the flux transmission condition,a novel mesh-free scheme,i.e.,the Dirichlet-Neumann learning algorithm,is studied in this work for solving the benchmark elliptic interface problems with high-contrast coefficients and irregular interfaces.By resorting to the variational principle,we carry out a rigorous error analysis to evaluate the discrepancy caused by the boundary penalty treatment for each decomposed subproblem,which paves the way for realizing the Dirichlet-Neumann algorithm using neural network extension operators.Through experimental validation on a series of testing problems in two and three dimensions,our methods demonstrate superior performance over other alternatives even in scenarios with inaccurate flux predictions at the interface. 展开更多
关键词 Elliptic interface problem high-contrast coefficient compensated deep Ritz method artificial neural networks deep learning
Convergence of Physics-Informed Neural Networks Applied to Linear Second-Order Elliptic Interface Problems 认领 引用 被引量：5
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作者 Sidi Wu Aiqing Zhu +1 位作者 Yifa Tang Benzhuo Lu 《Communications in Computational Physics》 SCIE 2023年第2期596-627,共32页
With the remarkable empirical success of neural networks across diverse scientific disciplines,rigorous error and convergence analysis are also being developed and enriched.However,there has been little theoretical wo... With the remarkable empirical success of neural networks across diverse scientific disciplines,rigorous error and convergence analysis are also being developed and enriched.However,there has been little theoretical work focusing on neural networks in solving interface problems.In this paper,we perform a convergence analysis of physics-informed neural networks(PINNs)for solving second-order elliptic interface problems.Specifically,we consider PINNs with domain decomposition technologies and introduce gradient-enhanced strategies on the interfaces to deal with boundary and interface jump conditions.It is shown that the neural network sequence obtained by minimizing a Lipschitz regularized loss function converges to the unique solution to the interface problem in H2 as the number of samples increases.Numerical experiments are provided to demonstrate our theoretical analysis. 展开更多
关键词 Elliptic interface problems generalization errors convergence analysis neural networks.
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 认领 引用 被引量：2
5
作者 Cheng Wang Pengtao Sun 《Communications in Computational Physics》 SCIE 2021年第6期124-143,共20页
.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... .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. 展开更多
关键词 Double saddle-point problem augmented Lagrangian Uzawa method elliptic interface problem distributed Lagrange multiplier/fictitious domain(DLM/FD)method
A FINITE DIFFERENCE METHOD FOR TWO DIMENSIONAL ELLIPTIC INTERFACE PROBLEMS WITH IMPERFECT CONTACT 认领 引用
6
作者 Fujun Cao Dongfang Yuan +1 位作者 Dongxu Jia Guangwei Yuan 《Journal of Computational Mathematics》 SCIE CSCD 2024年第5期1328-1355,共28页
In this paper two dimensional elliptic interface problem with imperfect contact is considered,which is featured by the implicit jump condition imposed on the imperfect contact interface,and the jumping quantity of the... In this paper two dimensional elliptic interface problem with imperfect contact is considered,which is featured by the implicit jump condition imposed on the imperfect contact interface,and the jumping quantity of the unknown is related to the flux across the interface.A finite difference method is constructed for the 2D elliptic interface problems with straight and curve interface shapes.Then,the stability and convergence analysis are given for the constructed scheme.Further,in particular case,it is proved to be monotone.Numerical examples for elliptic interface problems with straight and curve interface shapes are tested to verify the performance of the scheme.The numerical results demonstrate that it obtains approximately second-order accuracy for elliptic interface equations with implicit jump condition. 展开更多
关键词 Finite difference method Elliptic interface problem Imperfect contact
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An Efficient Neural-Network and Finite-Difference Hybrid Method for Elliptic Interface Problems with Applications 认领 引用 被引量：3
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作者 Wei-Fan Hu Te-Sheng Lin +1 位作者 Yu-Hau Tseng Ming-Chih Lai 《Communications in Computational Physics》 SCIE 2023年第4期1090-1105,共16页
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 lo... 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. 展开更多
关键词 Neural networks sharp interface method fast direct solver elliptic interface problem Stokes equations
A Parallel Second Order Cartesian Method for Elliptic Interface Problems 认领 引用 被引量：2
