A high-order multiscale discontinuous Galerkin method for two-dimensional Schrodinger equation in quantum transport

We develop and analyze a high-order multiscale discontinuous Galerkin (DG) method for two-dimensional stationary Schrodinger equations in quantum transport. The solution of the problem under consideration has oscillations mainly in one direction, so we include oscillatory non-polynomial basis functions in that direction and use polynomial basis in the other direction to approximate the solution. We prove that the resulting method converges with an optimal order when the mesh size is sufficiently small. Numerically we observe that the method converges on coarse meshes and achieves optimal higher-order convergence when the mesh size is refined to the scale of the wave length. Numerical results show that the method can capture highly oscillating solutions of Schrodinger equations more effectively than standard DG methods with polynomial basis. (c) 2022 Elsevier B.V. All rights reserved.