Adaptive discontinuous Galerkin approximations to fourth order parabolic problems
HTML articles powered by AMS MathViewer
- by Emmanuil H. Georgoulis and Juha M. Virtanen PDF
- Math. Comp. 84 (2015), 2163-2190 Request permission
Abstract:
An adaptive algorithm, based on residual type a posteriori indicators of errors measured in $L^\infty (L^2)$ and $L^2(L^2)$ norms, for a numerical scheme consisting of implicit Euler method in time and discontinuous Galerkin method in space for linear parabolic fourth order problems is presented. The a posteriori analysis is performed for convex domains in two and three space dimensions for local spatial polynomial degrees $r \ge 2$. The a posteriori estimates are then used within an adaptive algorithm, highlighting their relevance in practical computations, by resulting in substantial reduction of computational effort.References
- Slimane Adjerid, A posteriori error estimates for fourth-order elliptic problems, Comput. Methods Appl. Mech. Engrg. 191 (2002), no. 23-24, 2539–2559. MR 1902705, DOI 10.1016/S0045-7825(01)00412-1
- Garth A. Baker, Error estimates for finite element methods for second order hyperbolic equations, SIAM J. Numer. Anal. 13 (1976), no. 4, 564–576. MR 423836, DOI 10.1137/0713048
- Garth A. Baker, Finite element methods for elliptic equations using nonconforming elements, Math. Comp. 31 (1977), no. 137, 45–59. MR 431742, DOI 10.1090/S0025-5718-1977-0431742-5
- Ľubomír Baňas and Robert Nürnberg, A posteriori estimates for the Cahn-Hilliard equation with obstacle free energy, M2AN Math. Model. Numer. Anal. 43 (2009), no. 5, 1003–1026. MR 2559742, DOI 10.1051/m2an/2009015
- L. Beirão da Veiga, J. Niiranen, and R. Stenberg, A posteriori error estimates for the Morley plate bending element, Numer. Math. 106 (2007), no. 2, 165–179. MR 2291934, DOI 10.1007/s00211-007-0066-1
- Dietrich Braess, Finite elements, 2nd ed., Cambridge University Press, Cambridge, 2001. Theory, fast solvers, and applications in solid mechanics; Translated from the 1992 German edition by Larry L. Schumaker. MR 1827293
- Susanne C. Brenner, Thirupathi Gudi, and Li-yeng Sung, An a posteriori error estimator for a quadratic $C^0$-interior penalty method for the biharmonic problem, IMA J. Numer. Anal. 30 (2010), no. 3, 777–798. MR 2670114, DOI 10.1093/imanum/drn057
- Susanne C. Brenner and L. Ridgway Scott, The mathematical theory of finite element methods, 2nd ed., Texts in Applied Mathematics, vol. 15, Springer-Verlag, New York, 2002. MR 1894376, DOI 10.1007/978-1-4757-3658-8
- Susanne C. Brenner and Li-Yeng Sung, $C^0$ interior penalty methods for fourth order elliptic boundary value problems on polygonal domains, J. Sci. Comput. 22/23 (2005), 83–118. MR 2142191, DOI 10.1007/s10915-004-4135-7
- Susanne C. Brenner, Kening Wang, and Jie Zhao, Poincaré-Friedrichs inequalities for piecewise $H^2$ functions, Numer. Funct. Anal. Optim. 25 (2004), no. 5-6, 463–478. MR 2106270, DOI 10.1081/NFA-200042165
- Franco Brezzi and Michel Fortin, Mixed and hybrid finite element methods, Springer Series in Computational Mathematics, vol. 15, Springer-Verlag, New York, 1991. MR 1115205, DOI 10.1007/978-1-4612-3172-1
- C. Carstensen and Jun Hu, A posteriori error analysis for conforming MITC elements for Reissner-Mindlin plates, Math. Comp. 77 (2008), no. 262, 611–632. MR 2373172, DOI 10.1090/S0025-5718-07-02028-5
- Alain Charbonneau, Kokou Dossou, and Roger Pierre, A residual-based a posteriori error estimator for the Ciarlet-Raviart formulation of the first biharmonic problem, Numer. Methods Partial Differential Equations 13 (1997), no. 1, 93–111. MR 1426319, DOI 10.1002/(SICI)1098-2426(199701)13:1<93::AID-NUM7>3.3.CO;2-G
