SIAM Journal on Scientific Computing, Volume 43, Issue 4, Page A2714-A2736, January 2021.
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 us 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.

中文翻译：

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各向异性 Sobolev 空间抛物线初边界值问题的高效直接时空有限元求解器

SIAM 科学计算杂志，第 43 卷，第 4 期，第 A2714-A2736 页，2021 年 1 月。
我们考虑各向异性 Sobolev 空间中抛物线初边界值问题的时空变分公式，并结合希尔伯特型变换。这种变分设置是时空伽辽金有限元离散化的起点，它导致代数方程的大型全局线性系统。我们为该系统提出并研究了新的高效直接求解器。特别是，我们使用具有分段多项式、全局连续 ansatz 和测试函数的张量积方法。开发的求解器基于 Bartels--Stewart 方法和快速对角化方法，从而解决一系列空间子问题。基于快速对角化方法的求解器允许我们并行解决这些空间子问题，从而实现时间上的完全并行化。