On the optimal convergence rate of a Robin-Robin domain decomposition method

TL;DR

This paper proves that the convergence rate of the Robin-Robin domain decomposition method can be independent of mesh size, resolving a long-standing open problem and confirming the theory with numerical tests.

Contribution

The paper establishes that the convergence rate of the Robin-Robin DD method can be mesh-independent, solving a 20-year-old open problem.

Findings

Convergence rate can be mesh-independent.

Theoretical proof provided.

Numerical tests verify the theory.

Abstract

In this work, we solve a long-standing open problem: Is it true that the convergence rate of the Lions' Robin-Robin nonoverlapping domain decomposition(DD) method can be constant, independent of the mesh size $h$? We closed this twenty-year old problem with a positive answer. Our theory is also verified by numerical tests.