Space-time wavelet finite element method for parabolic equations

For a class of linear parabolic equations we propose a nonadaptive sparse space-time Galerkin least squares discretization. We formulate criteria on the trial and test spaces for the well-posedness of the corresponding Galerkin least squares solution. In order to obtain discrete stability uniformly in the discretization parameters, we allow test spaces which are suitably larger than the trial space. The problem is then reduced to a finite, overdetermined linear system of equations by a choice of bases. We present several strategies that render the resulting normal equations well-conditioned uniformly in the discretization parameters. The numerical solution is then shown to converge quasi-optimally to the exact solution in the natural space for the original equation. Numerical examples for the heat equation confirm the theory