Regular Article
A Cartesian Grid Embedded Boundary Method for Poisson's Equation on Irregular Domains

https://doi.org/10.1006/jcph.1998.5965Get rights and content

Abstract

We present a numerical method for solving Poisson's equation, with variable coefficients and Dirichlet boundary conditions, on two-dimensional regions. The approach uses a finite-volume discretization, which embeds the domain in a regular Cartesian grid. We treat the solution as a cell-centered quantity, even when those centers are outside the domain. Cells that contain a portion of the domain boundary use conservative differencing of second-order accurate fluxes on each cell volume. The calculation of the boundary flux ensures that the conditioning of the matrix is relatively unaffected by small cell volumes. This allows us to use multigrid iterations with a simple point relaxation strategy. We have combined this with an adaptive mesh refinement (AMR) procedure. We provide evidence that the algorithm is second-order accurate on various exact solutions and compare the adaptive and nonadaptive calculations.

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    Research supported at U.C. Berkeley by the U.S. Department of Energy Mathematical, Computing, and Information Sciences Division, Grants DE-FG03-94ER25205, DE-FG03-92ER25140; by the U.S. Air Force Office of Scientific Research, AASERT Grant F49620-93-1-0521; and at the Lawrence Berkeley National Laboratory by the U.S. Department of Energy Mathematical, Computing, and Information Sciences Division Contract DE-AC03-76SF00098. The U.S. Government's right to retain a nonexclusive royalty-free license in and to the copyright covering this paper, for governmental purposes, is acknowledged.

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    Corresponding author. E-mail: [email protected].

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