2020/08/05 by Ulrich Langer, Langer, Ulrich, Marco Zank +1 · 2 citations
Engineering · Mathematics · #65F05 #65M60 #Advanced Numerical Methods in Computational Mathematics #Differential Equations and Numerical Methods #FOS: Mathematics #Numerical Analysis (math.NA) #Numerical methods for differential equations
paper · pdf · doi:10.48550/arxiv.2008.01996
openalex publication_date 2020/08/05 · openalex created_date 2022/07/26 · openalex updated_date 2026/07/28
We consider a space-time variational formulation of parabolic\ninitial-boundary value problems in anisotropic Sobolev spaces in combination\nwith a Hilbert-type transformation. This variational setting is the starting\npoint for the space-time Galerkin finite element discretization that leads to a\nlarge global linear system of algebraic equations. We propose and investigate\nnew efficient direct solvers for this system. In particular, we use a\ntensor-product approach with piecewise polynomial, globally continuous ansatz\nand test functions. The developed solvers are based on the Bartels-Stewart\nmethod and on the Fast Diagonalization method, which result in solving a\nsequence of spatial subproblems. The solver based on the Fast Diagonalization\nmethod allows to solve these spatial subproblems in parallel leading to a full\nparallelization in time. We analyze the complexity of the proposed algorithms,\nand give numerical examples for a two-dimensional spatial domain, where sparse\ndirect solvers for the spatial subproblems are used.\n