Abstract

We present a finite difference method for solving two-dimensional Poisson problems where the solution, diffusion coefficient, source term, and flux are discontinuous across a one-dimensional interface. The interface is irregular, and an implicit, level set representation of the interface is assumed, as well as a Cartesian grid that is not fitted to the interface. The algorithm is based on a scheme presented by Towers (Contemp. Math., no. 526, 2010, pp. 359–389) for interface problems which captures the jump conditions via singular source terms. We adapt that method to deal with discontinuous coefficient problems by employing an iterative process. The advantages of this method are conceptual simplicity and ease of implementation. The system of equations that results at each iteration can be solved using a FFT-based fast Poisson solver. The algorithm is a sharp interface method — jumps in the solution are captured without smearing. Numerical examples indicate second-order accuracy (in the $L^{\infty }$ norm) for the solution, and a convergence rate for the gradient that is second order, or nearly so.

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