A posteriori error estimates for domain decomposition methods

TL;DR

This paper develops functional a posteriori error estimates for domain decomposition methods, enabling efficient, subdomain-wise error control in PDE solutions with guaranteed bounds using extended flux sets.

Contribution

It introduces a novel form of error majorant tailored for DDM, facilitating practical error estimation with minimal computational effort.

Findings

Efficient error bounds using extended flux sets.

Error estimates applicable to various norms.

Subdomain-wise error control achieved.

Abstract

Nowadays, a posteriori error control methods have formed a new important part of the numerical analysis. Their purpose is to obtain computable error estimates in various norms and error indicators that show distributions of global and local errors of a particular numerical solution. In this paper, we focus on a particular class of domain decomposition methods (DDM), which are among the most efficient numerical methods for solving PDEs. We adapt functional type a posteriori error estimates and construct a special form of error majorant which allows efficient error control of approximations computed via these DDM by performing only subdomain-wise computations. The presented guaranteed error bounds use an extended set of admissible fluxes which arise naturally in DDM.