A posteriori analysis and adaptive error control for operator decomposition solution of coupled semilinear elliptic systems

Abstract

SUMMARYIn this paper, we develop an a posteriori error analysis for operator decomposition iteration methods applied to systems of coupled semilinear elliptic problems. The goal is to compute accurate error estimates that account for the combined effects arising from numerical approximation (discretization) and operator decomposition iteration. In an earlier paper, we considered ‘triangular’ systems that can be solved without iteration. In contrast, operator decomposition iterative methods for fully coupled systems involve an iterative solution technique. We construct an error estimate for the numerical approximation error that specifically addresses the propagation of error between iterates and provide a computable estimate for the iteration error arising because of the decomposition of the operator. Finally, we develop an adaptive discretization strategy to systematically reduce the discretization error.Copyright © 2013 John Wiley & Sons, Ltd.

A Posteriori Error Analysis and Uncertainty Quantification for Adaptive Multiscale Operator Decomposition Methods for Multiphysics Problems

“…This turns out to be a crucial fact for a posteriori error analysis. (Carey et al, 2006), we solve a system -A«i =sin(47ri)sin(jrv), xeft .…”

“…These iterations are completely independent of the forward iterations. In (Estep et al, 2008a;Estep et al, 20086). we derive estimates that only require adjoint solutions of the two component problems.…”

“…This affects the analysis, but we do not discuss that here, sec (Estep et al, 2008a;Estep et al, 20086). …”

“…These issues are discussed in Estcp et al, 2008ft;Carey et al, 2008a;Estep et al, 2008a;Estcp et al, 2008ft). …”

Adaptive finite element solution of multiscale PDE–ODE systems

“…The error analysis is based on a posteriori error estimates that employ computable residuals and adjoint equations, see [6,7,8,9,10] for general information. For applications to multiscale systems, see [5,11,12,13,14,15,16]. We base the adaptive strategy on the block adaptive approach described in [17].…”