Uniform-in-time superconvergence of the HDG methods for the acoustic wave equation

Cockburn, Bernardo; Quenneville-Bélair, Vincent

doi:10.1090/S0025-5718-2013-02743-3

Uniform-in-time superconvergence of the HDG methods for the acoustic wave equation
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by Bernardo Cockburn and Vincent Quenneville-Bélair PDF

Math. Comp. 83 (2014), 65-85 Request permission

Abstract:

We present the first a priori error analysis of the hybridizable discontinuous Galerkin methods for the acoustic equation in the time-continuous case. We show that the velocity and the gradient converge with the optimal order of $k+1$ in the $L^2$-norm uniformly in time whenever polynomials of degree $k \geq 0$ are used. Finally, we show how to take advantage of this local postprocessing to obtain an approximation to the original scalar unknown also converging with order $k+2$ for $k \geq 1$. This puts on firm mathematical ground the numerical results obtained in J. Comput. Phys. 230 (2011), 3695–3718.

References

R. M. Alford, K. R. Kelly, and D. M. Boore, Accuracy of finite-difference modeling of the acoustic wave equation, Geophysics 39 (1974), no. 6, 834–842.
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
Peter Bastian and Béatrice Rivière, Superconvergence and $H(\textrm {div})$ projection for discontinuous Galerkin methods, Internat. J. Numer. Methods Fluids 42 (2003), no. 10, 1043–1057. MR 1991232, DOI 10.1002/fld.562
E. Bécache, P. Joly, and C. Tsogka, An analysis of new mixed finite elements for the approximation of wave propagation problems, SIAM J. Numer. Anal. 37 (2000), no. 4, 1053–1084. MR 1756415, DOI 10.1137/S0036142998345499
Oscar P. Bruno, New high-order integral methods in computational electromagnetism, CMES Comput. Model. Eng. Sci. 5 (2004), no. 4, 319–330. MR 2070392
Oscar P. Bruno and Michael C. Haslam, Efficient high-order evaluation of scattering by periodic surfaces: vector-parametric gratings and geometric singularities, Waves Random Complex Media 20 (2010), no. 4, 530–550. MR 2747419, DOI 10.1080/17455030.2010.499151
Fatih Celiker, Bernardo Cockburn, and Ke Shi, A projection-based error analysis of HDG methods for Timoshenko beams, Math. Comp. 81 (2012), no. 277, 131–151. MR 2833490, DOI 10.1090/S0025-5718-2011-02522-6
Fatih Celiker, Bernardo Cockburn, and Ke Shi, Hybridizable discontinuous Galerkin methods for Timoshenko beams, J. Sci. Comput. 44 (2010), no. 1, 1–37. MR 2647497, DOI 10.1007/s10915-010-9357-2
Brandon Chabaud and Bernardo Cockburn, Uniform-in-time superconvergence of HDG methods for the heat equation, Math. Comp. 81 (2012), no. 277, 107–129. MR 2833489, DOI 10.1090/S0025-5718-2011-02525-1
Yanlai Chen and Bernardo Cockburn, Analysis of variable-degree HDG methods for convection-diffusion equations. Part I: general nonconforming meshes, IMA J. Numer. Anal. 32 (2012), no. 4, 1267–1293. MR 2991828, DOI 10.1093/imanum/drr058
Yingda Cheng and Chi-Wang Shu, Superconvergence of discontinuous Galerkin and local discontinuous Galerkin schemes for linear hyperbolic and convection-diffusion equations in one space dimension, SIAM J. Numer. Anal. 47 (2010), no. 6, 4044–4072. MR 2585178, DOI 10.1137/090747701
Bernardo Cockburn and Jintao Cui, An analysis of HDG methods for the vorticity-velocity-pressure formulation of the Stokes problem in three dimensions, Math. Comp. 81 (2012), no. 279, 1355–1368. MR 2904582, DOI 10.1090/S0025-5718-2011-02575-5
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, Bo Dong, and Johnny Guzmán, A hybridizable and superconvergent discontinuous Galerkin method for biharmonic problems, J. Sci. Comput. 40 (2009), no. 1-3, 141–187. MR 2511731, DOI 10.1007/s10915-009-9279-z
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, Jayadeep Gopalakrishnan, Ngoc Cuong Nguyen, Jaume Peraire, and Francisco-Javier Sayas, Analysis of HDG methods for Stokes flow, Math. Comp. 80 (2011), no. 274, 723–760. MR 2772094, DOI 10.1090/S0025-5718-2010-02410-X
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, 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, Guido Kanschat, and Dominik Schotzau, A locally conservative LDG method for the incompressible Navier-Stokes equations, Math. Comp. 74 (2005), no. 251, 1067–1095. MR 2136994, DOI 10.1090/S0025-5718-04-01718-1
