Wavelet compressed, modified Hilbert transform in the space–time discretization of the heat equation

On a finite time interval $(0,T)$ we consider the multiresolution Galerkin discretization of a modified Hilbert transform ${\mathscr{H}}_{T}$ that arises in the space–time Galerkin discretization of the linear diffusion equation. To this end, we design spline-wavelet systems in $(0,T)$ consisting of piecewise polynomials of degree $\geq 1$ with sufficiently many vanishing moments that constitute Riesz bases in the Sobolev spaces $ H^{s}_{0,}(0,T)$ and $H^{s}_{,0}(0,T)$. These bases provide stable multilevel splittings of the temporal discretization spaces into ‘increment’ or ‘detail’ spaces. Furthermore, they allow to optimally compress the nonlocal integrodifferential operators that appear in stable space–time variational formulations of initial-boundary value problems, such as the heat equation and the acoustic wave equation. We then obtain sparse space–time tensor-product spaces via algebraic tensor-products of the temporal multilevel discretizations with standard, hierarchic finite element spaces in the spatial domain (with standard Lagrangian FE bases). Hence, the construction of multiresolutions in the spatial domain is not necessary. An efficient multilevel preconditioner is proposed that solves the linear system of equations resulting from the sparse space–time Galerkin discretization with essentially linear complexity (in work and memory). A substantial reduction in the number of the degrees of freedom and CPU time (compared with time-marching discretizations) is demonstrated in numerical experiments.