Post-Processing for Pointwise Local Error Bounds for Derivatives in Finite Element Solutions

This paper on "Post-Processing for Pointwise Local Error Bounds for Derivatives in Finite Element Solutions" was presented at the NAFEMS World Congress on Design, Simulation & Optimisation: Reliability & Applicability of Computational Methods - 9-11 April 1997, Stuttgart, Germany.

Introduction to Paper

Procedures developed by Kelly [6], and Ainsworth & Oden [2], can be used to determine a tight bound on the energy norm of the error in finite element solutions of linear elliptic problems. Procedures to localise the error bound to pointwise values of the solution or its derivatives are contained in the work on post-processing by Babuska & Miller [5], and developed by Yang, Kelly & Isles [8] and Wang, Sloan & Kelly [10]. The procedures have since been applied to a non-linear problem in Ainsworth, Kelly, Sloan & Wang [4].
The goal of this research is to produce tight guaranteed bounds on pointwise values of displacement and stress, as well as tight bounds in the energy norm for finite element solutions on a single mesh. The procedure requires the recovery of domain and interface residuals from a compatible finite element solution and identification of self equilibrating sets of residuals to define a complementary solution for the strain energy associated with the error. The localisation of the error bound for derivatives at a point requires the recovery of a post-processed solution determined by implementing a boundary integral defined using Green's identity, and the solution of a supplementary problem with boundary tractions defined from the appropriate singular fundamental solution to the governing equations. This supplementary solution corresponds to an extra right-hand-side in the parent finite element solution using the same mesh.
The procedures, using the approach taken by Ainsworth & Oden [1]-[3], are inexpensive to implement and the bounds on the energy norm are recommended by those authors to guide adaptive finite element schemes. In this paper we apply the bounding procedures to a problem of potential flow on a curved boundary and discuss the factors which affect the tightness of the bound.