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Šolín, P.; Demkowicz, L.

Goal-oriented \(hp\)-adaptivity for elliptic problems. (English) Zbl 1044.65082

Comput. Methods Appl. Mech. Eng. 193, No. 6-8, 449-468 (2004).
Summary: We propose and test a fully automatic, goal-oriented \(hp\)-adaptive strategy for elliptic problems. The method combines two techniques: the standard goal-oriented adaptivity based on a simultaneous solution of a dual problem, and a recently proposed \(hp\)-strategy based on minimizing the projection-based interpolation error of a reference solution. The proposed strategy is illustrated with two numerical examples: Laplace equation in L-shape domain, and an axisymmetric Maxwell problem involving radiation of a loop antenna wrapped around a metallic cylinder into a conductive medium.

Cited in 36 Documents

MSC:

65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
35J05 Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation
35Q60 PDEs in connection with optics and electromagnetic theory
78A40 Waves and radiation in optics and electromagnetic theory
78A50 Antennas, waveguides in optics and electromagnetic theory
78M10 Finite element, Galerkin and related methods applied to problems in optics and electromagnetic theory

Keywords:

\(hp\) finite elements; \(hp\)-adaptivity; Dual problem; Goal-oriented adaptivity; elliptic problems; numerical examples: Laplace equation; Maxwell problem; radiation; loop antenna
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\textit{P. Šolín} and \textit{L. Demkowicz}, Comput. Methods Appl. Mech. Eng. 193, No. 6--8, 449--468 (2004; Zbl 1044.65082)
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References:

[1] Ainsworth, M.; Senior, B., Aspects of an hp-adaptive finite element method: adaptive strategy, conforming approximation and efficient solvers, Comput. methods appl. math. engrg., 150, 65-87, (1997) · Zbl 0906.73057
[2] I. Babuška, T. Strouboulis, K. Copps, S.K. Gangaraj, C.S. Upadhyay, A-Posteriori Error Estimation for Finite Element and Generalized Finite Element Method, TICAM Report 98-01, The University of Texas at Austin, 1998
[3] R. Becker, R. Rannacher, Weighted A-Posteriori Error Control in FE Method, ENUMATH-95, Paris, September 1995 · Zbl 0968.65083
[4] Becker, R.; Rannacher, R., A feedback approach to error control in finite elements methods: basic analysis and examples, East – west J. numer. math., 4, 237-264, (1996) · Zbl 0868.65076
[5] Cirak, F.; Ramm, E., A-posteriori error estimation and adaptivity for linear elasticity using the reciprocal theorem, Comput. methods appl. mech. engrg., 156, 351-362, (1998) · Zbl 0947.74062
[6] 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. engrg., 77, 79-112, (1989) · Zbl 0723.73074
[7] L. Demkowicz, I. Babuška, Optimal p Interpolation Error Estimates for Edge Finite Elements of Variable Order in 2D, TIC AM Report 01-11
[8] Demkowicz, L.; Rachowicz, W.; Devloo, Ph., A fully automatic hp-adaptivity, J. sci. comput., 17, 1-3, 127-155, (2002) · Zbl 0999.65121
[9] Demkowicz, L., Asymptotic convergence in finite and boundary element methods. part 1: theoretical results, Comput. math. appl., 27, 12, 69-84, (1994) · Zbl 0807.65058
[10] Houston, P.; Senior, B.; Süli, E., Sobolev regularity estimation for hp-adaptive finite element methods, (), 619-644 · Zbl 1043.65114
[11] Lovell, J.R., Finite element methods in resistivity logging, (1993), Delft University of Technology, pp. 38-45
[12] Oden, J.T.; Prudhomme, S., Goal-oriented error estimation and adaptivity for the finite element method, Comput. math. appl., 41, 735-756, (2001) · Zbl 0987.65110
[13] Paraschivoiu, M.; Patera, A.T., A hierarchical duality approach to bounds for the outputs of partial differential equations, Comput. methods appl. mech. engrg., 158, 389-407, (1998) · Zbl 0953.76054
[14] Peraire, J.; Patera, A.T., Bounds for linear-functional outputs of coercive partial differential equations: local indicators and adaptive refinement, (), 199-215
[15] Rachowicz, W.; Oden, J.T.; Demkowicz, L., Toward a universal hp-adaptive finite element strategy. part 3: design of hp meshes, Comput. methods appl. math. engrg., 77, 2, 181-212, (1989) · Zbl 0723.73076
[16] Rachowicz, W.; Demkowicz, L., An hp-adaptive finite element method for electromagnetics. part II: A 3D implementation, Int. J. numer. methods engrg., 53, 147-180, (2002) · Zbl 0994.78012
[17] Rannacher, R.; Stuttmeier, F.T., A-posteriori error control in finite element methods via duality techniques: application to perfect plasticity, Comput. mech., 21, 123-133, (1998) · Zbl 0910.73064
[18] Schwab, C., p- and hp-finite element methods, (1998), Clarendon Press Oxford
[19] Šolı́n, P.; Segeth, K.; Doležel, I., Higher-order finite element methods, (2003), Chapman & Hall/CRC Press Boca Raton
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