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Mathematics of Computation

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Precise solution of Laplace's equation

Authors: Zhixin Shi and Brian Hassard
Journal: Math. Comp. 64 (1995), 515-536
MSC: Primary 35J05; Secondary 65N15, 65T10
MathSciNet review: 1270623
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Abstract: An approximate method is described for solving Laplace's equation

$\displaystyle \left\{ {\begin{array}{*{20}{c}} {\Delta u = 0} \\ {u{\vert _{\pa... ...,1),} \hfill \\ {{\text{on}}\;\partial \Omega } \hfill \\ \end{array} } \right.$

precisely in the sense of Aberth's 1988 monograph. The algorithm uses singularity extraction, Fourier series methods, Taylor series methods, and interval analysis to construct an approximation $ U(x,y)$ to the solution $ u(x,y)$ at points in the square, and a uniform bound on the error $ \vert U(x,y) - u(x,y)\vert$. The algorithm applies to problems in which the boundary data g is specified in terms of four elementary functions. The boundary data may be discontinuous at the corners.

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Article copyright: © Copyright 1995 American Mathematical Society

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