Parallel adaptive solution of a Poisson equation with multiwavelets

We present an adaptive algorithm for the solution of the Poisson equation. The domain is divided into subdomains. The resolution of each subdomain depends on the smoothness of the right-hand side of the Poisson equation. This determines the adaptivity of the algorithm. In each subdomain a particular solution is found. These solutions are patched by introducing double/single layers at the interfaces of the subdomains. The influence of these layers is effectively computed using multiwavelets. In the wavelet bases kernels of integrals which represent double layers are sparse. When the number of grid points increases as N, the number of essential wavelet coefficients, which represent a vector, increases as log N. Hence, using this sparsity reduces the number of operations from O(N-2) to O(N log N). The algorithm was implemented on parallel computers of SP2 and SGI types while each processor was assigned to each box. The efficiency of the algorithm was demonstrated.