Abstract
We consider a family of hp-version discontinuous Galerkin finite element methods with least-squares stabilization for symmetric systems of first-order partial differential equations. The family includes the classical discontinuous Galerkin finite element method, with and without streamline-diffusion stabilization, as well as the discontinuous version of the Galerkin least-squares finite element method. An hp-optimal error bound is derived in the associated DG-norm. If the solution of the problem is elementwise analytic, an exponential rate of convergence under p-refinement is proved. We perform numerical experiments both to illustrate the theoretical results and to compare the various methods within the family.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
REFERENCES
Barth, T. J. (2000). Simplified discontinuous Galerkin methods for systems of conservation laws with convex extension. In Cockburn, B., Karniadakis, G. E., and Shu, C.-W. (eds.), Discontinuous Galerkin Methods: Theory, Computation and Applications, Lecture Notes in Computational Science and Engineering, Vol. 11, Springer, Berlin, pp. 63–75.
Barth, T. J. (1999). Numerical methods for gas-dynamics systems on unstructured meshes. In Kröner, D., Ohlberger, M. and Rohde, C. (eds.), An Introduction to Recent Developments in Theory and Numerics of Conservation Laws, Lecture Notes in Computational Science and Engineering, Vol. 5, Springer, Berlin, pp. 195–285.
Bey, K. S., and Oden, J. T. (1996). hp-Version discontinuous Galerkin methods for hyperbolic conservation laws. Comput. Methods Appl. Mech. Engrg. 133, 259–286.
Bochev, P. B., and Gunzburger, M. D. (1998). Finite element methods of least-squares type. SIAM Rev. 40, 789–893.
Cockburn, B., Hou, S., and Shu, C.-W. (1990). TVB Runge-Kutta local projection discontinuous Galerkin finite elements for hyperbolic conservation laws. Math. Comp. 54, 545–581.
Cockburn, B., and Shu, C.-W. (1989). TVB Runge-Kutta local projection discontinuous Galerkin finite element method for scalar conservation laws II: General framework. Math. Comp. 52, 411–435.
Cockburn, B., and Shu, C.-W. (1991). The Runge-Kutta local projection P 1-discontinuous Galerkin method for scalar conservation laws. Modél. Math. Anal. Numér. 25, 337–361.
Cockburn, B., and Shu, C.-W. (1998). The local discontinuous Galerkin method for time-dependent reaction-diffusion systems. SIAM J. Numer. Anal. 35, 2440–2463.
Cockburn, B., and Shu, C.-W. (1998). The Runge-Kutta discontinuous Galerkin method for conservation laws: Multidimensional systems. J. Comput. Phys. 141, 199–244.
Falk, R. S., and Richter, G. R. (1999). Explicit finite element methods for symmetric hyperbolic equations. SIAM J. Numer. Anal. 36, 935–952.
Houston, P., Mackenzie, J., Süli, E., and Warnecke, G. (1999). A posteriori error analysis for numerical approximations of Friedrichs systems. Numer. Math. 82(3), 433–470.
Houston, P., Schwab Ch., and Süli, E. (2000). Stabilized hp-finite element methods for first-order hyperbolic problems. SIAM J. Numer. Anal. 37, 1618–1643.
Houston, P., Schwab, Ch., and Süli, E. Discontinuous hp-Finite Element Methods for Advection-Diffusion-Reaction Problems. SIAM J. Numer. Anal. (accepted for publication). Available from the URL: http://web.comlab.ox.ac.uk/oucl/publications/natr/ na–00–15.html
Houston, P., and Süli, E. (2001). Stabilized hp-finite element approximation of partial differential equations with non-negative characteristic form. Computing 66, 99–119.
Hughes, T. J. R., and Brooks, A. A. (1979). A multidimensional upwind scheme with no crosswind diffusion. Finite element methods for convection dominated flows (Papers, Winter Ann. Meeting Amer. Soc. Mech. Engrs., New York, 1979), pp. 19–35, AMD, 34, Amer. Soc. Mech. Engrs. (ASME), New York.
Johnson, C., Nävert, U., and Pitkäranta, J. (1984). Finite element methods for linear hyperbolic problems. Comp. Meth. Appl. Mech. Engrg. 45, 285–312.
Johnson, C., and Pitkäranta, J. (1986). An analysis of the discontinuous Galerkin method for a scalar hyperbolic equation. Math. Comp. 46, 1–26.
Lesaint, P., and Raviart, P. A. (1974). On a finite element method for solving the neutron transport equation. In deBoor, C. A. (ed.), Mathematical Aspects of Finite Elements in Partial Differential Equations, Academic Press, New York, pp. 89–123.
Richter, G. (1988). An optimal-order error estimate for the discontinuous Galerkin method. Math. Comp. 50, 75–88.
Reed, W. H., and Hill, T. R. (1973). Triangular Mesh Methods for the Neutron Transport Equation, Technical Report LA-UR–73–479, Los Alamos Scientific Laboratory, Los Alamos, New Mexico.
Schwab, Ch. (1998). p-and hp-Finite Element Methods. Theory and Applications to Solid and Fluid Mechanics, Oxford University Press, Oxford.
Süli, E., Houston, P., and Schwab, Ch. (2000). hp-Finite element methods for hyperbolic problems. In Whiteman, J. R. (ed.), The Mathematics of Finite Elements and Applications. MAFELAP X, Elsevier, Oxford, pp. 143–162.
Süli, E., Schwab, Ch., and Houston, P. (2000). hp-DGFEM for partial differential equations with nonnegative characteristic form. In Cockburn, B., Karniadakis, G. E., and Shu, C.-W. (eds.), Discontinuous Galerkin Methods: Theory, Computation and Applications, Lecture Notes in Computational Science and Engineering, Vol. 11, Springer, Berlin, pp. 221–230.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Houston, P., Jensen, M. & Süli, E. hp-Discontinuous Galerkin Finite Element Methods with Least-Squares Stabilization. Journal of Scientific Computing 17, 3–25 (2002). https://doi.org/10.1023/A:1015180009979
Issue Date:
DOI: https://doi.org/10.1023/A:1015180009979