Prewavelet solution to Poisson equations

Finite element method is one of powerful numerical methods to solve PDE. Usually, ifa finite element solution to a Poisson equation based on a triangulation of the underlyingdomain is not accurate enough, one will discard the solution and then refine the triangulationuniformly and compute a new finite element solution over the refined triangulation. It iswasteful to discard the original finite element solution. We propose a Prewavelet methodto save the original solution by adding a Prewavelet subsolution to obtain the refined levelfinite element solution. To increase the accuracy of numerical solution to Poisson equations,we can keep adding Prewavelet subsolutions.Our Prewavelets are orthogonal in the H1 norm and they are locally supported exceptfor one globally supported basis function in a rectangular domain. We have implementedthese Prewavelet basis functions in MATLAB and used them for numerical solution ofPoisson equation with Dirichlet boundary conditions. Numerical simulation demonstratesthat our Prewavelet solution is much more efficient than the standard finite element method.Prewavelets over other boundary domains, such as triangle, L-shape domain, are also constructed.