A Space-Time tensor-product B-spline discretization for Schrödinger Equation
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We introduce a Petrov–Galerkin discretization for a well posed space-time variational formulation of time-dependent Schrödinger equation as formulated in (2017, Demkowicz et al). We consider the trial space of maximum smoothness tensor product B-splines, while the test space is the image of the trials using the Schrödinger operator ${S} := i\frac{\partial{}}{\partial{t}}- \Delta$. The proposed discretization is inf-sup stable, and the matrix associated with the problem is sum of Kronecker products of space and time pencils, similar to the ultraweak discretization in (2022, Hain and Urban). Exploiting this structure, we propose a preconditioner for the resulting linear system, easily assembled and invertible with Fast Diagonalization (FD) techniques. We provide numerical tests showing high order approximation rates of the proposed method, and we highlight the competitive behavior concerning wall-clock time compared to classical algebraic preconditioners.
