Adaptive Wavelet Schemes for Parabolic Problems: Sparse Matrices and Numerical Results

A simultaneous space-time variational formulation of a parabolic evolution problem is solved with an adaptive wavelet method. This method is shown to converge with the best possible rate in linear complexity. Thanks to the use of tensor product bases, there is no penalty in complexity due to the additional time dimension. Special wavelets are designed such that the bi-infinite system matrix is sparse. This sparsity largely simplifies the implementation and improves the quantitative properties of the adaptive wavelet method. Numerical results for an ODE and the heat equation are presented.