Interior penalty bilinear IFE discontinuous Galerkin methods for elliptic equations with discontinuous coefficient

He, Xiaoming; Lin, Tao; Lin, Yanping

doi:10.1007/s11424-010-0141-z

Cited by 49 publications

(22 citation statements)

References 69 publications

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“…Babuška constructed one of the first FEMs for elliptic interface problems in 1970 [61]. Since then, elliptic interface problems have attracted extensive effort in the FEM community as well [62, 63, 64, 65]. Recently, non-conforming FEM [63] and discontinuous Galerkin (DG) FEM [66, 64] have been developed for elliptic equations with discontinuous coefficients.…”

Section: Introductionmentioning

confidence: 99%

“…Since then, elliptic interface problems have attracted extensive effort in the FEM community as well [62, 63, 64, 65]. Recently, non-conforming FEM [63] and discontinuous Galerkin (DG) FEM [66, 64] have been developed for elliptic equations with discontinuous coefficients. According to the topological relation between discrete elements and the interface, one can classify FEM based interface methods into two types: interface-fitted FEMs and immersed FEMs.…”

Section: Introductionmentioning

confidence: 99%

“…According to the topological relation between discrete elements and the interface, one can classify FEM based interface methods into two types: interface-fitted FEMs and immersed FEMs. In the interface-fitted FEMs, or body-fitted FEMs, element meshes are aligned with the interface [67, 68, 63, 66, 64]. Two major factors that determine the performance of interface-fitted FEMs are respectively the Galerkin formulation and the quality of element meshes near the interface.…”

Section: Introductionmentioning

confidence: 99%

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MIB Galerkin method for elliptic interface problems

Xia

Zhan

Wei

2014

Journal of Computational and Applied Mathematics

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Summary Material interfaces are omnipresent in the real-world structures and devices. Mathematical modeling of material interfaces often leads to elliptic partial differential equations (PDEs) with discontinuous coefficients and singular sources, which are commonly called elliptic interface problems. The development of high-order numerical schemes for elliptic interface problems has become a well defined field in applied and computational mathematics and attracted much attention in the past decades. Despite of significant advances, challenges remain in the construction of high-order schemes for nonsmooth interfaces, i.e., interfaces with geometric singularities, such as tips, cusps and sharp edges. The challenge of geometric singularities is amplified when they are associated with low solution regularities, e.g., tip-geometry effects in many fields. The present work introduces a matched interface and boundary (MIB) Galerkin method for solving two-dimensional (2D) elliptic PDEs with complex interfaces, geometric singularities and low solution regularities. The Cartesian grid based triangular elements are employed to avoid the time consuming mesh generation procedure. Consequently, the interface cuts through elements. To ensure the continuity of classic basis functions across the interface, two sets of overlapping elements, called MIB elements, are defined near the interface. As a result, differentiation can be computed near the interface as if there is no interface. Interpolation functions are constructed on MIB element spaces to smoothly extend function values across the interface. A set of lowest order interface jump conditions is enforced on the interface, which in turn, determines the interpolation functions. The performance of the proposed MIB Galerkin finite element method is validated by numerical experiments with a wide range of interface geometries, geometric singularities, low regularity solutions and grid resolutions. Extensive numerical studies confirm the designed second order convergence of the MIB Galerkin method in the L∞ and L2 errors. Some of the best results are obtained in the present work when the interface is C1 or Lipschitz continuous and the solution is C2 continuous.

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Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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MIB Galerkin method for elliptic interface problems

Xia

Zhan

Wei

2014

Journal of Computational and Applied Mathematics

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“…In the past forty years, a variety of new FEM approaches has been developed for elliptic interface problems [18, 62, 31, 11, 37]. With extra number of degrees of freedom, Non-conforming FEM [62] and discontinuous Galerkin (DG) FEM [16, 31] appear naturally suitable for solving elliptic equations with discontinuous coefficients. Recently, a novel Galerkin formulation, the Wang-Ye Galerkin FEM, has been proposed for solving PDEs [68].…”

Section: Introductionmentioning

confidence: 99%

“…According to the topological relation between elements and the interface, FEM based elliptic interface methods can be categorized into two major classes: interface-fitted FEMs and immersed FEMs. Interface-fitted FEMs, or body-fitted FEMs, allocate unstructured element meshes to align with the interface [8, 9, 62, 16, 31]. This is a natural approach to complex interfaces and irregular boundaries as standard h-refinement techniques, such as a priori and/or a posteriori error estimation based local mesh refinements, can be employed to improve the accuracy.…”

Section: Introductionmentioning

confidence: 99%

A Galerkin formulation of the MIB method for three dimensional elliptic interface problems

Xia

Wei

2014

Computers & Mathematics with Applications

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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 C1 continuous or H2 continuous solutions associated with a Lipschitz continuous interface in a 3D setting.

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A selective immersed discontinuous Galerkin method for elliptic interface problems

Lin

2013

Math. Meth. Appl. Sci.

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This article proposes a selective immersed discontinuous Galerkin method based on bilinear immersed finite elements (IFE) for solving second‐order elliptic interface problems. This method applies the discontinuous Galerkin formulation wherever selected, such as those elements around an interface or a singular source, but the regular Galerkin formulation everywhere else. A selective bilinear IFE space is constructed and applied to the selective immersed discontinuous Galerkin method based on either the symmetric or nonsymmetric interior penalty discontinuous Galerkin formulation. The new method can solve an interface problem by a rectangular mesh with local mesh refinement independent of the interface even if its geometry is nontrivial. Meanwhile, if desired, its computational cost can be maintained very close to that of the standard Galerkin IFE method. It is shown that the selective bilinear IFE space has the optimal approximation capability expected from piecewise bilinear polynomials. Numerical examples are provided to demonstrate features of this method, including the effectiveness of local mesh refinement around the interface and the sensitivity to the penalty parameters. Copyright © 2013 John Wiley & Sons, Ltd.

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Interior penalty bilinear IFE discontinuous Galerkin methods for elliptic equations with discontinuous coefficient

Cited by 49 publications

References 69 publications

MIB Galerkin method for elliptic interface problems

MIB Galerkin method for elliptic interface problems

A Galerkin formulation of the MIB method for three dimensional elliptic interface problems

A selective immersed discontinuous Galerkin method for elliptic interface problems

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