A Space-Time Continuous Galerkin Finite Element Method for Linear Schrödinger Equations
Marco Zank
TL;DR
The paper presents an $H^1$-conforming space-time Galerkin finite element method for the linear Schrödinger equation $i \partial_t \psi - \Delta_x \psi = f$ with Dirichlet data, using a tensor-product discretization in time and space. The global system $$(i B_t \otimes M_x + M_t \otimes A_x) \boldsymbol{\psi} = \boldsymbol{f}$$ is solved via Kronecker-structured direct solvers, either Bartels–Stewart (Schur decomposition) or the fast diagonalization method, enabling parallelization across time through independent spatial solves. Numerical experiments on a 2D unit square demonstrate second-order convergence in $L^2(Q)$ and first-order convergence in $H^1(Q)$, with significant speedups when employing the fast diagonalization approach, even on nonuniform temporal meshes. The findings indicate that time-parallel direct solvers for space-time FEMs offer a practical route to efficient quantum-dynamics simulations on Lipschitz domains.
Abstract
We introduce a space-time finite element method for the linear time-dependent Schrödinger 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 Schrödinger 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.