Convergence of a discontinuous Galerkin multiscale method

TL;DR

This paper proves convergence properties of a discontinuous Galerkin multiscale method for elliptic problems with heterogeneous coefficients, demonstrating exponential decay of error with patch size and confirming results through numerical experiments.

Contribution

It establishes convergence rates for a multiscale method without scale separation assumptions, using localized corrected basis functions for elliptic PDEs with highly variable coefficients.

Findings

Error decreases exponentially with patch size.

Linear convergence in energy norm achieved.

Quadratic convergence in L2 norm confirmed.

Abstract

A convergence result for a discontinuous Galerkin multiscale method for a second order elliptic problem is presented. We consider a heterogeneous and highly varying diffusion coefficient in $L^\infty(\Omega,\mathbb{R}^{d\times d}_{sym})$ with uniform spectral bounds and 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 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 as $\mathcal{O}(H|\log(H^{-1})|)$. We also discuss a way to further localize the corrected basis to element-wise support leading to a slight increase of the dimension of the space. Improved convergence rate can be achieved depending on the piecewise regularity of the forcing function. Linear convergence in energy norm and quadratic convergence in $L^2$-norm is obtained independently of the forcing function. A series of numerical experiments confirms the theoretical rates of convergence.