High order matched interface and boundary methods for the Helmholtz equation in media with arbitrarily curved interfaces

Zhao, Shan

doi:10.1016/j.jcp.2009.12.034

Cited by 72 publications

(41 citation statements)

References 41 publications

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“…However, most of these methods typically require tools not frequently available in standard finite element and finite difference software packages. Examples of such approaches include the extended and composite finite element methods (e.g., [31,12,23,13,32,55,7,4]), immersed interface methods (e.g., [40,43,60,44,65]), virtual node methods with embedded boundary conditions (e.g., [3,73,34]), matched interface and boundary methods (e.g., [71,68,69,67,72]), modified finite volume/embedded boundary/cut-cell methods/ghost-fluid methods (e.g., [27,36,19,25,26,35,47,70,48,37,46,64,49,9,10,52,53,33,63]). In another approach, known as the fictitious domain method (e.g., [28,29,56,45]), the original system is either augmented with equations for Lagrange multipliers to enforce the boundary conditions, or the penalty method is used to enforce the boundary condi-tions weakly.…”

Section: Introductionmentioning

confidence: 99%

Analysis of the diffuse-domain method for solving PDEs in complex geometries

Lervåg

Lowengrub

2015

Communications in Mathematical Sciences

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Abstract. In recent work, Li et al. (Comm. Math. Sci., 7:81-107, 2009) developed a diffusedomain method (DDM) for solving partial differential equations in complex, dynamic geometries with Dirichlet, Neumann, and Robin boundary conditions. The diffuse-domain method uses an implicit representation of the geometry where the sharp boundary is replaced by a diffuse layer with thickness that is typically proportional to the minimum grid size. The original equations are reformulated on a larger regular domain and the boundary conditions are incorporated via singular source terms. The resulting equations can be solved with standard finite difference and finite element software packages.Here, we present a matched asymptotic analysis of general diffuse-domain methods for Neumann and Robin boundary conditions. Our analysis shows that for certain choices of the boundary condition approximations, the DDM is second-order accurate in . However, for other choices the DDM is only first-order accurate. This helps to explain why the choice of boundary-condition approximation is important for rapid global convergence and high accuracy. Our analysis also suggests correction terms that may be added to yield more accurate diffuse-domain methods. Simple modifications of first-order boundary condition approximations are proposed to achieve asymptotically second-order accurate schemes. Our analytic results are confirmed numerically in the L 2 and L ∞ norms for selected test problems.

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Section: Introductionmentioning

confidence: 99%

Analysis of the diffuse-domain method for solving PDEs in complex geometries

Lervåg

Lowengrub

2015

Communications in Mathematical Sciences

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“…However, for elliptic PDEs with discontinuous coefficients and singular sources, i.e., elliptic interface problems, ordinary numerical methods do not work, and special interface schemes are required to deal with interface jump conditions. Elliptic interface problems have a variety of applications in many scientific and engineering disciplines, including fluid dynamics [1, 2, 3, 4, 5, 6, 7, 8], electromagnetic wave propagation [9, 10, 11, 12, 13, 14], materials science, [15, 16] and biological systems [17, 18, 19, 20, 21]. The past few decades have witnessed intensive research activity in interface problems [22, 23, 24, 25, 26, 27, 16, 28, 29, 30, 31, 32, 12, 6, 7, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43].…”

Section: Introductionmentioning

confidence: 99%

“…It takes much effort to achieve the second order convergence in solving the PBE for arbitrarily complex protein interfaces with geometric singularities and multiple material interfaces [58]. Recently, Zhao has developed impressive MIB schemes for the Helmholtz equation [14, 59]. Note that interface conditions in this problem are more complex.…”

Section: Introductionmentioning

confidence: 99%

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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“…A comparison of the GFM, IIM and MIB approaches is discussed in our earlier work [87, 86]. The development of the MIB methodology is motivated by the practical needs in scientific and engineering applications, such as optical molecular imaging [14], nano-electronic devices, [13], vibration analysis of plates [78], wave propagation [81, 80], geodynamics [85] and electrostatic potential in proteins [84, 76, 25, 12]. …”

Section: Introductionmentioning

confidence: 99%

“…Indeed, elliptic interface problems have a variety of applications in science and engineering, including fluid dynamics [19, 22, 23, 40, 57, 47, 46], electromagnetic wave propagation [28, 33, 43, 44, 82, 81], materials science, [35, 39] and biological systems [52, 84, 76, 25, 12]. The past few decades have witnessed intensive research effort in interface problems [2, 6, 7, 17, 21, 26, 39, 36, 41, 42, 44, 47, 51, 54, 63, 64, 65, 66, 58, 72, 10].…”

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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High order matched interface and boundary methods for the Helmholtz equation in media with arbitrarily curved interfaces

Cited by 72 publications

References 41 publications

Analysis of the diffuse-domain method for solving PDEs in complex geometries

Analysis of the diffuse-domain method for solving PDEs in complex geometries

MIB Galerkin method for elliptic interface problems

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

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