Optimized Schwarz methods for heterogeneous heat transfer problems

TL;DR

This paper develops and analyzes optimized Schwarz methods with local transmission conditions for heterogeneous heat transfer problems, demonstrating improved efficiency and robustness through theoretical analysis and numerical experiments.

Contribution

It introduces three local approximation strategies for transmission operators in Schwarz methods and provides a detailed continuous-level analysis for heterogeneous heat transfer problems.

Findings

Local transmission conditions improve efficiency in heterogeneous scenarios.

Proper scaling of transmission parameters enhances robustness.

Numerical results confirm the theoretical advantages of local approximations.

Abstract

We present here nonoverlapping optimized Schwarz methods applied to heat transfer problems with heterogeneous diffusion coefficients. After a Laplace transform in time, we derive the error equation and obtain the convergence factor. The optimal transmission operators are nonlocal, and thus inconvenient to use in practice. We introduce three versions of local approximations for the transmission parameter, and provide a detailed analysis at the continuous level in each case to identify the best local transmission conditions. Numerical experiments are presented to illustrate the performance of each local transmission condition. As shown in our analysis, local transmission conditions, which are scaled appropriately with respect to the heterogeneous diffusion coefficients, are more efficient and robust especially when the discontinuity of the diffusion coefficient is large.