Convergence analysis of the nonoverlapping Robin-Robin method for nonlinear elliptic equations

• We prove convergence for the nonoverlapping Robin-Robin method applied to nonlinear elliptic equations with a p-structure, including degenerate diffusion equations governed by the p-Laplacian. This nonoverlapping domain decomposition is commonly encountered when discretizing elliptic equations, as it enables the usage of parallel and distributed hardware. Convergence has been derived in various linear contexts, but little has been proven for nonlinear equations. Hence, we develop a new theory for nonlinear Steklov-Poincaré operators based on the p-structure and the L^p-generalization of the Lions-Magenes spaces. This framework allows the reformulation of the Robin-Robin method into a Peaceman-Rachford splitting on the interfaces of the subdomains, and the convergence analysis then follows by employing elements of the abstract theory for monotone operators. The analysis is performed on Lipschitz domains and without restrictive regularity assumptions on the solutions.

Subject headings

NATURVETENSKAP  -- Matematik -- Beräkningsmatematik (hsv//swe)

NATURAL SCIENCES  -- Mathematics -- Computational Mathematics (hsv//eng)

Keyword

Robin-Robin method

Nonoverlapping domain decomposition

Nonlinear elliptic equation

Convergence

Steklov-Poincaré operator

Publication and Content Type

art (subject category)

ref (subject category)