Computational multiscale methods for nondivergence-form elliptic partial differential equations

TL;DR

This paper develops multiscale computational methods for nondivergence-form elliptic PDEs with heterogeneous coefficients, extending the localized orthogonal decomposition approach to handle these complex equations effectively.

Contribution

It introduces a novel multiscale method based on LOD for nondivergence-form elliptic PDEs, with rigorous error analysis confirming its effectiveness beyond traditional assumptions.

Findings

Method achieves accurate approximations without scale separation.

Error bounds comparable to divergence-form PDEs.

Applicable to heterogeneous coefficients satisfying Cordes condition.

Abstract

This paper proposes novel computational multiscale methods for linear second-order elliptic partial differential equations in nondivergence-form with heterogeneous coefficients satisfying a Cordes condition. The construction follows the methodology of localized orthogonal decomposition (LOD) and provides operator-adapted coarse spaces by solving localized cell problems on a fine scale in the spirit of numerical homogenization. The degrees of freedom of the coarse spaces are related to nonconforming and mixed finite element methods for homogeneous problems. The rigorous error analysis of one exemplary approach shows that the favorable properties of the LOD methodology known from divergence-form PDEs, i.e., its applicability and accuracy beyond scale separation and periodicity, remain valid for problems in nondivergence-form.