Convergence of a Discontinuous Galerkin Multiscale Method

Abstract

We present a discontinuous Galerkin multiscale method for second order elliptic problems and prove convergence. We consider a heterogeneous and highly varying diffusion coefficient in $L^\infty(\Omega,\mathbb{R}^{d\times d}_{sym})$ with uniform spectral bounds without any assumption on scale separation or periodicity. The multiscale method uses a corrected basis that is computed on patches/subdomains. The error, due to truncation of the corrected basis, decreases exponentially with the size of the patches. Hence, to achieve an algebraic convergence rate of the multiscale solution on a uniform mesh with mesh size $H$ to a reference solution, it is sufficient to choose the patch sizes $\mathcal{O}(H|\log H|)$. We also discuss a way to further localize the corrected basis to elementwise support. Improved convergence rate can be achieved depending on the piecewise regularity of the forcing function. Linear convergence in energy norm and quadratic convergence in the $L^2$-norm is obtained independently of the forcing function. A series of numerical experiments confirms the theoretical rates of convergence.