Convergence analysis of Schwarz-like methods for degenerate elliptic-parabolic equations

TL;DR

This paper proves convergence of Schwarz-like domain decomposition methods for nonlinear, degenerate elliptic-parabolic PDEs, enabling parallel solutions for complex diffusion models.

Contribution

It introduces a novel convergence proof for Schwarz-like methods applied to nonlinear degenerate PDEs using a nonlinear framework based on monotone operator theory.

Findings

Proves convergence of Schwarz-like methods for degenerate elliptic-parabolic equations.

Uses a nonlinear framework based on monotone operators for the proof.

Applicable to nonlinear diffusion processes with a $p$-structure.

Abstract

Convergence is proven for Schwarz-like methods applied to degenerate elliptic-parabolic equations with a $p$-structure. This family of PDEs, e.g., arises when modelling nonlinear diffusion processes. The Schwarz-like approximation methods are based on decomposing the space-time domain into overlapping subdomains, which enables parallel implementations. The methods are derived by introducing a pseudo-time component and applying time integrators of splitting type, which are time stepped towards infinity. This approach of decomposing the space-time domain is related to Schwarz waveform relaxation methods, but the methods considered here have the advantage that they can be proven to converge when applied to nonlinear parabolic, or even degenerate elliptic-parabolic, PDEs. We prove convergence by deriving a nonlinear framework based on the abstract theory for monotone operators and the existence theory for degenerate elliptic-parabolic equations.