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作者 Marco Cisternino Lisl Weynans 《Communications in Computational Physics》 SCIE 2012年第10期1562-1587,共26页
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 ... 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. 展开更多
关键词 Elliptic interface problem Cartesian method second-order scheme interface unknowns
AN EMBEDDED BOUNDARY METHOD FOR ELLIPTIC AND PARABOLIC PROBLEMS WITH INTERFACES AND APPLICATION TO MULTI-MATERIAL SYSTEMS WITH PHASE TRANSITIONS 认领 引用 被引量：1
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作者 Shuqiang Wang Roman Samulyak Tongfei Guo 《Acta Mathematica Scientia》 SCIE 2010年第2期499-521,共23页
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 ellipti... 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. 展开更多
关键词 embedded boundary method elliptic interface problem front tracking Ste-fan problem
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An Edge-Based Anisotropic Mesh Refinement Algorithm and its Application to Interface Problems 认领 引用 被引量：1
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作者 Duan Wang Ruo Li Ningning Yan 《Communications in Computational Physics》 SCIE 2010年第8期511-540,共30页
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 estimat... 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. 展开更多
关键词 Adaptive finite element method anisotropic mesh refinement elliptic interface problem non-homogeneous jump a posteriori error estimate
Mathematical model and numerical method for spontaneous potential log in heterogeneous formations 认领 引用 被引量：1
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作者 潘克家 谭永基 胡宏伶 《Applied Mathematics and Mechanics(English Edition)》 SCIE EI 2009年第2期209-219,共11页
This paper introduces a new spontaneous potential log model for the case in which formation resistivity is not piecewise constant. The spontaneous potential satisfies an elliptic boundary value problem with jump condi... This paper introduces a new spontaneous potential log model for the case in which formation resistivity is not piecewise constant. The spontaneous potential satisfies an elliptic boundary value problem with jump conditions on the interfaces. It has beer/ shown that the elliptic interface problem has a unique weak solution. Furthermore, a jump condition capturing finite difference scheme is proposed and applied to solve such elliptic problems. Numerical results show validity and effectiveness of the proposed method. 展开更多
关键词 spontaneous potential log elliptic interface problems mathematical model,numerical simulation
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A Fourth-Order Kernel-Free Boundary Integral Method for Interface Problems 认领 引用 被引量：1
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作者 Yaning Xie Shuwang Li Wenjun Ying 《Communications in Computational Physics》 SCIE 2023年第3期764-794,共31页
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 explic... 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. 展开更多
关键词 Elliptic interface problem compact scheme finite difference method Cartesian grid method kernel-free boundary integral method boundary integral equation
An Adaptive Mesh Refinement Strategy for Immersed Boundary/Interface Methods 认领 引用 被引量：3
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作者 Zhilin Li Peng Song 《Communications in Computational Physics》 SCIE 2012年第7期515-527,共13页
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 repre... 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. 展开更多
关键词 Adaptive mesh refinement immersed boundary method immersed interface method elliptic interface problem Cartesian grid method level set representation singular sources
Unfitted Spectral Element Method for Interfacial Models 认领 引用
14
作者 Nicolas Gonzalez Hailong Guo Xu Yang 《Communications in Computational Physics》 SCIE CSCD 2025年第9期987-1016,共30页
In this paper,we propose an unfitted spectral element method for solving elliptic interface and corresponding eigenvalue problems.The novelty of the proposed method lies in its combination of the spectral accuracy of ... In this paper,we propose an unfitted spectral element method for solving elliptic interface and corresponding eigenvalue problems.The novelty of the proposed method lies in its combination of the spectral accuracy of the spectral element method and the flexibility of the unfitted Nitsche’s method.We also use tailored ghost penalty terms to enhance its robustness.We establish optimal hp convergence rates for both elliptic interface problems and interface eigenvalue problems.Additionally,we demonstrate spectral accuracy for model problems in terms of polynomial degree. 展开更多
关键词 Elliptic interface problem interface eigenvalue problem unfitted Nitsche’s method hp estimate ghost penalty
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