Steklov approximations of harmonic boundary value problems on planar regions

Error estimates for approximations of solutions of Laplace's equation with Dirichlet, Robin or Neumann boundary value conditions are described. The solutions are represented by orthogonal series using the harmonic Steklov eigenfunctions. Error bounds for partial sums involving the lowest eigenfunctions are found. When the region is a rectangle, explicit formulae for the Steklov eigenfunctions and eigenvalues are known. These were used to find approximations for problems with known explicit solutions. Results about the accuracy of these solutions, as a function of the number of eigenfunctions used, are given.