Abstract

We present the hierarchical matrix compression algorithms to speed up the computations to solve unstable space-time finite element method. Namely, we focus on the non-stationary time-dependent advection dominated diffusion problem solved by using space-time finite element method. We formulate the problem on the space-time mesh, where two axes of coordinates system denote the spatial dimension, and the third axis denotes the temporal dimension. By employing the space-time mesh, we avoid time iterations, and we solve the problem “at once” by calling a solver once for the entire mesh. This problem, however, is challenging, and it requires the application of special stabilization methods. We propose the stabilization method based on least-squares. We derive the space-time formulation, and solve it using adaptive finite element method. To speed up the solution process, we compress the matrix of the space-time formulation using the low-rank compression algorithm. We show that the compressed matrix allows for quasi-linear computational cost matrix-vector multiplication. Thus, we apply the GMRES solver with hierarchical matrix-vector multiplications. Summing up, we propose a quasi-linear computational cost solver for stabilized space-time formulations of advection dominated diffusion problem.