Functional A Posteriori Error Control for Conforming Mixed Approximations of Coercive Problems with Lower Order Terms

Research output: Contribution to journalResearch articleContributedpeer-review

Contributors

Abstract

The results of this contribution are derived in the framework of functional type a posteriori error estimates. The error is measured in a combined norm which takes into account both the primal and dual variables denoted by x and y, respectively. Our first main result is an error equality for all equations of the class A∗Ax + x = f or in mixed formulation A∗y + x = f, Ax = y, where the exact solution (x, y) is in D(A) × D(A∗). Here A is a linear, densely defined and closed (usually a differential) operator and A∗its adjoint. In this paper we deal with very conforming mixed approximations, i.e., we assume that the approximation (x, y) belongs to D(A) × D(A∗). In order to obtain the exact global error value of this approximation one only needs the problem data and the mixed approximation itself, i.e., we have the equality |x - x|2 + |A(x - x)|2 + |y - y|2 + |A∗ (y - y)|2 = M(x, y), where M(x, y) := |f - x - A∗y|2 + |y - Ax|2 contains only known data. Our second main result is an error estimate for all equations of the class A∗Ax + ix = f or in mixed formulation A∗y + ix = f, Ax = y, where i is the imaginary unit. For this problem we have the two-sided estimate (equation presented) where Mi(x, y) := |f - ix - A∗y|2 + |y - Ax|2 contains only known data. We will point out a motivation for the study of the latter problems by time discretizations or time-harmonic ansatz of linear partial differential equations and we will present an extensive list of applications including the reaction-diffusion problem and the eddy current problem.

Details

Original languageEnglish
Pages (from-to)609-631
Number of pages23
JournalComputational methods in applied mathematics
Volume16
Issue number4
Publication statusPublished - 1 Oct 2016
Peer-reviewedYes

External IDs

ORCID /0000-0003-4155-7297/work/145224234

Keywords

Keywords

  • Combined Norm, Error Equalities, Functional A Posteriori Error Estimates, Mixed Formulations