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A source term method for Poisson problems with a discontinuous diffusion coefficient

John D. Towers

Vol. 18 (2023), No. 1, 153–176
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 norm) for the solution, and a convergence rate for the gradient that is second order, or nearly so.

Keywords
Poisson problem, discontinuous coefficient, level set, irregular interface
Mathematical Subject Classification
Primary: 35J05, 65M06, 65N06
Milestones
Received: 26 February 2023
Revised: 25 July 2023
Accepted: 30 July 2023
Published: 21 December 2023
Authors
John D. Towers
Mathematics Department
MiraCosta College
Oceanside, CA
United States