- Philippe G. Ciarlet, The finite element method for elliptic problems, Classics in Applied Mathematics, vol. 40, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2002. Reprint of the 1978 original [North-Holland, Amsterdam; MR0520174 (58 #25001)]. MR 1930132, DOI 10.1137/1.9780898719208
- Philippe Destuynder and Michel Salaun, Mathematical analysis of thin plate models, Mathématiques & Applications (Berlin) [Mathematics & Applications], vol. 24, Springer-Verlag, Berlin, 1996 (English, with French summary). MR 1422248, DOI 10.1007/978-3-642-51761-7
- Todd Dupont and Ridgway Scott, Polynomial approximation of functions in Sobolev spaces, Math. Comp. 34 (1980), no. 150, 441–463. MR 559195, DOI 10.1090/S0025-5718-1980-0559195-7
- G. Engel, K. Garikipati, T. J. R. Hughes, M. G. Larson, L. Mazzei, and R. L. Taylor, Continuous/discontinuous finite element approximations of fourth-order elliptic problems in structural and continuum mechanics with applications to thin beams and plates, and strain gradient elasticity, Comput. Methods Appl. Mech. Engrg. 191 (2002), no. 34, 3669–3750. MR 1915664, DOI 10.1016/S0045-7825(02)00286-4
- Kenneth Eriksson and Claes Johnson, Adaptive finite element methods for parabolic problems. I. A linear model problem, SIAM J. Numer. Anal. 28 (1991), no. 1, 43–77. MR 1083324, DOI 10.1137/0728003
- Kenneth Eriksson and Claes Johnson, Adaptive finite element methods for parabolic problems. II. Optimal error estimates in $L_\infty L_2$ and $L_\infty L_\infty$, SIAM J. Numer. Anal. 32 (1995), no. 3, 706–740. MR 1335652, DOI 10.1137/0732033
- Kenneth Eriksson and Claes Johnson, Adaptive finite element methods for parabolic problems. IV. Nonlinear problems, SIAM J. Numer. Anal. 32 (1995), no. 6, 1729–1749. MR 1360457, DOI 10.1137/0732078
- Kenneth Eriksson and Claes Johnson, Adaptive finite element methods for parabolic problems. V. Long-time integration, SIAM J. Numer. Anal. 32 (1995), no. 6, 1750–1763. MR 1360458, DOI 10.1137/0732079
- Kenneth Eriksson, Claes Johnson, and Stig Larsson, Adaptive finite element methods for parabolic problems. VI. Analytic semigroups, SIAM J. Numer. Anal. 35 (1998), no. 4, 1315–1325. MR 1620144, DOI 10.1137/S0036142996310216
- Alexandre Ern and Martin Vohralík, A posteriori error estimation based on potential and flux reconstruction for the heat equation, SIAM J. Numer. Anal. 48 (2010), no. 1, 198–223. MR 2608366, DOI 10.1137/090759008
- Xiaobing Feng and Ohannes A. Karakashian, Two-level non-overlapping Schwarz preconditioners for a discontinuous Galerkin approximation of the biharmonic equation, J. Sci. Comput. 22/23 (2005), 289–314. MR 2142199, DOI 10.1007/s10915-004-4141-9
- Xiaobing Feng and Ohannes A. Karakashian, Fully discrete dynamic mesh discontinuous Galerkin methods for the Cahn-Hilliard equation of phase transition, Math. Comp. 76 (2007), no. 259, 1093–1117. MR 2299767, DOI 10.1090/S0025-5718-07-01985-0
- Xiaobing Feng and Haijun Wu, A posteriori error estimates for finite element approximations of the Cahn-Hilliard equation and the Hele-Shaw flow, J. Comput. Math. 26 (2008), no. 6, 767–796. MR 2464735
- Emmanuil H. Georgoulis and Paul Houston, Discontinuous Galerkin methods for the biharmonic problem, IMA J. Numer. Anal. 29 (2009), no. 3, 573–594. MR 2520159, DOI 10.1093/imanum/drn015
- Emmanuil H. Georgoulis, Paul Houston, and Juha Virtanen, An a posteriori error indicator for discontinuous Galerkin approximations of fourth-order elliptic problems, IMA J. Numer. Anal. 31 (2011), no. 1, 281–298. MR 2755946, DOI 10.1093/imanum/drp023
- Emmanuil H. Georgoulis, Omar Lakkis, and Juha M. Virtanen, A posteriori error control for discontinuous Galerkin methods for parabolic problems, SIAM J. Numer. Anal. 49 (2011), no. 2, 427–458. MR 2784879, DOI 10.1137/080722461
