NUMERICAL SOLUTION OF THE DIRICHLET PROBLEM FOR LAPLACE'S EQUATION,
JOHNS HOPKINS UNIV SILVER SPRING MD APPLIED PHYSICS LAB
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The author gives asymptotic estimates for the minimum number of arithmetic operations needed to numerically solve the Dirichlet problem for Laplaces equation as a function of accuracy. In the course of proving these estimates, he gives error estimates that exhibit an increase in accuracy near the boundary, for difference schemes which are of positive type and satisfy certain additional conditions. The author constructs difference schemes which are locally low-order accurate on a fine mesh near the boundary and locally high-order accurate on a coarse mesh away from the boundary. He then shows that the number of arithmetic operations required to solve the resulting system of equations by iteration to the desired accuracy is asymptotically of the same order as the minimum possible, and thus is in that sense optimal.
- Theoretical Mathematics