Identification and a posteriori estimation of transported errors in finite element analysis

Abstract

A systematic approach for developing reliable and efficient a posteriori error estimators for quite general classes of problems is presented. A major difficulty to be overcome is the definition and identification of the transported error. For certain classes of problem, there is no significant transported error and the only issue is how to develop appropriate local estimators for the error. For other types of problem, there is no alternative but to estimate the transported error. A general framework is presented for achieving this goal. The theory is reminiscent of the additive Schwarz methods that have proved successful in the development of effective preconditioners. The general theory is illustrated by applying it a singularly perturbed reaction-diffusion problem to derive estimators similar to those recently developed, and to derive new estimators for the radial part of the Laplace operator in cylindrical polar coordinates that are reliable and efficient. The results obtained indicate the theory gives sharp and practical estimates.