The h, p and h-p versions of the finite element method in 1 dimension. III. The adaptive h-p version. (English) Zbl 0614.65090

In this final paper [for part I and II see ibid. 49,577-612 and 613-657 (1986; reviewed above)] the authors analyze the theoretical aspects of the adaptive h-p version of FEM and based on it they give concrete algorithms for the one-dimensional problem. They are concerned here only with the convergence and its rate in energy norm. Some numerical features and implementational aspects are presented too. The authors hope that the principles exposed in these three papers could be also successfully applied in the higher dimensional case. So in the final section they summarize the major properties of the three basic versions of the FEM.
Reviewer: C.-I.Gheorghiu


65L10 Numerical solution of boundary value problems involving ordinary differential equations
65L60 Finite element, Rayleigh-Ritz, Galerkin and collocation methods for ordinary differential equations
34B05 Linear boundary value problems for ordinary differential equations
Full Text: DOI EuDML


[1] Gui, W., Babuška, I.: Theh, p andh-p versions of, the Finite Element Method for One Dimensional Problem. Part I: The error Analysis of thep-Version Numer. Math.49, 577–612 (1986) · Zbl 0614.65088
[2] Gui, W., Babuška, I.: Theh, p andh-p Version of the Finite Element Method for One Dimensional Problem. Part II. The Error Analysis of theh andh-p Versions. Numer. Math.49, 613–657 (1986) · Zbl 0614.65089
[3] Babuška, I., Miller, A., Vogelius, M.: Adaptive Methods and Error Estimation for Elliptic Problems of Structural Mechanics. In: Adaptive Computational Methods for Patial Differential Equations (I. Babuška, J. Chandra, J.E. Flaherty, eds.) SIAM 33-56 (1983)
[4] Babuška, I., Vogelius, M.: Feedback and Adaptive Finite Element Solution for One-dimensional Boundary Value Problems. Numer. Math.44, 75–107 (1984) · Zbl 0574.65098
[5] Zienkiewicz, O.C., Craig, A.W.: Adaptive Mesh Refinement and A Posteriori Error Estimation for thep Version of the Finite Element Method. In: Adaptive Computational Method for P.D.E. (I. Babuška, J. Chandra, J.E. Flaherty, eds.). SIAM 33-56 (1983) · Zbl 0564.65070
[6] Mesztenyi, C., Szymczak, a W.: FEARS User’s Manual for UNIVAC 1100. Tech. Note BN991, IPST, University of Maryland, College Park 1982
[7] Bank, R.E.: PLTMG User’s Guide, Edition 4.0, Tech. Report Department of Mathematics, Univ. of UCSD 1985
[8] Babuška, I., Dorr, M.: Error Estimate for the Combinedh andp Versions of the Finite Element Method. Numer. Math.37, 257–277 (1981) · Zbl 0487.65058
[9] Weiser, A.: Local-Mesh, Local-Order, Adaptive Finite Element Methods with A Posteriori Error Estimates for Elliptic Partial Differential Equations. Ph.D. Thesis, Yale Univ. Tech. Report 213, 1981
[10] Traub, J.F., Wasilkowski, G.W., Wozniakowski, H.: Information, Uncertainty, Complexity. London: Addison-Wesley 1983
[11] Wasilkowski, G.W., Wozniakowski, H.: Can adaption help on the average? Numer. Math.44, 169–190 (1984) · Zbl 0555.65030
[12] Babuška, A.: Feedback, Adaptivity and A Posteriori Estimates in Finite Element Methods: Aims, Theory and Experience. In: Accuracy Estimates and Adaptive Refinements in Finite Element Computations. (I. Babuška, O.C. Zienkiewicz, J. Gago, E.R. de Oliveira, eds.), pp. 3–24. New York: John Wiley 1986
[13] Rheinbolt, W.C.: Feedback System and Adaptivity for Numerical Computations. In: Adaptive Computational Methods for Partial Differential Equations (I. Babuška, J. Chandra, J.E. Flaherty, eds.). SIAM 3–19 (1983)
[14] Babuška, I.: Thep andh-p Versions of the Finite Element Method. A Survey. In: Proceedings of the Workshop on Theory and Applications of Finite Elements (R. Voigt, M.Y. Hussaini, eds.). Berlin, Heidelberg, New York: Springer 1987
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