Finite Element Approximation of Elliptic Homogenization Problems in Nondivergence-Form
It provides a rigorous numerical framework for solving a class of homogenization problems that are challenging due to their nondivergence structure, benefiting researchers in computational mathematics and materials science.
The paper develops and analyzes a finite element scheme for elliptic homogenization problems in nondivergence-form, proving corrector results using uniform W^{2,p} estimates and demonstrating the method's performance through numerical experiments.
We use uniform $W^{2,p}$ estimates to obtain corrector results for periodic homogenization problems of the form $A(x/\varepsilon):D^2 u_{\varepsilon} = f$ subject to a homogeneous Dirichlet boundary condition. We propose and rigorously analyze a numerical scheme based on finite element approximations for such nondivergence-form homogenization problems. The second part of the paper focuses on the approximation of the corrector and numerical homogenization for the case of nonuniformly oscillating coefficients. Numerical experiments demonstrate the performance of the scheme.