Abstract
We present an algorithm, and a 2D implementation for a fully automatic hp-adaptive strategy for elliptic problems. Given a mesh, the next, optimally refined mesh, is determined by maximizing the rate of decrease of the hp-interpolation error for a reference solution. Numerical results confirm optimal, exponential convergence rates predicted by the theory of hp methods.
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REFERENCES
Ainsworth, M., and Senior, B. (1997). Aspects of an adaptive hp-finite element method: Adaptive strategy, conforming approximation, and efficient solvers. Comput. Methods Appl. Mech. Engrg. 150, 65–87.
Babuška, I., and Guo, B. Q. (1996). Approximation properties of the hp version of the finite element method. In Babuška, I., and Oden, J. T. (eds.), Computer Methods in Applied Mechanics and Engineering, Special Issue on p and hp-Methods, Vol. 133, pp. 319–346.
Babuška, I., Strouboulis, T., and Copps, K. (1997). hp Optimization of finite element approximations: Analysis of the optimal mesh sequences in one dimension. Comput. Methods Appl. Mech. Engrg. 150, 89–108.
Cugnon, F. (2000). Automatisation des Caculs Éléments Finis dans le Cadre de la Méthode p, Ph.D. Dissertation, Universite de Liege.
Demkowicz, L., Oden, J. T., and Devloo, Ph. (1985). On h-type mesh refinement strategy based on a minimization of interpolation error. Comput. Methods Appl. Mech. Engrg. 53, 67–89.
Demkowicz, L., and Oden, J. T. (1996). Application of hp-adaptive BE/FE methods to elastic scattering. Comput. Methods Appl. Mech. Engrg. 133(3/4), 287–318.
Demkowicz, L., Monk, P., Vardapetyan, L., and Rachowicz, W. (2000). De Rham diagram for hp finite element spaces. Mathematics and Computers with Applications 39(7/8), 29–38.
Demkowicz, L., and Babuška, I. (2001). Optimal p Interpolation Error Estimates for Edge Finite Elements of Variable Order in 2D, TICAM Report 01–11, submitted to SIAM J. Numer. Anal.
Demkowicz, L., and Pardo, D. The Ultimate Data Structure for Three Dimensional, Anisotropic hp Refinements, TICAM Report, in preparation.
Houston, P., and Suli, E. hp-Adaptive discontinuous Galerkin finite element methods for first order hyperbolic problems. SIAM J. Numer. Anal., to appear.
Ingerman, D., Druskin, V., and Knizhnerman, L. (2000). Optimal finite difference grids and rational approximations of the square root. I. Elliptic problems. Comm. Pure Appl. Math. 8, 1039–1066.
Liszka, T., Tworzydlo, W., Bass, J., Sharma, S., Westermann, T., and Yavari, B. (1997). ProPHLEX-An hp adaptive finite element kernel for solving coupled systems of partial differential equations in computational mechanics. Comput. Methods Appl. Mech. Engrg. 150, 251–271.
Melenk, J. M. (1997). On the robust exponential convergence of hp finite element methods for problems with boundary layers. IMA J. Numer. Anal. 17, 577–601.
Novotny, A. A., Pereira, J. T., Fancello, E. A., and de Barcellos, C. S. A fast hp adaptive finite element mesh design for 2D elliptic boundary value problems. Comput. Methods Appl. Mech. Engrg., in print.
Oden, J. T., Wu, W., and Ainsworth, M. (1995). Three step hp adaptive strategy for the incompressible Navier–Stokes equations. In Babuška, I., and Flahertyin, J. E. (eds.), Modeling, Mesh Generation and Adaptive Numerical Methods for Partial Differential Equations, IMA Minnesota.
Oden, J. T., Patra, A., and Feng, Y. (1992). An hp adaptive strategy. In Noor, A. K. (ed.), Adaptive Multilevel and Hierarchical Computational Strategies, ASME Publication, Vol. 157, pp. 23–46.
Patra, A., and Gupta, A. (2001). A systematic strategy for simultaneous adaptive hp finite element mesh modification using nonlinear programming. Comput. Methods Appl. Mech. Engrg. 190, 3797–3818.
Rachowicz, W., Oden, J. T., and Demkowicz, L. (1989). Toward a universal h-p adaptive finite element strategy, Part 3. Design of h-p meshes. Comput. Methods Appl. Mech. Engrg. 77, 181–212.
Schmidt, A., and Siebert, K. G. (2000). A posteriori estimators for the hp version of the finite element method in 1D. Appl. Numer. Math. 35, 143–66.
Schwab, Ch. (1998). p and hp-Finite Element Methods, Clarendon Press, Oxford.
Zumbusch, G. (1996). Simultaneous hp Adaption in Multilevel finite Elements, Ph.D. Dissertation, Freie Universtität Berlin, 1995, published by Shaker Verlag, Aachen.
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Demkowicz, L., Rachowicz, W. & Devloo, P. A Fully Automatic hp-Adaptivity. Journal of Scientific Computing 17, 117–142 (2002). https://doi.org/10.1023/A:1015192312705
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DOI: https://doi.org/10.1023/A:1015192312705