  Finite element method is one of powerful numerical methods to solve PDE.
Usually, if a finite element solution to a Poisson equation based on a
triangulation of the underlying domain is not accurate enough, one will discard
the solution and then refine the triangulation uniformly and compute a new
finite element solution over the refined triangulation. It is wasteful to
discard the original finite element solution. We propose a prewavelet method to
save the original solution by adding a prewavelet subsolution to obtain the
refined level finite element solution. To increase the accuracy of numerical
solution to Poisson equations, we can keep adding prewavelet subsolutions.
  Our prewavelets are orthogonal in the $H^1$ norm and they are compactly
supported except for one globally supported basis function in a rectangular
domain. We have implemented these prewavelet basis functions in MATLAB and used
them for numerical solution of Poisson equation with Dirichlet boundary
conditions. Numerical simulation demonstrates that our prewavelet solution is
much more efficient than the standard finite element method.
