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The Quarterly Journal of Mechanics and Applied Mathematics 1959 12(1):117-123; doi:10.1093/qjmam/12.1.117
© 1959 by Oxford University Press
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ON THE NUMERICAL SOLUTION OF DIRICHLET PROBLEMS

DONALD GREENSPAN

( Purdue University Mathematics Dept., Lafayette, Indiana, U.S.A. )

Generalizations to a rectangular grid are given of some fundamental lemmas and theorems of S. Gerschgorin. The popular five-point technique for the numerical resolution of boundary-value problems associated with the Laplace equation is extended. The Dirichlet problem is replaced by the problem of solving a system of linear, algebraic equations, which, it is shown, has a unique solution. It is proved finally that the numerical solution converges to the analytic solution as the mesh constants converge to zero.


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