Linearized Localized Orthogonal Decomposition for Quasilinear Nonmonotone Elliptic PDE

TL;DR

This paper introduces a multiscale method based on the Localized Orthogonal Decomposition for solving complex quasilinear nonmonotone elliptic PDEs without requiring periodicity or scale separation, supported by theoretical analysis and numerical validation.

Contribution

It develops a novel multiscale approach for nonmonotone quasilinear elliptic PDEs that does not rely on traditional structural assumptions, with rigorous analysis and practical linearization strategies.

Findings

Method achieves convergence in the $H^1$-semi norm.

Linearization techniques are effective for different problem settings.

Numerical experiments confirm theoretical convergence and applicability.

Abstract

In this paper, we propose and analyze a multiscale method for a class of quasilinear elliptic problems of nonmonotone type with spatially multiscale coefficient. The numerical approach is inspired by the Localized Orthogonal Decomposition (LOD), so that we do not require structural assumptions such as periodicity or scale separation and only need minimal regularity assumptions on the coefficient.To construct the multiscale space, we solve linear fine-scale problems on small local subdomains, for which we consider two different linearization techniques. For both, we present a rigorous well-posedness analysis and convergence estimates in the $H^1$-semi norm. We compare and discuss theoretically and numerically the performance of our strategies for different linearization points. Numerical experiments underline the theoretical findings and illustrate the applicability of the method.