Prewavelet analysis of the heat equation

Summary. We consider the heat equation in a smooth domain of R $^2$ with Dirichlet and Neumann boundary conditions. It is solved by using its integral formulation with double-layer potentials, where the unknown $\mu=[\Phi]$, the jump of the solution through the boundary, belongs to an anisotropic Sobolev space. We approximate $\mu$ by the Galerkin method and use a prewavelet basis on $\Sigma _T=\Gamma \times (0,T)$, which characterizes the anisotropic space. The use of prewavelets allows to compress the stiffness matrix from $O(N^2)$ to $O(N \log N)$ when N is the size of the matrix, and the condition number of the compressed matrix is uniformly bounded as the initial one in the prewavelet basis. Finally we show that the compressed scheme converges as fast as the Galerkin one, even for the Dirichlet problem which does not admit a coercive variational formulation.