# Finite Element Approximation of Elliptic Homogenization Problems in Nondivergence-Form

## TL;DR

This paper develops and analyzes a finite element numerical scheme for solving elliptic homogenization problems in nondivergence form, providing error estimates and demonstrating effectiveness through numerical experiments.

## Contribution

It introduces a new finite element approach for nondivergence-form homogenization problems with rigorous analysis and numerical validation.

## Key findings

- The scheme achieves accurate approximation of solutions.
- Numerical experiments confirm the scheme's effectiveness.
- Error estimates are established for the proposed method.

## Abstract

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.