- P. Grisvard, Elliptic problems in nonsmooth domains, Monographs and Studies in Mathematics, vol. 24, Pitman (Advanced Publishing Program), Boston, MA, 1985. MR 775683
- Thirupathi Gudi, Residual-based a posteriori error estimator for the mixed finite element approximation of the biharmonic equation, Numer. Methods Partial Differential Equations 27 (2011), no. 2, 315–328. MR 2752872, DOI 10.1002/num.20524
- Ohannes A. Karakashian and Frederic Pascal, Convergence of adaptive discontinuous Galerkin approximations of second-order elliptic problems, SIAM J. Numer. Anal. 45 (2007), no. 2, 641–665. MR 2300291, DOI 10.1137/05063979X
- Omar Lakkis and Charalambos Makridakis, Elliptic reconstruction and a posteriori error estimates for fully discrete linear parabolic problems, Math. Comp. 75 (2006), no. 256, 1627–1658. MR 2240628, DOI 10.1090/S0025-5718-06-01858-8
- Omar Lakkis and Tristan Pryer, Gradient recovery in adaptive finite-element methods for parabolic problems, IMA J. Numer. Anal. 32 (2012), no. 1, 246–278. MR 2875251, DOI 10.1093/imanum/drq019
- S. Larsson, and A. Mesforush, A posteriori error analysis for the Cahn-Hilliard equation, Chalmers University of Technology Preprint 2010:19 (2010).
- Charalambos Makridakis and Ricardo H. Nochetto, Elliptic reconstruction and a posteriori error estimates for parabolic problems, SIAM J. Numer. Anal. 41 (2003), no. 4, 1585–1594. MR 2034895, DOI 10.1137/S0036142902406314
- Igor Mozolevski and Endre Süli, A priori error analysis for the $hp$-version of the discontinuous Galerkin finite element method for the biharmonic equation, Comput. Methods Appl. Math. 3 (2003), no. 4, 596–607. MR 2048235, DOI 10.2478/cmam-2003-0037
- P. Neittaanmäki and S. I. Repin, A posteriori error estimates for boundary-value problems related to the biharmonic operator, East-West J. Numer. Math. 9 (2001), no. 2, 157–178. MR 1836871
- B. Rivière and M. F. Wheeler, A posteriori error estimates for a discontinuous Galerkin method applied to elliptic problems. Log number: R74, Comput. Math. Appl. 46 (2003), no. 1, 141–163. $p$-FEM2000: $p$ and $hp$ finite element methods—mathematics and engineering practice (St. Louis, MO). MR 2015276, DOI 10.1016/S0898-1221(03)90086-1
- Roy H. Stogner and Graham F. Carey, $C^1$ macroelements in adaptive finite element methods, Internat. J. Numer. Methods Engrg. 70 (2007), no. 9, 1076–1095. MR 2322460, DOI 10.1002/nme.1912
- Endre Süli and Igor Mozolevski, $hp$-version interior penalty DGFEMs for the biharmonic equation, Comput. Methods Appl. Mech. Engrg. 196 (2007), no. 13-16, 1851–1863. MR 2298696, DOI 10.1016/j.cma.2006.06.014
- R. Verfürth, A Review of A Posteriori Error Estimation and Adaptive Mesh-Refinement Techniques, Wiley-Teubner, Chichester-Stuttgart, 1996.
- J. M. Virtanen, Adaptive discontinuous Galerkin methods for fourth order problems, Ph.D. Thesis, University of Leicester (2010).
Additional Information
- Emmanuil H. Georgoulis
- Affiliation: Department of Mathematics, University of Leicester, University Road, Leicester LE1 7RH, United Kingdom.
- Email: Emmanuil.Georgoulis@le.ac.uk
- Juha M. Virtanen
- Affiliation: Department of Mathematics, University of Leicester, University Road, Leicester LE1 7RH, United Kingdom
- Email: jv77@le.ac.uk
- Received by editor(s): March 7, 2013
- Received by editor(s) in revised form: October 28, 2013
- Published electronically: February 24, 2015
- © Copyright 2015 American Mathematical Society
- Journal: Math. Comp. 84 (2015), 2163-2190
- MSC (2010): Primary 65M15, 65M50, 65M60
- DOI: https://doi.org/10.1090/mcom/2936
- MathSciNet review: 3356023