An Adaptive Mesh Refinement Strategy for Immersed Boundary/Interface Methods.

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 function φ(x,y). Our adaptive mesh refinement is done within a small tube of |φ(x,y)|≤ δ with 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.

[1]  Isidore Rigoutsos,et al.  An algorithm for point clustering and grid generation , 1991, IEEE Trans. Syst. Man Cybern..

[2]  C. Peskin.  The immersed boundary method , 2002, Acta Numerica.

[3]  P. Colella,et al.  Local adaptive mesh refinement for shock hydrodynamics , 1989 .

[4]  Li Ge-wen.  Fast Tree-Based Redistancing for Level Set Computations , 2003 .

[5]  K. Ito,et al.  An augmented approach for Stokes equations with a discontinuous viscosity and singular forces , 2007 .

[6]  Kazufumi Ito,et al.  Higher-Order, Cartesian Grid Based Finite Difference Schemes for Elliptic Equations on Irregular Domains , 2005, SIAM J. Sci. Comput..

[7]  C. Peskin,et al.  A three-dimensional computational method for blood flow in the heart. II. contractile fibers , 1989 .

[8]  I-Liang Chern,et al.  New Formulations for Interface Problems in Polar Coordinates , 2003, SIAM J. Sci. Comput..

[9]  Pingwen Zhang,et al.  Frontiers and prospects of contemporary applied mathematics , 2006 .

[10]  R. LeVeque,et al.  A comparison of the extended finite element method with the immersed interface method for elliptic equations with discontinuous coefficients and singular sources , 2006 .

[11]  Kazufumi Ito,et al.  Three-dimensional elliptic solvers for interface problems and applications , 2003 .

[12]  Zhilin Li,et al.  The immersed interface method for the Navier-Stokes equations with singular forces , 2001 .

[13]  Boyce E. Griffith,et al.  An adaptive, formally second order accurate version of the immersed boundary method , 2007, J. Comput. Phys..

[14]  J. Thomas Beale,et al.  ON THE ACCURACY OF FINITE DIFFERENCE METHODS FOR ELLIPTIC PROBLEMS WITH INTERFACES , 2006 .

[15]  John K. Hunter,et al.  Autophobic spreading of drops , 1999 .

[16]  Ann S. Almgren,et al.  An adaptive level set approach for incompressible two-phase flows , 1997 .

[17]  M. Berger,et al.  An Adaptive Version of the Immersed Boundary Method , 1999 .

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

[19]  C. Peskin,et al.  A three-dimensional computational method for blood flow in the heart. 1. Immersed elastic fibers in a viscous incompressible fluid , 1989 .

[20]  Boyce E. Griffith,et al.  On the order of accuracy of the immersed boundary method: Higher order convergence rates for sufficiently smooth problems , 2005 .

[21]  Boyce E. Griffith,et al.  A Comparison of Two Adaptive Versions of the Immersed Boundary Method , 2009 .

[22]  Stanley Osher,et al.  A Hybrid Method for Moving Interface Problems with Application to the Hele-Shaw Flow , 1997 .

[23]  Charles S. Peskin,et al.  Improved Volume Conservation in the Computation of Flows with Immersed Elastic Boundaries , 1993 .

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

[25]  J. K. Hunter,et al.  Reactive autophobic spreading of drops , 2002 .

[26]  J. W. Ruge,et al.  4. Algebraic Multigrid , 1987 .

[27]  Ming-Chih Lai,et al.  ADAPTIVE MESH REFINEMENT FOR ELLIPTIC INTERFACE PROBLEMS USING THE NON-CONFORMING IMMERSED FINITE ELEMENT METHOD , 2011 .

[28]  C S Peskin,et al.  A general method for the computer simulation of biological systems interacting with fluids. , 1995, Symposia of the Society for Experimental Biology.

[29]  S. Osher,et al.  Spatially adaptive techniques for level set methods and incompressible flow , 2006 .