# A linear Uzawa-type solver for nonlinear transmission problems

## TL;DR

This paper introduces a linear Uzawa-type iterative solver for nonlinear transmission problems, combining BEM and FEM methods, and proves its linear convergence without restrictive assumptions on ellipticity or monotonicity.

## Contribution

The paper presents a novel Uzawa-type iteration that efficiently solves nonlinear transmission problems using combined BEM and FEM, avoiding previous restrictive conditions.

## Key findings

- Proves linear convergence of the proposed method.
- Avoids restrictions on ellipticity and strong monotonicity.
- Combines BEM and FEM for efficient nonlinear problem solving.

## Abstract

We propose an Uzawa-type iteration for the Johnson-N\'ed\'elec formulation of a Laplace-type transmission problem with possible (strongly monotone) nonlinearity in the interior domain. In each step, we sequentially solve one BEM for the weakly-singular integral equation associated with the Laplace-operator and one FEM for the linear Yukawa equation. In particular, the nonlinearity is only evaluated to build the right-hand side of the Yukawa equation. We prove that the proposed method leads to linear convergence with respect to the number of Uzawa iterations. Moreover, while the current analysis of a direct FEM-BEM discretization of the Johnson-N\'ed\'elec formulation requires some restrictions on the ellipticity (resp. strong monotonicity constant) in the interior domain, our Uzawa-type solver avoids such assumptions.