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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[1] Adams, R.A. (1975): Sobolev spaces, Academic Press, New York · Zbl 0314.46030
[2] Ainsworth, M., Craig, A.W. (1992): A posteriori error estimators in the finite element method. Numer. Math.60, 429-463 · Zbl 0757.65109
[3] Babu?ka, I., Miller, A.D. (1987): A feedback finite element method with a posteriori error estimation: Part 1. Comput. Methods Appl. Mech. Eng.61, pp. 1-40 · Zbl 0593.65064
[4] Babu?ka, I., Rheinboldt, W.C. (1978): A posteriori error estimates for adaptive finite element computations. SIAM J. Numer. Anal.15, pp. 736-754 · Zbl 0398.65069
[5] Bank, R.E., Weiser, A. (1985): Some a posteriori error estimators for elliptic partial differential equations, Math. Comput.44, 283-301 · Zbl 0569.65079
[6] Demkowicz, L., Oden, J.T., Strouboulis, T. (1985): An adaptivep-version finite element method for transient flow problems with moving boundaries. In: R.H.Gallagher, G. Carey, J.T. Oden, O.C. Zienkiewicz, eds., Finite elements in fluids, Vol. 6. Wiley, Chichester, pp. 291-306
[7] Fraejis de Veubeke, B. (1974): Variational principles and the patch test, Int. J. Numer. Methods Eng.8, 783-801 · Zbl 0284.73043
[8] Girault, V., Raviart, P.A. (1986): Finite element methods for Navier Stokes equations: Theory and algorithms Springer Series in Computational Mathematics5. Springer, Berlin Heidelberg New York · Zbl 0585.65077
[9] Kelly, D.W. (1984): The self equilibration of residuals and complementary a posteriori error estimates in the finite element method. Int. J. Numer. Methods Eng.20, 1491-1506 · Zbl 0575.65100
[10] Ladeveze, D., Leguillon (1983): Error estimate procedure in the finite element method and applications. SIAM J. Numer. Anal.20, 485-509 · Zbl 0582.65078
[11] Oden, J.T., Demkowicz, L., Rachowicz, W., Westermann, T.A. (1989): Towards a universalh-p finite element strategy. Part II. A posteriori error estimation, Comput. Methods Appl. Mech. Eng.77, 113-180 · Zbl 0723.73075
[12] Oden, J.T., Demkowicz, L., Strouboulis, T., Devloo, P. (1986): Adaptive methods for problems in solid and fluid mechanics. In: I. Babu?ka, O.C. Zienkiewicz, J. Gago, E.R. de A. Oliveira, eds., Accuracy estimates and adaptive refinements in finite element computations, Wiley Chichester, pp. 249-280.
[13] Percell, P., Wheeler, M.F. (1978): A local finite element procedure for elliptic equation., SIAM J. Numer. Anal.15, pp. 705-714 · Zbl 0396.65067
[14] Raviart, P.A., Thomas, J.M. (1977): Primal hybrid finite element methods for 2nd order elliptic equations. Math. Comput.31, 391-413 · Zbl 0364.65082
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