Linearized localized orthogonal decomposition for quasilinear nonmonotone elliptic PDE

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. Both numerical experiments and theoretical analysis demonstrate the validity and applicability of the method.