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High-order multiscale discontinuous Galerkin methods for the one-dimensional stationary Schrödinger equation. Dong, Bo


We develop high-order multiscale discontinuous Galerkin (DG) methods for one-dimensional stationary Schr\"{o}dinger equations with oscillating solutions. We propose two types of multiscale finite element spaces, and prove that the resulting DG methods converge optimally with respect to the mesh size $h$ in $L^2$ norm when $h$ is small enough. In the lowest order case, we prove that the second order multiscale DG method has the optimal convergence even when the mesh size is larger than the wave length. Numerically we observe that all these multiscale DG methods have at least the second-order convergence on coarse meshes and optimal high-order convergence on fine meshes.

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