论文标题
有效的直接时空有限元求解器，用于抛物线的初始有限价值问题
Efficient Direct Space-Time Finite Element Solvers for Parabolic Initial-Boundary Value Problems in Anisotropic Sobolev Spaces
论文作者
论文摘要
我们考虑了各向异性SOBOLEV空间中抛物线寄生虫初始值问题的时空变异公式与希尔伯特型转换结合使用。这种变分设置是时空Galerkin有限元离散化的起点，它导致了代数方程的大全局线性系统。我们建议并研究该系统的新有效直接求解器。特别是，我们使用分段多项式，全球连续的ANSATZ和测试功能的张量产品方法。开发的求解器基于Bartels-Stewart方法和快速对角线化方法，从而解决了一系列空间子问题。基于快速对角线化方法的求解器允许并行求解这些空间子问题，从而使时间平行。我们分析了所提出的算法的复杂性，并为二维空间结构域提供了数值示例，其中使用了空间子问题的稀疏直接求解器。
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.
