# Numerical homogenization for nonlinear strongly monotone problems

## TL;DR

This paper introduces a multiscale finite element method for nonlinear monotone equations that achieves optimal error estimates without requiring structural assumptions, demonstrated through numerical examples including Richards equation.

## Contribution

A novel multiscale method for nonlinear monotone problems that constructs problem-adapted spaces via local linear problems, avoiding structural assumptions.

## Key findings

- Method achieves optimal a priori error estimates
- Efficient due to linearity and localization of local problems
- Validated by numerical examples including Richards equation

## Abstract

In this work we introduce and analyze a new multiscale method for strongly nonlinear monotone equations in the spirit of the Localized Orthogonal Decomposition. A problem-adapted multiscale space is constructed by solving linear local fine-scale problems which is then used in a generalized finite element method. The linearity of the fine-scale problems allows their localization and, moreover, makes the method very efficient to use. The new method gives optimal a priori error estimates up to linearization errors. The results neither require structural assumptions on the coefficient such as periodicity or scale separation nor higher regularity of the solution. The effect of different linearization strategies is discussed in theory and practice. Several numerical examples including stationary Richards equation confirm the theory and underline the applicability of the method.