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

Kelin Xia1, Guo-Wei Wei2

  • 1Department of Mathematics, Michigan State University, East Lansing, MI 48824, USA.

Computers & Mathematics with Applications (Oxford, England : 1987)
|October 14, 2014
PubMed
Summary

We developed a 3D Galerkin formulation of the matched interface and boundary (MIB) method to solve elliptic partial differential equations (PDEs) with discontinuous coefficients. This new method achieves near second-order accuracy for complex interfaces, including realistic protein surfaces.

Keywords:
Elliptic interface problemsGalerkin formulationLow regularity solutionsMatched interface and boundaryNonsmooth interfacesProteins and multiprotein complex

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Area of Science:

  • Numerical Analysis
  • Computational Science
  • Partial Differential Equations

Background:

  • Solving elliptic partial differential equations (PDEs) with discontinuous coefficients, known as elliptic interface problems, is challenging.
  • Conventional finite element methods (FEMs) require complex mesh generation, especially for intricate geometries.
  • Enforcing interface jump conditions accurately is crucial for reliable numerical solutions.

Purpose of the Study:

  • To develop a novel three-dimensional (3D) Galerkin formulation of the matched interface and boundary (MIB) method.
  • To efficiently solve elliptic interface problems without the need for body-fitted meshes.
  • To achieve high-order accuracy for complex and realistic interface geometries.

Main Methods:

  • A three-dimensional (3D) Galerkin formulation of the matched interface and boundary (MIB) method is developed.
  • Two sets of overlapping elements are used on extended subdomains, with fictitious solutions defined on the overlap.
  • Interface jump conditions are enforced by determining coefficients of polynomial basis functions on overlapping elements.
  • Cartesian meshes are utilized, avoiding complex mesh generation inherent in traditional FEMs.
  • The method is implemented using rectangular prism, five-tetrahedron, and six-tetrahedron elements compatible with Cartesian grids.

Main Results:

  • The proposed 3D MIB Galerkin method demonstrates near second-order accuracy for elliptic interface problems.
  • Accuracy, stability, and robustness are validated across analytically defined, protein surface, and multiprotein complex interfaces.
  • The method achieves near second-order convergence for the Poisson equation with realistic protein surfaces, a first for FEMs.
  • It provides the first known near second-order accurate method for C1 or H2 continuous solutions with Lipschitz continuous interfaces in 3D.

Conclusions:

  • The 3D MIB Galerkin method offers an efficient and accurate approach for solving elliptic interface problems.
  • The utilization of Cartesian meshes significantly simplifies the numerical workflow.
  • This method sets a new benchmark for accuracy in solving PDEs with complex interfaces, particularly in biological applications.