ON THE ACCURACY OF FINITE DIFFERENCE METHODS FOR ELLIPTIC PROBLEMS WITH INTERFACES

In problems with interfaces, the unknown or its derivatives may have jump discontinuities. Finite difference methods, including the method of A. Mayo and the immersed interface method of R. LeVeque and Z. Li, maintain accuracy by adding corrections, found from the jumps, to the difference operator at grid points near the interface and by modifying the operator if necessary. It has long been observed that the solution can be computed with uniform O.h 2 / accuracy even if the truncation error is O.h/ at the interface, while O.h 2 / in the interior. We prove this fact for a class of static interface problems of elliptic type using discrete analogues of estimates for elliptic equations. Moreover, we show that the gradient is uniformly accurate to O.h 2 log.1=h//. Various implications are discussed, including the accuracy of these methods for steady fluid flow governed by the Stokes equations. Two-fluid problems can be handled by first solving an integral equation for an unknown jump. Numerical examples are presented which confirm the analytical conclusions, although the observed error in the gradient is O.h 2 /.

[1]  P. G. Ciarlet,et al.  Basic error estimates for elliptic problems , 1991 .

[2]  Anita Mayo,et al.  Fast High Order Accurate Solution of Laplace’s Equation on Irregular Regions , 1985 .

[3]  C. Pozrikidis,et al.  Boundary Integral and Singularity Methods for Linearized Viscous Flow: The boundary integral equations , 1992 .

[4]  Anne Greenbaum,et al.  Fast Parallel Iterative Solution of Poisson's and the Biharmonic Equations on Irregular Regions , 2011, SIAM J. Sci. Comput..

[5]  B. E. Hubbard,et al.  On the formulation of finite difference analogues of the Dirichlet problem for Poisson's equation , 1962 .

[6]  Vidar Thomée,et al.  Convergence estimates for essentially positive type discrete Dirichlet problems , 1969 .

[7]  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 .

[8]  O. A. Ladyzhenskai︠a︡,et al.  Linear and quasilinear elliptic equations , 1968 .

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

[10]  Klaus Böhmer,et al.  Asymptotic expansion for the discretization error in linear elliptic boundary value problems on general regions , 1981 .

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

[12]  Zhilin Li.  AN OVERVIEW OF THE IMMERSED INTERFACE METHOD AND ITS APPLICATIONS , 2003 .

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

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

[15]  K. Uchiyama,et al.  Potential kernel for two-dimensional random walk , 1996 .

[16]  G. Akrivis.  A First Course In The Numerical Analysis Of Differential Equations [Book News & Reviews] , 1998, IEEE Computational Science and Engineering.

[17]  J. Thomas Beale,et al.  A Method for Computing Nearly Singular Integrals , 2000, SIAM J. Numer. Anal..

[18]  P. Brandimarte.  Finite Difference Methods for Partial Differential Equations , 2006 .

[19]  Xu-Dong Liu,et al.  Convergence of the ghost fluid method for elliptic equations with interfaces , 2003, Math. Comput..

[20]  Anita Mayo,et al.  The rapid evaluation of volume integrals of potential theory on general regions , 1992 .

[21]  V. Thomée,et al.  Pointwise Bounds for Discrete Green’s Functions , 1969 .

[22]  J. Zou,et al.  Some New A Priori Estimates for Second-Order Elliptic and Parabolic Interface Problems , 2002 .

[23]  Z. Jomaa,et al.  The embedded finite difference method for the Poisson equation in a domain with an irregular boundary and Dirichlet boundary conditions , 2005 .

[24]  A. Baker,et al.  A concise introduction to the theory of numbers , 1984 .

[25]  Andreas Wiegmann,et al.  The Explicit-Jump Immersed Interface Method: Finite Difference Methods for PDEs with Piecewise Smooth Solutions , 2000, SIAM J. Numer. Anal..

[26]  A. Mayo.  The Fast Solution of Poisson’s and the Biharmonic Equations on Irregular Regions , 1984 .

[27]  Zhilin Li,et al.  Convergence analysis of the immersed interface method , 1999 .

[28]  J. Zou,et al.  Finite Element Methods and Their Convergencefor Elliptic and Parabolic Interface , 1996 .

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

[30]  Anita Mayo.  A Decomposition Finite Difference Method for the Fourth Order Accurate Solution of Poisson's equation on General Regions , 1991, Int. J. High Speed Comput..

[31]  Ricardo Cortez,et al.  The Method of Regularized Stokeslets , 2001, SIAM J. Sci. Comput..

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

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

[34]  L. R. Scott,et al.  The Mathematical Theory of Finite Element Methods , 1994 .

[35]  Randall J. LeVeque,et al.  Immersed Interface Methods for Stokes Flow with Elastic Boundaries or Surface Tension , 1997, SIAM J. Sci. Comput..

[36]  R. LeVeque,et al.  Analysis of a one-dimensional model for the immersed boundary method , 1992 .