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Arbitrary-level hanging nodes and automatic adaptivity in the \(hp\)-FEM. (English) Zbl 1135.65394

Summary: We present a new automatic adaptivity algorithm for the \(hp\)-finite element method (FEM) which is based on arbitrary-level hanging nodes and local element projections. The method is very simple to implement compared to other existing \(hp\)-adaptive strategies, while its performance is comparable or superior. This is demonstrated on several numerical examples which include the \(L\)-shape domain problem, a problem with internal layer, and the Girkmann problem of linear elasticity. With appropriate simplifications, the proposed technique can be applied to standard lower-order and spectral finite element methods.

MSC:

65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
65N50 Mesh generation, refinement, and adaptive methods for boundary value problems involving PDEs
35J05 Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation
74B05 Classical linear elasticity
74S05 Finite element methods applied to problems in solid mechanics

Software:

par2Dhp
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Full Text: DOI

References:

[1] Aksoylu, B.; Bond, S.; Holst, M., An Odyssey into local refinement and multilevel preconditioning III: implementation and numerical experiments, SIAM J. Sci. Comput., 25, 478-498 (2003) · Zbl 1048.65104
[2] Babuška, I.; Griebel, M.; Pitkaranta, J., The problem of selecting the shape functions for \(p\)-type elements, Int. J. Num. Methods Eng., 28, 1891-1908 (1989) · Zbl 0705.73246
[3] Babuška, I.; Gui, W., The h, p and hp-versions of the finite element method in 1 dimension—Part I. The error analysis of the \(p\)-version, Numer. Math., 49, 577-612 (1986) · Zbl 0614.65088
[4] Babuška, I.; Gui, W., The h, p and hp-versions of the finite element method in 1 dimension—Part II. The error analysis of the \(h\) and hp-versions, Numer. Math., 49, 613-657 (1986) · Zbl 0614.65089
[5] Babuška, I.; Gui, W., The h, p and hp-versions of the finite element method in 1 dimension—Part III. The adaptive hp-version, Numer. Math., 49, 659-683 (1986) · Zbl 0614.65090
[6] Babuška, I.; Guo, B. Q., Approximation properties of the hp version of the finite element method, Comput. Methods Appl. Mech. Eng., 133, 319-346 (1996) · Zbl 0882.65096
[7] Babuška, I.; Szabo, B.; Katz, I. N., The \(p\)-version of the finite element method, SIAM J. Numer. Anal., 18, 515-545 (1981) · Zbl 0487.65059
[8] Babuska, I.; Izadpanah, K.; Szabo, B., The postprocessing technique in the finite element method. The theory and experience, (Kardestuncer, H., Unification of Finite Element Methods (1984), Elsevier Science Publishers B.V.: Elsevier Science Publishers B.V. North-Holland), 97-121 · Zbl 0547.73048
[9] Bauer, A. C.; Patra, A. K., Performance of parallel preconditioners for adaptive hp-FEM discretizations of incompressible flows, Commun. Numer. Methods Eng., 18, 305-313 (2002) · Zbl 1005.76054
[10] Bauer, A. C.; Patra, A. K., Robust and efficient domain decomposition preconditioners for adaptive hp finite element approximation for linear elasticity with and without discontinuous coefficients, Int. J. Numer. Methods Eng., 59, 337-364 (2004) · Zbl 1047.74052
[11] Bertolazzi, E., Discrete conservation and discrete maximum principle for elliptic PDEs, Math. Models Methods Appl. Sci., 8, 685-711 (1998) · Zbl 0939.65123
[12] Demkowicz, L.; Oden, J. T.; Rachowicz, W.; Hardy, O., Toward a universal hp-adaptive finite element strategy. Part 1: constrained approximation and data structure, Comput. Methods Appl. Math. Eng., 77, 79-112 (1989) · Zbl 0723.73074
[13] L. Demkowicz, W. Rachowicz, P. Devloo, A fully automatic hp; L. Demkowicz, W. Rachowicz, P. Devloo, A fully automatic hp
[14] Karniadakis, G. E.; Sherwin, S. J., Spectral/hp Element Methods for CFD (1999), Oxford University Press: Oxford University Press Oxford · Zbl 0954.76001
[15] Laszloffy, A.; Long, J.; Patra, A. K., Simple data management, scheduling and solution strategies for managing the irregularities in parallel adaptive hp finite element simulations, Parallel Comput., 26, 1765-1788 (2000) · Zbl 0948.68013
[16] Melenk, J. M., hp-Finite Element Methods for Singular Perturbations (2002), Springer-Verlag: Springer-Verlag Berlin, (Lecture Notes in Mathematics 1796) · Zbl 1021.65055
[17] Pardo, D.; Demkowicz, L., Integration of hp-adaptivity and a two grid solver for elliptic problems, Comput. Methods Appl. Mech. Eng., 195, 7/8 (2006) · Zbl 1093.65112
[18] M. Paszynski, J. Kurtz, L. Demkowicz, Parallel, fully automatic hp-adaptive 2D finite element package, TICAM Report 04-07, The University of Texas at Austin, 2004.; M. Paszynski, J. Kurtz, L. Demkowicz, Parallel, fully automatic hp-adaptive 2D finite element package, TICAM Report 04-07, The University of Texas at Austin, 2004. · Zbl 1093.65113
[19] Peano, A. G., Hierarchies of conforming finite elements for plane elasticity and plate bending, Comput. Math. Appl., 2, 211-224 (1976) · Zbl 0369.73071
[20] Rachowicz, W.; Demkowicz, L., An hp-adaptive finite element method for electromagnetics. Part II: A 3D implementation, Int. J. Numer. Methods Eng., 53, 147-180 (2002) · Zbl 0994.78012
[21] Schwab, C., \(p\)- and hp-Finite Element Methods, Theory and Applications to Solid and Fluid Mechanics (1998), Oxford University Press: Oxford University Press New York · Zbl 0910.73003
[22] Šolín, P., Partial Differential Equations and the Finite Element Method (2005), J. Wiley & Sons
[23] Šolín, P.; Demkowicz, L., Goal-oriented hp-adaptivity for elliptic problems, Comput. Methods Appl. Mech. Eng., 193, 449-468 (2004) · Zbl 1044.65082
[24] Šolín, P.; Segeth, K.; Doležel, I., Higher-Order Finite Element Methods (2003), Chapman & Hall/CRC Press: Chapman & Hall/CRC Press Boca Raton
[25] Šolín, P.; Vejchodský, T., On a weak discrete maximum principle for hp-FEM, J. Comput. Appl. Math., 209, 54-65 (2007) · Zbl 1153.35313
[26] Zítka, M.; Šolín, P.; Vejchodský, T.; Avila, F., Imposing orthogonality to hierarchic higher-order finite elements, Math. Comput. Simul., 76, 211-217 (2007) · Zbl 1135.65396
[27] Szabo, B.; Babuška, I., Finite Element Analysis (1991), J. Wiley & Sons: J. Wiley & Sons New York, p. 368
[28] T. Vejchodský, P. Šolín, Discrete maximum principle for higher-order finite elements in 1D, Math. Comput., November 2006, in press.; T. Vejchodský, P. Šolín, Discrete maximum principle for higher-order finite elements in 1D, Math. Comput., November 2006, in press.
[29] Vejchodský, T.; Šolín, P.; Zítka, M., Modular hp-FEM system HERMES and its application to the Maxwell’s equations, Math. Comput. Simul., 76, 223-228 (2007) · Zbl 1157.78356
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