A post-processed higher-order multiscale method for nondivergence-form elliptic equations

TL;DR

This paper introduces a multiscale finite element method with post-processing for nondivergence-form elliptic equations, achieving higher-order convergence despite low regularity of data.

Contribution

It develops a novel post-processing strategy within a multiscale framework to improve convergence rates for complex elliptic PDEs with heterogeneous coefficients.

Findings

The method attains higher-order convergence rates.

Numerical experiments confirm the effectiveness of the approach.

Abstract

We study the finite element approximation of linear second-order elliptic partial differential equations in nondivergence form with highly heterogeneous diffusion and drift coefficients. A generalized Cordes condition is imposed to guarantee that a suitably renormalized version of the nondivergence-form differential operator is near the Laplacian. Based on a stabilized symmetric formulation for the gradient that enables the use of $H^1$-conforming approximation spaces, we construct a multiscale method following the methodology of the localized orthogonal decomposition with coarse basis functions tailored to the heterogeneous coefficients. We employ a novel post-processing strategy to obtain higher-order convergence rates, overcoming previous limitations imposed by the low regularity of the load functional. Numerical experiments demonstrate the performance of the method.