Linearly convergent nonoverlapping domain decomposition methods for quasilinear parabolic equations

TL;DR

This paper establishes linear convergence of new nonoverlapping domain decomposition methods for quasilinear parabolic equations, extending previous elliptic results to more general time-dependent problems without strong regularity assumptions.

Contribution

It introduces a novel analytical framework for domain decomposition methods applied to quasilinear parabolic equations, proving convergence without restrictive regularity conditions.

Findings

Proves linear convergence of Dirichlet--Neumann and Robin--Robin methods for quasilinear parabolic equations.

Extends convergence results from continuous to discrete space-time finite element methods.

Provides a new framework based on fractional derivatives and time-dependent operators.

Abstract

We prove linear convergence for a new family of modified Dirichlet--Neumann methods applied to quasilinear parabolic equations, as well as the convergence of the Robin--Robin method. Such nonoverlapping domain decomposition methods are commonly employed for the parallelization of partial differential equation solvers. Convergence has been extensively studied for elliptic equations, but in the case of parabolic equations there are hardly any convergence results that are not relying on strong regularity assumptions. Hence, we construct a new framework for analyzing domain decomposition methods applied to quasilinear parabolic problems, based on fractional time derivatives and time-dependent Steklov--Poincar\'e operators. The convergence analysis is conducted without assuming restrictive regularity assumptions on the solutions or the numerical iterates. We also prove that these continuous convergence results extend to the discrete case obtained when combining domain decompositions with space-time finite elements.