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

Highly accurate finite difference schemes are developed for Laplace’s equation with the Dirichlet boundary condition on general bounded regions in R n {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 ) O({h5.5}) in L 2 {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.