Efficient Direct Space-Time Finite Element Solvers for Parabolic Initial-Boundary Value Problems in Anisotropic Sobolev Spaces

TL;DR

This paper develops efficient direct solvers for large space-time finite element systems arising from parabolic PDEs in anisotropic Sobolev spaces, enabling parallel computation in time and improving solution efficiency.

Contribution

It introduces novel tensor-product based direct solvers using Bartels-Stewart and Fast Diagonalization methods for space-time discretized parabolic problems.

Findings

Fast Diagonalization solver enables parallel spatial subproblem solutions.

The proposed algorithms have analyzed computational complexity.

Numerical examples demonstrate solver efficiency in 2D spatial domains.

Abstract

We consider a space-time variational formulation of parabolic initial-boundary value problems in anisotropic Sobolev spaces in combination with a Hilbert-type transformation. This variational setting is the starting point for the space-time Galerkin finite element discretization that leads to a large global linear system of algebraic equations. We propose and investigate new efficient direct solvers for this system. In particular, we use a tensor-product approach with piecewise polynomial, globally continuous ansatz and test functions. The developed solvers are based on the Bartels-Stewart method and on the Fast Diagonalization method, which result in solving a sequence of spatial subproblems. The solver based on the Fast Diagonalization method allows to solve these spatial subproblems in parallel leading to a full parallelization in time. We analyze the complexity of the proposed algorithms, and give numerical examples for a two-dimensional spatial domain, where sparse direct solvers for the spatial subproblems are used.