Hybridizable discontinuous Galerkin and mixed finite element methods for elliptic problems on surfaces

Cockburn, Bernardo; Demlow, Alan

doi:10.1090/mcom/3093

Hybridizable discontinuous Galerkin and mixed finite element methods for elliptic problems on surfaces
HTML articles powered by AMS MathViewer

by Bernardo Cockburn and Alan Demlow PDF

Math. Comp. 85 (2016), 2609-2638 Request permission

Abstract:

We define and analyze hybridizable discontinuous Galerkin methods for the Laplace-Beltrami problem on implicitly defined surfaces. We show that the methods can retain the same convergence and superconvergence properties they enjoy in the case of flat surfaces. Special attention is paid to the relative effect of approximation of the surface and that introduced by discretizing the equations. In particular, we show that when the geometry is approximated by polynomials of the same degree of those used to approximate the solution, although the optimality of the approximations is preserved, the superconvergence is lost. To recover it, the surface has to be approximated by polynomials of one additional degree. We also consider mixed surface finite element methods as a natural part of our presentation. Numerical experiments verifying and complementing our theoretical results are shown.

References

P. Antonietti, A. Dedner, P. Madhavan, S. Stangalino, B. Stinner, and M. Verani, High order discontinuous Galerkin methods on surfaces, arXiv e-prints (2014).
Douglas N. Arnold, Daniele Boffi, and Richard S. Falk, Approximation by quadrilateral finite elements, Math. Comp. 71 (2002), no. 239, 909–922. MR 1898739, DOI 10.1090/S0025-5718-02-01439-4
Douglas N. Arnold, Daniele Boffi, and Richard S. Falk, Quadrilateral $H(\textrm {div})$ finite elements, SIAM J. Numer. Anal. 42 (2005), no. 6, 2429–2451. MR 2139400, DOI 10.1137/S0036142903431924
Douglas N. Arnold, Franco Brezzi, Bernardo Cockburn, and L. Donatella Marini, Unified analysis of discontinuous Galerkin methods for elliptic problems, SIAM J. Numer. Anal. 39 (2001/02), no. 5, 1749–1779. MR 1885715, DOI 10.1137/S0036142901384162
Douglas N. Arnold, Richard S. Falk, and Ragnar Winther, Finite element exterior calculus: from Hodge theory to numerical stability, Bull. Amer. Math. Soc. (N.S.) 47 (2010), no. 2, 281–354. MR 2594630, DOI 10.1090/S0273-0979-10-01278-4
W. Bangerth, R. Hartmann, and G. Kanschat, deal.II—a general-purpose object-oriented finite element library, ACM Trans. Math. Software 33 (2007), no. 4, Art. 24, 27. MR 2404402, DOI 10.1145/1268776.1268779
W. Bangerth, T. Heister, G. Kanschat et al., deal.II, Differential Equations Analysis Library, Technical Reference. http://www.dealii.org.
A. Bendali, Numerical analysis of the exterior boundary value problem for the time-harmonic Maxwell equations by a boundary finite element method. II. The discrete problem, Math. Comp. 43 (1984), no. 167, 47–68. MR 744924, DOI 10.1090/S0025-5718-1984-0744924-3
Andrea Bonito and Joseph E. Pasciak, Convergence analysis of variational and non-variational multigrid algorithms for the Laplace-Beltrami operator, Math. Comp. 81 (2012), no. 279, 1263–1288. MR 2904579, DOI 10.1090/S0025-5718-2011-02551-2
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
Snorre H. Christiansen, Discrete Fredholm properties and convergence estimates for the electric field integral equation, Math. Comp. 73 (2004), no. 245, 143–167. MR 2034114, DOI 10.1090/S0025-5718-03-01581-3
L. Chen, iFEM: An innovative finite element method package in Matlab, Technical Report, University of California-Irvine (2009).
Yanlai Chen and Bernardo Cockburn, Analysis of variable-degree HDG methods for convection-diffusion equations. Part II: Semimatching nonconforming meshes, Math. Comp. 83 (2014), no. 285, 87–111. MR 3120583, DOI 10.1090/S0025-5718-2013-02711-1
Bernardo Cockburn, Bo Dong, and Johnny Guzmán, A superconvergent LDG-hybridizable Galerkin method for second-order elliptic problems, Math. Comp. 77 (2008), no. 264, 1887–1916. MR 2429868, DOI 10.1090/S0025-5718-08-02123-6
Bernardo Cockburn, Jayadeep Gopalakrishnan, and Raytcho Lazarov, Unified hybridization of discontinuous Galerkin, mixed, and continuous Galerkin methods for second order elliptic problems, SIAM J. Numer. Anal. 47 (2009), no. 2, 1319–1365. MR 2485455, DOI 10.1137/070706616
Bernardo Cockburn, Johnny Guzmán, and Haiying Wang, Superconvergent discontinuous Galerkin methods for second-order elliptic problems, Math. Comp. 78 (2009), no. 265, 1–24. MR 2448694, DOI 10.1090/S0025-5718-08-02146-7
Bernardo Cockburn, Jayadeep Gopalakrishnan, and Francisco-Javier Sayas, A projection-based error analysis of HDG methods, Math. Comp. 79 (2010), no. 271, 1351–1367. MR 2629996, DOI 10.1090/S0025-5718-10-02334-3
