Abstract
We analyze the discretization errors of discontinuous Galerkin solutions of steady two-dimensional hyperbolic conservation laws on unstructured meshes. We show that the leading term of the error on each element is a linear combination of orthogonal polynomials of degrees p and p+1. We further show that there is a strong superconvergence property at the outflow edge(s) of each element where the average discretization error converges as O(h 2p+1) compared to a global rate of O(h p+1). Our analyses apply to both linear and nonlinear conservation laws with smooth solutions. We show how to use our theory to construct efficient and asymptotically exact a posteriori discretization error estimates and we apply these to some examples.
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References
M. Abramowitz and I. Stegun, eds., Handbook of Mathematical Functions (Dover, New York, 1965).
S. Adjerid, K. Devine, J. Flaherty and L. Krivodonova, A posteriori error estimation for discontinuous Galerkin solutions of hyperbolic problems, Computer Methods Appl. Mech. Engrg. 191 (2002) 1097–1112.
M. Ainsworth and J. Oden, A Posteriori Error Estimation in Finite Element Analysis, Computational and Applied Mathematics (Wiley-Interscience, New York, 2000).
O. Axelsson and V. Barker, Finite Element Solution of Boundary Value Problems: Theory and Computation (Academic Press, Orlando, FL 1984).
I. Babuška and. Strouboulis, Finite Element Method and its Reliability (Oxford Univ. Press, Oxford, 2001).
R. Biswas, K. Devine and J. Flaherty, Parallel adaptive finite element methods for conservation laws, Appl. Numer. Math. 14 (1994) 255–284.
B. Cockburn, A simple introduction to error estimation for nonlinear hyperbolic conservation laws. Some ideas, techiques and promising results, in: Proc.of the 1998 EPSRC Summer School in Numerical Analysis, SSCM, The Graduate Student's Guide to Numerical Analysis (Springer, Berlin, 1999) pp. 1–46.
B. Cockburn and P. Gremaud, Error estimates for finite element methods for nonlinear conservation laws, SIAM J. Numer. Anal. 33 (1996) 522–554.
J. Flaherty, L. Krivodonova, J.-F. Remacle and M. Shephard, Some aspects of dicontinuous Galerkin methods for hyperbolic conservation laws, Finite Elements in Analysis and Design 38 (2002) 889–908.
J. Flaherty, R. Loy, M. Shephard, B. Szymanski, J. Teresco and L. Ziantz, Adaptive local refinement with octree load-balancing for the parallel solution of three-dimensional conservation laws, J. Parallel Distributed Comput. 47 (1997) 139–152.
J. Flaherty, R. Loy, M. Shephard and J. Teresco, Software for the parallel adaptive solution of conservation laws by discontinuous Galerkin methods, in: Discontinous Galerkin Methods: Theory, Computation and Applications, eds. B. Cockburn, G. Karniadakis and S.-W. Shu, Lecture Notes in Computational Science and Engineering, Vol. 11 (Springer, Berlin, 2000) pp. 113–124.
P. Houston, E. Süli and C. Schwab, Stabilized hp-finite element methods for first-order hyperbolic problems, SIAM J. Numer. Anal. 37 (2000) 1618–1643.
M. Larson and T. Barth, A posteriori error estimation for discontinuous Galerkin approximations of hyperbolic systems, in: Discontinous Galerkin Methods: Theory, Computation and Applications, eds. B. Cockburn, G. Karniadakis and C.-W. Shu, Lecture Notes in Computational Science and Engineering, Vol. 11 (Springer, Berlin, 2000) pp. 363–368.
N. Pierce and M. Giles, Adjoint recovery of superconvergent functionals from PDE approximation, SIAM Rev. 42 (2000) 247–264.
W. Reed and T. Hill, Triangular mesh methods for the neutron transport equation, Technical Report LA-UR-73-479, Los Alamos Scientific Laboratory, Los Alamos (1973).
J.-F. Remacle, J. Flaherty and M. Shephard, An adaptive discontinuous Galerkin technique with an orthogonal basis applied to compressible flow problems, SIAM Rev. (2000) submitted.
J.-F. Remacle, J. Flaherty and M. Shephard, An efficient local time stepping scheme for transient adaptive computation (2001) in preparation.
P. L. Saint and P. Raviart, On a finite element method for solving the neutron transport equation, in: Mathematical Aspects of Finite Elements in Partial Differential Equations, ed. C. de Boor(Academic Press, New York, 1974) pp. 89–145.
E. Süli, A posteriori error analysis and adaptivity for finite element approximations of hyperbolic problems, in: An Introduction to Recent Developments in Theory and Numerics for Conservation Laws, eds. D. Kroner, M. Ohlberger and C. Rhode, Lecture Notes in Computational Science and Engineering, Vol. 5 (Springer, Berlin, 1999) pp. 123–194.
E. Süli, C. Schwab and P. Houston, Hp-DGFEM for partial differential equations with non-negative characteristic form, in: Discontinous Galerkin Methods: Theory, Computation and Applications, eds. B. Cockburn, G. Karniadakis and C.-W. Shu, Lecture Notes in Computational Science and Engineering, Vol. 11 (Springer, Berlin, 2000) pp. 221–230.
B. Szabó and I. Babuška, Finite Element Analysis (Wiley, New York, 1991).
R. Verfürth, A Review of a Posteriori Error Estimation and Adaptive Mesh-Refinement Techniques (Teubner, Stuttgart, 1996).
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Krivodonova, L., Flaherty, J.E. Error Estimation for Discontinuous Galerkin Solutions of Two-Dimensional Hyperbolic Problems. Advances in Computational Mathematics 19, 57–71 (2003). https://doi.org/10.1023/A:1022894504834
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DOI: https://doi.org/10.1023/A:1022894504834