Enhanced accuracy by post-processing for finite element methods for hyperbolic equations

Authors:
Bernardo Cockburn, Mitchell Luskin, Chi-Wang Shu and Endre Süli

Journal:
Math. Comp. **72** (2003), 577-606

MSC (2000):
Primary 65M60, 65N30, 35L65

Published electronically:
November 20, 2002

MathSciNet review:
1954957

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Abstract | References | Similar Articles | Additional Information

Abstract: We consider the enhancement of accuracy, by means of a simple post-processing technique, for finite element approximations to transient hyperbolic equations. The post-processing is a convolution with a kernel whose support has measure of order one in the case of arbitrary unstructured meshes; if the mesh is locally translation invariant, the support of the kernel is a cube whose edges are of size of the order of only. For example, when polynomials of degree are used in the discontinuous Galerkin (DG) method, and the exact solution is globally smooth, the DG method is of order in the -norm, whereas the post-processed approximation is of order ; if the exact solution is in only, in which case no order of convergence is available for the DG method, the post-processed approximation converges with order in , where is a subdomain over which the exact solution is smooth. Numerical results displaying the sharpness of the estimates are presented.

**1.**Slimane Adjerid, Mohammed Aiffa, and Joseph E. Flaherty,*Computational methods for singularly perturbed systems*, Analyzing multiscale phenomena using singular perturbation methods (Baltimore, MD, 1998) Proc. Sympos. Appl. Math., vol. 56, Amer. Math. Soc., Providence, RI, 1999, pp. 47–83. MR**1718897**, 10.1090/psapm/056/1718897**2.**Slimane Adjerid, Mohammed Aiffa, and Joseph E. Flaherty,*High-order finite element methods for singularly perturbed elliptic and parabolic problems*, SIAM J. Appl. Math.**55**(1995), no. 2, 520–543. Perturbation methods in physical mathematics (Troy, NY, 1993). MR**1322771**, 10.1137/S0036139993269345**3.**M.Y.T. Apelkrans,*Some properties of difference schemes for hyperbolic equations with discontinuities and a new method with almost quadratic convergence*, Tech. Report 15A, Uppsala University, Dept. of Computer Science, 1969.**4.**L.A. Bales,*Some remarks on post-processing and negative norm estimates for approximations to nonsmooth solutions of hyperbolic equations*, Comm. Numer. Methods Engrg.**9**(1993), 701-710. CMP**93:17****5.**J. H. Bramble and A. H. Schatz,*Higher order local accuracy by averaging in the finite element method*, Math. Comp.**31**(1977), no. 137, 94–111. MR**0431744**, 10.1090/S0025-5718-1977-0431744-9**6.**Philip Brenner, Vidar Thomée, and Lars B. Wahlbin,*Besov spaces and applications to difference methods for initial value problems*, Lecture Notes in Mathematics, Vol. 434, Springer-Verlag, Berlin-New York, 1975. MR**0461121****7.**Hans Forrer and Marsha Berger,*Flow simulations on Cartesian grids involving complex moving geometries*, Hyperbolic problems: theory, numerics, applications, Vol. I (Zürich, 1998), Internat. Ser. Numer. Math., vol. 129, Birkhäuser, Basel, 1999, pp. 315–324. MR**1717201****8.**Bernardo Cockburn and Chi-Wang Shu,*TVB Runge-Kutta local projection discontinuous Galerkin finite element method for conservation laws. II. General framework*, Math. Comp.**52**(1989), no. 186, 411–435. MR**983311**, 10.1090/S0025-5718-1989-0983311-4**9.**Bernardo Cockburn and Chi-Wang Shu,*The local discontinuous Galerkin method for time-dependent convection-diffusion systems*, SIAM J. Numer. Anal.**35**(1998), no. 6, 2440–2463 (electronic). MR**1655854**, 10.1137/S0036142997316712**10.**Jim Douglas Jr.,*Superconvergence in the pressure in the simulation of miscible displacement*, SIAM J. Numer. Anal.**22**(1985), no. 5, 962–969. MR**799123**, 10.1137/0722058**11.**Bjorn Engquist and Björn Sjögreen,*The convergence rate of finite difference schemes in the presence of shocks*, SIAM J. Numer. Anal.**35**(1998), no. 6, 2464–2485. MR**1655855**, 10.1137/S0036142997317584**12.**R. P. Fedorenko,*Application of high-accuracy difference schemes to the numerical solution of hyperbolic equations*, Ž. Vyčisl. Mat. i Mat. Fiz.**2**(1962), 1122–1128 (Russian). MR**0148231****13.**David Gottlieb and Eitan Tadmor,*Recovering pointwise values of discontinuous data within spectral accuracy*, Progress and supercomputing in computational fluid dynamics (Jerusalem, 1984), Progr. Sci. Comput., vol. 6, Birkhäuser Boston, Boston, MA, 1985, pp. 357–375. MR**935160****14.**Bertil Gustafsson, Heinz-Otto Kreiss, and Joseph Oliger,*Time dependent problems and difference methods*, Pure and Applied Mathematics (New York), John Wiley & Sons, Inc., New York, 1995. A Wiley-Interscience Publication. MR**1377057****15.**Claes Johnson and Uno Nävert,*An analysis of some finite element methods for advection-diffusion problems*, Analytical and numerical approaches to asymptotic problems in analysis (Proc. Conf., Univ. Nijmegen, Nijmegen, 1980) North-Holland Math. Stud., vol. 47, North-Holland, Amsterdam-New York, 1981, pp. 99–116. MR**605502****16.**C. Johnson and J. Pitkäranta,*An analysis of the discontinuous Galerkin method for a scalar hyperbolic equation*, Math. Comp.**46**(1986), no. 173, 1–26. MR**815828**, 10.1090/S0025-5718-1986-0815828-4**17.**Boško S. Jovanović, Lav D. Ivanović, and Endre E. Süli,*Convergence of a finite-difference scheme for second-order hyperbolic equations with variable coefficients*, IMA J. Numer. Anal.**7**(1987), no. 1, 39–45. MR**967833**, 10.1093/imanum/7.1.39**18.**R.B. Lowrie,*Compact higher-order numerical methods for hyperbolic conservation laws*, Ph.D. thesis, University of Michigan, 1996.**19.**Andrew Majda, James McDonough, and Stanley Osher,*The Fourier method for nonsmooth initial data*, Math. Comp.**32**(1978), no. 144, 1041–1081. MR**501995**, 10.1090/S0025-5718-1978-0501995-4**20.**Andrew Majda and Stanley Osher,*Propagation of error into regions of smoothness for accurate difference approximations to hyperbolic equations*, Comm. Pure Appl. Math.**30**(1977), no. 6, 671–705. MR**0471345****21.**Michael S. Mock and Peter D. Lax,*The computation of discontinuous solutions of linear hyperbolic equations*, Comm. Pure Appl. Math.**31**(1978), no. 4, 423–430. MR**0468216****22.**Vidar Thomée,*High order local approximations to derivatives in the finite element method*, Math. Comp.**31**(1977), no. 139, 652–660. MR**0438664**, 10.1090/S0025-5718-1977-0438664-4**23.**Vidar Thomée,*Negative norm estimates and superconvergence in Galerkin methods for parabolic problems*, Math. Comp.**34**(1980), no. 149, 93–113. MR**551292**, 10.1090/S0025-5718-1980-0551292-5**24.**Lars B. Wahlbin,*Superconvergence in Galerkin finite element methods*, Lecture Notes in Mathematics, vol. 1605, Springer-Verlag, Berlin, 1995. MR**1439050**

