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High order fast Laplace solvers for the Dirichlet problem on general regions

Authors: Victor Pereyra, Wlodzimierz Proskurowski and Olof Widlund
Journal: Math. Comp. 31 (1977), 1-16
MSC: Primary 65N15; Secondary 65B05
MathSciNet review: 0431736
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Abstract: Highly accurate finite difference schemes are developed for Laplace's equation with the Dirichlet boundary condition on general bounded regions in $ {R^n}$. A second order accurate scheme is combined with a deferred correction or Richardson extrapolation method to increase the accuracy. The Dirichlet condition is approximated by a method suggested by Heinz-Otto Kreiss. A convergence proof of his, previously not published, is given which shows that, for the interval size h, one of the methods has an accuracy of at least $ O({h^{5.5}})$ in $ {L_2}$. The linear systems of algebraic equations are solved by a capacitance matrix method. The results of our numerical experiments show that highly accurate solutions are obtained with only a slight additional use of computer time when compared to the results obtained by second order accurate methods.

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Article copyright: © Copyright 1977 American Mathematical Society

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