Bernardo Cockburn, Weifeng Qiu, and Ke Shi, Conditions for superconvergence of HDG methods for second-order elliptic problems, Math. Comp. 81 (2012), no. 279, 1327–1353. MR 2904581, DOI 10.1090/S0025-5718-2011-02550-0
Bernardo Cockburn, Weifeng Qiu, and Ke Shi, Superconvergent HDG methods on isoparametric elements for second-order elliptic problems, SIAM J. Numer. Anal. 50 (2012), no. 3, 1417–1432. MR 2970749, DOI 10.1137/110840790
Bernardo Cockburn and Wujun Zhang, A posteriori error estimates for HDG methods, J. Sci. Comput. 51 (2012), no. 3, 582–607. MR 2914423, DOI 10.1007/s10915-011-9522-2
Bernardo Cockburn and Wujun Zhang, A posteriori error analysis for hybridizable discontinuous Galerkin methods for second order elliptic problems, SIAM J. Numer. Anal. 51 (2013), no. 1, 676–693. MR 3033028, DOI 10.1137/120866269
Andreas Dedner, Pravin Madhavan, and Björn Stinner, Analysis of the discontinuous Galerkin method for elliptic problems on surfaces, IMA J. Numer. Anal. 33 (2013), no. 3, 952–973. MR 3081490, DOI 10.1093/imanum/drs033
Alan Demlow, Higher-order finite element methods and pointwise error estimates for elliptic problems on surfaces, SIAM J. Numer. Anal. 47 (2009), no. 2, 805–827. MR 2485433, DOI 10.1137/070708135
Alan Demlow and Gerhard Dziuk, An adaptive finite element method for the Laplace-Beltrami operator on implicitly defined surfaces, SIAM J. Numer. Anal. 45 (2007), no. 1, 421–442. MR 2285862, DOI 10.1137/050642873
François Dubois, Discrete vector potential representation of a divergence-free vector field in three-dimensional domains: numerical analysis of a model problem, SIAM J. Numer. Anal. 27 (1990), no. 5, 1103–1141. MR 1061122, DOI 10.1137/0727065
Gerhard Dziuk, Finite elements for the Beltrami operator on arbitrary surfaces, Partial differential equations and calculus of variations, Lecture Notes in Math., vol. 1357, Springer, Berlin, 1988, pp. 142–155. MR 976234, DOI 10.1007/BFb0082865
Charles M. Elliott and Björn Stinner, Modeling and computation of two phase geometric biomembranes using surface finite elements, J. Comput. Phys. 229 (2010), no. 18, 6585–6612. MR 2660322, DOI 10.1016/j.jcp.2010.05.014
R. S. Falk and J. E. Osborn, Error estimates for mixed methods, RAIRO Anal. Numér. 14 (1980), no. 3, 249–277 (English, with French summary). MR 592753
David Gilbarg and Neil S. Trudinger, Elliptic partial differential equations of second order, Classics in Mathematics, Springer-Verlag, Berlin, 2001. Reprint of the 1998 edition. MR 1814364, DOI 10.1007/978-3-642-61798-0
Sven Groß, Volker Reichelt, and Arnold Reusken, A finite element based level set method for two-phase incompressible flows, Comput. Vis. Sci. 9 (2006), no. 4, 239–257. MR 2280477, DOI 10.1007/s00791-006-0024-y
R. Hiptmair and C. Schwab, Natural boundary element methods for the electric field integral equation on polyhedra, SIAM J. Numer. Anal. 40 (2002), no. 1, 66–86. MR 1921910, DOI 10.1137/S0036142901387580
Michael Holst and Ari Stern, Geometric variational crimes: Hilbert complexes, finite element exterior calculus, and problems on hypersurfaces, Found. Comput. Math. 12 (2012), no. 3, 263–293. MR 2915563, DOI 10.1007/s10208-012-9119-7
Robert M. Kirby, Spencer J. Sherwin, and Bernardo Cockburn, To CG or to HDG: a comparative study, J. Sci. Comput. 51 (2012), no. 1, 183–212. MR 2891951, DOI 10.1007/s10915-011-9501-7
U. Langer and S. E. Moore, Discontinuous Galerkin isogeometric analysis of elliptic pde on surfaces, Tech. Rep. 2014-01, Johann Radon Institute for Computational and Applied Mathematics, 2014.
K. Larsson and M. G. Larson, A continuous/discontinuous Galerkin method and a priori error estimates for the biharmonic problem on surfaces, arXiv e-prints (2013).
Khamron Mekchay, Pedro Morin, and Ricardo H. Nochetto, AFEM for the Laplace-Beltrami operator on graphs: design and conditional contraction property, Math. Comp. 80 (2011), no. 274, 625–648. MR 2772090, DOI 10.1090/S0025-5718-2010-02435-4
J.-C. Nédélec, Computation of eddy currents on a surface in $\textbf {R}^{3}$ by finite element methods, SIAM J. Numer. Anal. 15 (1978), no. 3, 580–594. MR 495761, DOI 10.1137/0715038
Maxim A. Olshanskii, Arnold Reusken, and Jörg Grande, A finite element method for elliptic equations on surfaces, SIAM J. Numer. Anal. 47 (2009), no. 5, 3339–3358. MR 2551197, DOI 10.1137/080717602
M. Reuter, F.-E. Wolter, M. Shenton, and M. Niethammer, Laplace-Beltrami eigenvalues and topological features of eigenfunctions for statistical shape analysis, Computer-Aided Design 41 (2009), 739–755.
Paul Steinmann, On boundary potential energies in deformational and configurational mechanics, J. Mech. Phys. Solids 56 (2008), no. 3, 772–800. MR 2394163, DOI 10.1016/j.jmps.2007.07.001
Rolf Stenberg, Postprocessing schemes for some mixed finite elements, RAIRO Modél. Math. Anal. Numér. 25 (1991), no. 1, 151–167 (English, with French summary). MR 1086845, DOI 10.1051/m2an/1991250101511