B. Cockburn, N. C. Nguyen, and J. Peraire, A comparison of HDG methods for Stokes flow, J. Sci. Comput. 45 (2010), no. 1-3, 215–237. MR 2679797, DOI 10.1007/s10915-010-9359-0
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
B. Cockburn and F.J. Sayas, Divergence-conforming HDG methods for Stokes flow, submitted.
Bernardo Cockburn and Ke Shi, Conditions for superconvergence of HDG methods for Stokes flow, Math. Comp. 82 (2013), no. 282, 651–671. MR 3008833, DOI 10.1090/S0025-5718-2012-02644-5
Ramon Codina, Finite element approximation of the hyperbolic wave equation in mixed form, Comput. Methods Appl. Mech. Engrg. 197 (2008), no. 13-16, 1305–1322. MR 2387006, DOI 10.1016/j.cma.2007.11.006
Gary Cohen and Patrick Joly, Construction analysis of fourth-order finite difference schemes for the acoustic wave equation in nonhomogeneous media, SIAM J. Numer. Anal. 33 (1996), no. 4, 1266–1302. MR 1403546, DOI 10.1137/S0036142993246445
David Colton and Rainer Kress, Inverse acoustic and electromagnetic scattering theory, 2nd ed., Applied Mathematical Sciences, vol. 93, Springer-Verlag, Berlin, 1998. MR 1635980, DOI 10.1007/978-3-662-03537-5
Lawrence C. Cowsar, Todd F. Dupont, and Mary F. Wheeler, A priori estimates for mixed finite element methods for the wave equation, Comput. Methods Appl. Mech. Engrg. 82 (1990), no. 1-3, 205–222. Reliability in computational mechanics (Austin, TX, 1989). MR 1077657, DOI 10.1016/0045-7825(90)90165-I
Lawrence C. Cowsar, Todd F. Dupont, and Mary F. Wheeler, A priori estimates for mixed finite element approximations of second-order hyperbolic equations with absorbing boundary conditions, SIAM J. Numer. Anal. 33 (1996), no. 2, 492–504. MR 1388485, DOI 10.1137/0733026
M. A. Dablain, The application of high-order differencing to the scalar wave equation, Geophysics 51 (1986), no. 1, 54–66.
Todd Dupont, $L^{2}$-estimates for Galerkin methods for second order hyperbolic equations, SIAM J. Numer. Anal. 10 (1973), 880–889. MR 349045, DOI 10.1137/0710073
Alexandre Ern and Jean-Luc Guermond, Theory and practice of finite elements, Applied Mathematical Sciences, vol. 159, Springer-Verlag, New York, 2004. MR 2050138, DOI 10.1007/978-1-4757-4355-5
Lawrence C. Evans, Partial differential equations, Graduate Studies in Mathematics, vol. 19, American Mathematical Society, Providence, RI, 1998. MR 1625845
Abdelaâziz Ezziani and Patrick Joly, Local time stepping and discontinuous Galerkin methods for symmetric first order hyperbolic systems, J. Comput. Appl. Math. 234 (2010), no. 6, 1886–1895. MR 2644184, DOI 10.1016/j.cam.2009.08.094
Richard S. Falk and Gerard R. Richter, Explicit finite element methods for symmetric hyperbolic equations, SIAM J. Numer. Anal. 36 (1999), no. 3, 935–952. MR 1688992, DOI 10.1137/S0036142997329463
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
Lucia Gastaldi and Ricardo H. Nochetto, Sharp maximum norm error estimates for general mixed finite element approximations to second order elliptic equations, RAIRO Modél. Math. Anal. Numér. 23 (1989), no. 1, 103–128 (English, with French summary). MR 1015921, DOI 10.1051/m2an/1989230101031
Tunc Geveci, On the application of mixed finite element methods to the wave equations, RAIRO Modél. Math. Anal. Numér. 22 (1988), no. 2, 243–250 (English, with French summary). MR 945124, DOI 10.1051/m2an/1988220202431
L. Greengard and V. Rokhlin, A fast algorithm for particle simulations, J. Comput. Phys. 73 (1987), no. 2, 325–348. MR 918448, DOI 10.1016/0021-9991(87)90140-9
Marcus J. Grote, Anna Schneebeli, and Dominik Schötzau, Discontinuous Galerkin finite element method for the wave equation, SIAM J. Numer. Anal. 44 (2006), no. 6, 2408–2431. MR 2272600, DOI 10.1137/05063194X
Eleanor W. Jenkins, Numerical solution of the acoustic wave equation using Raviart-Thomas elements, J. Comput. Appl. Math. 206 (2007), no. 1, 420–431. MR 2337454, DOI 10.1016/j.cam.2006.08.003
Eleanor W. Jenkins, Béatrice Rivière, and Mary F. Wheeler, A priori error estimates for mixed finite element approximations of the acoustic wave equation, SIAM J. Numer. Anal. 40 (2002), no. 5, 1698–1715. MR 1950619, DOI 10.1137/S0036142901388068
Heinz-Otto Kreiss, N. Anders Petersson, and Jacob Yström, Difference approximations of the Neumann problem for the second order wave equation, SIAM J. Numer. Anal. 42 (2004), no. 3, 1292–1323. MR 2113686, DOI 10.1137/S003614290342827X