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Additional Information

**Bernardo Cockburn**

Affiliation:
School of Mathematics, University of Minnesota, Minneapolis, Minnesota 55455

Email:
cockburn@math.umn.edu

**Mitchell Luskin**

Affiliation:
School of Mathematics, University of Minnesota, Minneapolis, Minnesota 55455

Email:
luskin@math.umn.edu

**Chi-Wang Shu**

Affiliation:
Division of Applied Mathematics, Brown University, Providence, Rhode Island 02912

Email:
shu@cfm.brown.edu

**Endre Süli**

Affiliation:
Oxford University Computing Laboratory, Wolfson Building, Parks Road, Oxford OX1 3QD, United Kingdom

Email:
Endre.Suli@comlab.ox.ac.uk

DOI:
https://doi.org/10.1090/S0025-5718-02-01464-3

Keywords:
Post-processing,
finite element methods,
hyperbolic problems

Received by editor(s):
November 14, 2000

Published electronically:
November 20, 2002

Additional Notes:
The first author was supported in part by NSF Grant DMS-9807491 and by the University of Minnesota Supercomputing Institute

The second author was supported in part by NSF Grant DMS 95-05077, by AFOSR Grant F49620-98-1-0433, by ARO Grant DAAG55-98-1-0335, by the Institute for Mathematics and its Applications, and by the Minnesota Supercomputing Institute

The third author was supported in part by ARO Grant DAAG55-97-1-0318 and DAAD19-00-1-0405, NSF Grant DMS-9804985, NASA Langley Grant NCC1-01035 and and Contract NAS1-97046 while this author was in residence at ICASE, NASA Langley Research Center, and by AFOSR Grant F49620-99-1-0077

The fourth author is grateful to the Institute for Mathematics and Its Applications at the University of Minnesota and the University of Minnesota Supercomputing Institute for their generous support

Article copyright:
© Copyright 2002
American Mathematical Society