Adaptive Mesh Refinement and Superconvergence for Two-Dimensional Interface Problems

Adaptive mesh refinement and the Borgers algorithm are combined to generate a body-fitted mesh which can resolve the interface with fine geometric details. Standard linear finite element method based on such body-fitted meshes is applied to the elliptic interface problem and proven to be superclose to the linear interpolation of the exact solution. Based on this superconvergence result, a maximal norm error estimate of order one and half is obtained without using the discrete maximum principle. The data structure and meshing algorithms, including local refinement and coarsening, are very simple. In particular, no tree structure is needed. An efficient solver for solving the resulting linear algebraic systems is also developed and shown be robust with respect to both the problem size and the jump of the diffusion coefficients.

[1]  Jean-Luc Guermond,et al.  A grid-alignment finite element technique for incompressible multicomponent flows , 2008, J. Comput. Phys..

[2]  Pascal J. Frey,et al.  Fast Adaptive Quadtree Mesh Generation , 1998, IMR.

[3]  Ronald Fedkiw,et al.  The immersed interface method. Numerical solutions of PDEs involving interfaces and irregular domains , 2007, Math. Comput..

[4]  Patrick M. Knupp,et al.  Algebraic Mesh Quality Metrics , 2001, SIAM J. Sci. Comput..

[5]  Hui Xie.  Finite element methods for interface problems with locally modified triangulations , 2009 .

[6]  Zhiming Chen,et al.  The adaptive immersed interface finite element method for elliptic and Maxwell interface problems , 2009, J. Comput. Phys..

[7]  Bo Li,et al.  Immersed-Interface Finite-Element Methods for Elliptic Interface Problems with Nonhomogeneous Jump Conditions , 2007, SIAM J. Numer. Anal..

[8]  I-Liang Chern,et al.  A coupling interface method for elliptic interface problems , 2007, J. Comput. Phys..

[9]  Christoph Pflaum,et al.  Semi-Unstructured Grids , 2001, Computing.

[10]  Chen-Song Zhang,et al.  A COARSENING ALGORITHM ON ADAPTIVE GRIDS BY NEWEST VERTEX BISECTION AND ITS APPLICATIONS , 2010 .

[11]  Jinchao Xu,et al.  Asymptotically Exact A Posteriori Error Estimators, Part I: Grids with Superconvergence , 2003, SIAM J. Numer. Anal..

[12]  Per-Olof Persson,et al.  A Simple Mesh Generator in MATLAB , 2004, SIAM Rev..

[13]  Jinchao Xu,et al.  Robust and accurate algorithms for solving anisotropic singularities , 2005 .

[14]  Ja. A. Roitberg,et al.  a Theorem on Homeomorphisms for Elliptic Systems and its Applications , 1969 .

[15]  Christoph Börgers,et al.  A triangulation algorithm for fast elliptic solvers based on domain imbedding , 1990 .

[16]  Long Chen,et al.  Mesh Smoothing Schemes Based on Optimal Delaunay Triangulations , 2004, IMR.

[17]  James H. Bramble,et al.  A finite element method for interface problems in domains with smooth boundaries and interfaces , 1996, Adv. Comput. Math..

[18]  Eftychios Sifakis,et al.  ' s personal copy A second order virtual node method for elliptic problems with interfaces and irregular domains , 2010 .

[19]  Tao Lin,et al.  New Cartesian grid methods for interface problems using the finite element formulation , 2003, Numerische Mathematik.

[20]  Mark Yerry,et al.  A Modified Quadtree Approach To Finite Element Mesh Generation , 1983, IEEE Computer Graphics and Applications.

[21]  Jonathan Richard Shewchuk,et al.  Delaunay refinement algorithms for triangular mesh generation , 2002, Comput. Geom..

[22]  Michael Holst,et al.  Efficient mesh optimization schemes based on Optimal Delaunay Triangulations , 2011 .

[23]  J. Cooper.  SINGULAR INTEGRALS AND DIFFERENTIABILITY PROPERTIES OF FUNCTIONS , 1973 .

[24]  Kazufumi Ito,et al.  Maximum Principle Preserving Schemes for Interface Problems with Discontinuous Coefficients , 2001, SIAM J. Sci. Comput..

[25]  I. Babuska,et al.  ON THE ANGLE CONDITION IN THE FINITE ELEMENT METHOD , 1976 .

[26]  Mohamed S. Ebeida,et al.  A new fast hybrid adaptive grid generation technique for arbitrary two‐dimensional domains , 2010 .

[27]  Deborah Greaves,et al.  Quadtree grid generation: Information handling, boundary fitting and CFD applications , 1996 .

[28]  E. Stein.  Singular Integrals and Di?erentiability Properties of Functions , 1971 .

[29]  Hui Xie,et al.  A FINITE ELEMENT METHOD FOR ELASTICITY INTERFACE PROBLEMS WITH LOCALLY MODIFIED TRIANGULATIONS. , 2011, International journal of numerical analysis and modeling.

[30]  Zhilin Li.  A Fast Iterative Algorithm for Elliptic Interface Problems , 1998 .

[31]  J. Zou,et al.  Finite element methods and their convergence for elliptic and parabolic interface problems , 1998 .

[32]  I. Fried.  Condition of finite element matrices generated from nonuniform meshes. , 1972 .