Timo Lähivaara, Matti Malinen, Jari P. Kaipio, and Tomi Huttunen, Computational aspects of the discontinuous Galerkin method for the wave equation, J. Comput. Acoust. 16 (2008), no. 4, 507–530. MR 2523080, DOI 10.1142/S0218396X08003762
Yang Liu and Mrinal K. Sen, A new time-space domain high-order finite-difference method for the acoustic wave equation, J. Comput. Phys. 228 (2009), no. 23, 8779–8806. MR 2558777, DOI 10.1016/j.jcp.2009.08.027
Peter Monk and Gerard R. Richter, A discontinuous Galerkin method for linear symmetric hyperbolic systems in inhomogeneous media, J. Sci. Comput. 22/23 (2005), 443–477. MR 2142205, DOI 10.1007/s10915-004-4132-5
N. C. Nguyen, J. Peraire, and B. Cockburn, A hybridizable discontinuous Galerkin method for the incompressible Navier-Stokes equations (AIAA Paper 2010-362), Proceedings of the 48th AIAA Aerospace Sciences Meeting and Exhibit (Orlando, Florida), January 2010.
N. C. Nguyen, J. Peraire, and B. Cockburn, High-order implicit hybridizable discontinuous Galerkin methods for acoustics and elastodynamics, J. Comput. Phys. 230 (2011), no. 10, 3695–3718. MR 2783813, DOI 10.1016/j.jcp.2011.01.035
N. C. Nguyen, J. Peraire, and B. Cockburn, An implicit high-order hybridizable discontinuous Galerkin method for the incompressible Navier-Stokes equations, J. Comput. Phys. 230 (2011), no. 4, 1147–1170. MR 2753354, DOI 10.1016/j.jcp.2010.10.032
N. C. Nguyen, J. Peraire, and B. Cockburn, Hybridizable discontinuous Galerkin methods, Proceedings of the International Conference on Spectral and High Order Methods (Trondheim, Norway), Lect. Notes Comput. Sci. Engrg., Springer Verlag, June 2009.
N. C. Nguyen, J. Peraire, and B. Cockburn, An implicit high-order hybridizable discontinuous Galerkin method for linear convection-diffusion equations, J. Comput. Phys. 228 (2009), no. 9, 3232–3254. MR 2513831, DOI 10.1016/j.jcp.2009.01.030
N. C. Nguyen, J. Peraire, and B. Cockburn, An implicit high-order hybridizable discontinuous Galerkin method for nonlinear convection-diffusion equations, J. Comput. Phys. 228 (2009), no. 23, 8841–8855. MR 2558780, DOI 10.1016/j.jcp.2009.08.030
N. C. Nguyen, J. Peraire, and B. Cockburn, A hybridizable discontinuous Galerkin method for Stokes flow, Comput. Methods Appl. Mech. Engrg. 199 (2010), no. 9-12, 582–597. MR 2796169, DOI 10.1016/j.cma.2009.10.007
N. Nishimura, Fast multipole accelerated boundary integral equation methods, Applied Mechanics Reviews 55 (2002), no. 4, 299–324.
J. Peraire, N. C. Nguyen, and B. Cockburn, A hybridizable discontinuous Galerkin method for the compressible Euler and Navier-Stokes equations (AIAA Paper 2010-363), Proceedings of the 48th AIAA Aerospace Sciences Meeting and Exhibit (Orlando, Florida), January 2010.
Steffen Petersen, Charbel Farhat, and Radek Tezaur, A space-time discontinuous Galerkin method for the solution of the wave equation in the time domain, Internat. J. Numer. Methods Engrg. 78 (2009), no. 3, 275–295. MR 2542506, DOI 10.1002/nme.2485
P.-A. Raviart and J. M. Thomas, A mixed finite element method for 2nd order elliptic problems, Mathematical aspects of finite element methods (Proc. Conf., Consiglio Naz. delle Ricerche (C.N.R.), Rome, 1975) Lecture Notes in Math., Vol. 606, Springer, Berlin, 1977, pp. 292–315. MR 0483555
S.-C. Soon, Hybridizable discontinuous Galerkin methods for solid mechanics, Ph.D. thesis, University of Minnesota, 2008.
S.-C. Soon, B. Cockburn, and Henryk K. Stolarski, A hybridizable discontinuous Galerkin method for linear elasticity, Internat. J. Numer. Methods Engrg. 80 (2009), no. 8, 1058–1092. MR 2589528, DOI 10.1002/nme.2646
Rolf Stenberg, A family of mixed finite elements for the elasticity problem, Numer. Math. 53 (1988), no. 5, 513–538. MR 954768, DOI 10.1007/BF01397550
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
Jacek Szarski, Differential inequalities, Monografie Matematyczne, Tom 43, Państwowe Wydawnictwo Naukowe, Warsaw, 1965. MR 0190409
Liu Y. and Wei X., Finite-difference numerical modeling with even-order accuracy in two-phase anisotropic media, Applied Geophysics 5 (2008), 107–114.
Abraham Zemui, Fourth order symmetric finite difference schemes for the acoustic wave equation, BIT 45 (2005), no. 3, 627–651. MR 2189769, DOI 10.1007/s10543-005-0021-4