Additive Schwarz methods for semilinear elliptic problems with convex energy functionals: Convergence rate independent of nonlinearity

TL;DR

This paper demonstrates that additive Schwarz methods for semilinear elliptic problems with convex energy functionals have convergence rates unaffected by nonlinearity, ensuring efficient and scalable solutions comparable to linear problems.

Contribution

The paper proves that additive Schwarz methods maintain convergence rates independent of nonlinearity for semilinear elliptic problems with convex energy functionals, and establishes their scalability.

Findings

Convergence rates are independent of the nonlinear term.

Two-level method's convergence depends only on mesh and overlap parameters.

Numerical results confirm theoretical convergence bounds.

Abstract

We investigate additive Schwarz methods for semilinear elliptic problems with convex energy functionals, which have wide scientific applications. A key observation is that the convergence rates of both one- and two-level additive Schwarz methods have bounds independent of the nonlinear term in the problem. That is, the convergence rates do not deteriorate by the presence of nonlinearity, so that solving a semilinear problem requires no more iterations than a linear problem. Moreover, the two-level method is scalable in the sense that the convergence rate of the method depends on $H/h$ and $H/\delta$ only, where $h$ and $H$ are the typical diameters of an element and a subdomain, respectively, and $\delta$ measures the overlap among the subdomains. Numerical results are provided to support our theoretical findings.