Numerische Mathematik

, Volume 65, Issue 1, pp 23–50

A unified approach to a posteriori error estimation using element residual methods

  • Mark Ainsworth
  • J. Tinsley Oden

DOI: 10.1007/BF01385738

Cite this article as:
Ainsworth, M. & Oden, J.T. Numer. Math. (1993) 65: 23. doi:10.1007/BF01385738


This paper deals with the problem of obtaining numerical estimates of the accuracy of approximations to solutions of elliptic partial differential equations. It is shown that, by solving appropriate local residual type problems, one can obtain upper bounds on the error in the energy norm. Moreover, in the special case of adaptiveh-p finite element analysis, the estimator will also give a realistic estimate of the error. A key feature of this is the development of a systematic approach to the determination of boundary conditions for the local problems. The work extends and combines several existing methods to the case of fullh-p finite element approximation on possibly irregular meshes with, elements of non-uniform degree. As a special case, the analysis proves a conjecture made by Bank and Weiser [Some A Posteriori Error Estimators for Elliptic Partial Differential Equations, Math. Comput.44, 283–301 (1985)].

Mathematics Subject Classification (1991)


Copyright information

© Springer-Verlag 1993

Authors and Affiliations

  • Mark Ainsworth
    • 1
  • J. Tinsley Oden
    • 1
  1. 1.Texas Institute for Computational MechanicsThe University of Texas at AustinAustinUSA
  2. 2.Mathematics DepartmentLeicester UniversityLeicesterUK