Marchuk identity-type second order difference schemes of 2-d and 3-d elliptic problems with intersected interfaces

This article presents second-order difference schemes of 2-D and 3-D elliptic problems with intersecting interfaces. The discretization is made using new Marchuk identities. It possesses the typical for the method advantages as conservatism, second-order accuracy even at low smoothness of the differential problem solution. The convergence and accuracy are discussed theoretically and experimentally. Numerical tests show the feasiblity of the schemes.

[1]  Lubin G. Vulkov,et al.  The immersed interface method for two-dimensional heat-diffusion equations with singular own sources , 2007 .

[2]  Uri M. Ascher,et al.  Finite Difference Methods for Boundary Value Problems , 1998 .

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

[4]  G. T. McAllister.  Difference methods for a nonlinear elliptic system of partial differential equations , 1966 .

[5]  P. Grisvard.  Elliptic Problems in Nonsmooth Domains , 1985 .

[6]  Houde Han.  The numerical solutions of interface problems by infinite element method , 1982 .

[7]  Mikhail Shashkov,et al.  The sensitivity and accuracy of fourth order finite-difference schemes on nonuniform grids in one dimension , 1995 .

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

[9]  D. Knees.  Regularity results for quasilinear elliptic systems of power-law growth in nonsmooth domains : boundary, transmission and crack problems , 2005 .

[10]  Richard E. Ewing,et al.  A Modified Finite Volume Approximation of Second-Order Elliptic Equations with Discontinuous Coefficients , 2001, SIAM J. Sci. Comput..

[11]  Lubin G. Vulkov,et al.  Finite Difference Approximation of an Elliptic Interface Problem with Variable Coefficients , 2004, NAA.

[12]  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 .

[13]  S. K. Godunov,et al.  Theory of difference schemes : an introduction , 1964 .

[14]  Lubin G. Vulkov,et al.  Singularly perturbed differential equations with discontinuous coefficients and concentrated factors , 2004, Appl. Math. Comput..

[15]  R. Bruce Kellogg,et al.  On the poisson equation with intersecting interfaces , 1974 .

[16]  Lubin G. Vulkov,et al.  Numerical Solution of a Reaction-Diffusion Elliptic Interface Problem with Strong Anisotropy , 2003, Computing.

[17]  Lubin G. Vulkov,et al.  High-order difference schemes based on new Marchuk integral identities for one-dimensional interface problems , 2005, J. Num. Math..