A Fast Iterative Algorithm for Elliptic Interface Problems

A fast, second-order accurate iterative method is proposed for the elliptic equation \[ \grad\cdot(\beta(x,y) \grad u) =f(x,y) \] in a rectangular region $\Omega$ in two-space dimensions. We assume that there is an irregular interface across which the coefficient $\beta$, the solution u and its derivatives, and/or the source term f may have jumps. We are especially interested in the cases where the coefficients $\beta$ are piecewise constant and the jump in $\beta$ is large. The interface may or may not align with an underlying Cartesian grid. The idea in our approach is to precondition the differential equation before applying the immersed interface method proposed by LeVeque and Li [ SIAM J. Numer. Anal., 4 (1994), pp. 1019--1044]. In order to take advantage of fast Poisson solvers on a rectangular region, an intermediate unknown function, the jump in the normal derivative across the interface, is introduced. Our discretization is equivalent to using a second-order difference scheme for a corresponding Poisson equation in the region, and a second-order discretization for a Neumann-like interface condition. Thus second-order accuracy is guaranteed. A GMRES iteration is employed to solve the Schur complement system derived from the discretization. A new weighted least squares method is also proposed to approximate interface quantities from a grid function. Numerical experiments are provided and analyzed. The number of iterations in solving the Schur complement system appears to be independent of both the jump in the coefficient and the mesh size.