A unified approach to a posteriori error estimation using element residual methods. (English) Zbl 0797.65080

The paper studies the problem of obtaining numerical estimates of the accuracy of finite element approximations to solutions of second order elliptic differential equations. By solving appropriate local residual type problems, one obtains realistic upper bounds on the error in the energy norm.
A fundamental difference between the method proposed here and existing methods is that the local problem involves only the Laplacian operator whilst other methods have local problems based on the actual operator. While it might seem more advantageous to have local problems based on the actual operator, in the authors’ opinion this actually does not appear to be the case. This analysis is similar in type to those of R. E. Bank and A. Weiser [Math. Comput. 44, 283-301 (1985; Zbl 0569.65079)], who had conjectured the upper bound property.
The paper further focuses on the determination of boundary conditions used in the local problems, especially on the choice of splitting which determines the boundary conditions. Some earlier results as that of P. Percell and M. F. Wheeler on local refinement procedures [SIAM J. Numer. Anal. 15, 705-714 (1978; Zbl 0396.65067)] are generalized. The recent work provides theoretical support for the heuristic results of D. W. Kelly [Int. J. Numer. Methods Eng. 20, 1491-1506 (1984; Zbl 0575.65100)].


65N15 Error bounds for boundary value problems involving PDEs
65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
35J25 Boundary value problems for second-order elliptic equations
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