An efficient algorithm for the variational-difference Dirichlet problem

An efficient algorithm is described for solving the sets of algebraic equations that arise in the finite element method for the Dirichlet problem, in a domain composed of rectangles with sides parallel to the axes. The algorithm is based on the method with capacitance matrix and reduces the problem to the solution of problems in rectangles and a system with capacitance matrix $C$. A problem in rectangles is solved by means of a fast Fourier transformation involving $\sim N^2\log_2N$, $N=1/h$, operations, and the system with matrix $C$, by an iterative method involving $\sim N\log_2N\ln\varepsilon^{-1}$ operations.