Simplified Immersed Interface Methods for Elliptic Interface Problems with Straight Interfaces

In this article, we propose simplified immersed interface methods for elliptic partial/ordinary differential equations with discontinuous coefficients across interfaces that are few isolated points in 1D, and straight lines in 2D. For one-dimensional problems or two-dimensional problems with circular interfaces, we propose a conservative second-order finite difference scheme whose coefficient matrix is symmetric and definite. For two-dimensional problems with straight interfaces, we first propose a conservative first-order finite difference scheme, then use the Richardson extrapolation technique to get a second-order method. In both cases, the finite difference coefficients are almost the same as those for regular problems. Error analysis is given along with numerical example. (C) 2010 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 28: 188-203, 2012