标题： 用于线性薛定谔方程的时空连续伽辽金有限元方法 显示英文标题

标题： A Space-Time Continuous Galerkin Finite Element Method for Linear Schrödinger Equations

摘要： 我们引入了一种时空有限元方法，用于有界Lipschitz区域中具有Dirichlet条件的线性时变薛定谔方程。所提出的离散化方案基于时变薛定谔方程的时空变分公式。特别是，时空方法是符合的，并且是Galerkin类型，即试验空间和测试空间相等。我们考虑了时间和空间的张量积方法，使用分段多项式、连续的试验和测试函数。在这种情况下，我们给出了全局线性系统以及基于利用全局系统矩阵的Kronecker结构的高效直接时空求解器。这导致了Bartels-Stewart方法和快速对角化方法。这两种方法都导致解决一系列空间子问题。特别是，快速对角化方法允许并行求解空间子问题，即可以进行时间并行化。二维空间域的数值例子说明了时空范数中的收敛性，并展示了所提出求解器的潜力。 显示英文摘要

摘要： We introduce a space-time finite element method for the linear time-dependent Schr\"odinger equation with Dirichlet conditions in a bounded Lipschitz domain. The proposed discretization scheme is based on a space-time variational formulation of the time-dependent Schr\"odinger equation. In particular, the space-time method is conforming and is of Galerkin-type, i.e., trial and test spaces are equal. We consider a tensor-product approach with respect to time and space, using piecewise polynomial, continuous trial and test functions. In this case, we state the global linear system and efficient direct space-time solvers based on exploiting the Kronecker structure of the global system matrix. This leads to the Bartels-Stewart method and the fast diagonalization method. Both methods result in solving a sequence of spatial subproblems. In particular, the fast diagonalization method allows for solving the spatial subproblems in parallel, i.e., a time parallelization is possible. Numerical examples for a two-dimensional spatial domain illustrate convergence in space-time norms and show the potential of the proposed solvers.