A New Finite Volume Scheme for a Linear Schrödinger Evolution Equation

Abstract

We consider the linear Schrödinger evolution equation with a time dependent potential in several space dimension. We provide a new implicit  time finite volume scheme, using the general nonconforming meshes of [2] as discretization in space. We prove that the convergence order is $h_{\fancyscript{D}}+k$, where $h_{\fancyscript{D}}$ (resp. $k$) is the mesh size of the spatial (resp. time) discretization, in discrete norms ${\mathbb {L}}^{\infty }(0,T;H^1_0(\varOmega ))$ and ${\fancyscript{W}}^{1,\infty }(0,T;L^2(\varOmega ))$. These error estimates are useful because they allow to obtain approximations to the exact solution and its first derivatives of order $h_{\fancyscript{D}}+k$.