A fast multi-level multi-grid method for the Laplace equation

This chapter develops a fast, accurate, and efficient multilevel iterative method to solve general boundary value problems arising in computational mechanics. It concentrates on problems of two-dimensional steady potential flow and presents a fast direct boundary element method. This novel method extends the work on multilevel multiintegration (MLMI) in several important ways to address problems with arbitrary geometry and mixed boundary conditions. Researchers use biconjugate gradient methods and implement the MLMI approach for fast matrix and matrix transpose multiplication for every iteration loop. A multigrid algorithm is introduced to find that the number of iterations for the bi-conjugate gradient methods is independent of the boundary-element mesh discretization for problems of steady-state heat diffusion discussed in this chapter. The method can be extended in a straightforward manner to the solution of many problems in science and engineering that result in very large sets of matrix equations when the associated integral equations are discretized.