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Abstract
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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
norm) for the solution, and a convergence rate for the gradient that is second order,
or nearly so.
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Keywords
Poisson problem, discontinuous coefficient, level set,
irregular interface
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Mathematical Subject Classification
Primary: 35J05, 65M06, 65N06
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Milestones
Received: 26 February 2023
Revised: 25 July 2023
Accepted: 30 July 2023
Published: 21 December 2023
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