Discontinuous Galerkin finite element approximation of quasilinear elliptic boundary value problems I: The scalar case

IMA Journal of Numerical Analysis

Volumn 25, Issue 4, 2005, Pages 726-749

  Houston, Paul ^a   Robson, Janice ^b   Süli, Endre ^b  

^a UNIVERSITY OF LEICESTER   (United Kingdom)

^b UNIVERSITY OF OXFORD   (United Kingdom)

Author keywords

Discontinuous Galerkin methods; hp finite element methods; Quasilinear elliptic PDEs

Indexed keywords

BOUNDARY CONDITIONS; BOUNDARY VALUE PROBLEMS; FINITE ELEMENT METHOD; NUMERICAL METHODS;

DIRICHLET-NEUMANN BOUNDARY CONDITION; DISCONTINUOUS GALERKIN; DISCONTINUOUS GALERKIN FINITE-ELEMENT METHOD; DISCONTINUOUS GALERKIN METHODS; HP-FINITE ELEMENT METHODS; QUASI-LINEAR ELLIPTIC; QUASILINEAR ELLIPTIC EQUATIONS; REAL-VALUED FUNCTIONS;

GALERKIN METHODS;

EID: 26444484915     PISSN: 02724979     EISSN: 14643642     Source Type: Journal    
DOI: 10.1093/imanum/dri014     Document Type: Article

References (33)

1. ARNOLD, D. N. (1982) An interior penalty finite element method with discontinuous elements. SIAM J. Numer. Anal., 19, 742-760.
2. ARNOLD, D. N., BREZZI, F., COCKBURN, B. & MARINI, L. D. (2002) Unified analysis of discontinuous Galerkin methods for elliptic problems. SIAM J. Numer. Anal., 39, 1749-1779.
3. 2AN Math. Model. Numer. Anal., 21, 199-238.
4. BARRETT, J. W. & LIU, W. (1994) Quasi-error bounds for the finite element approximation of a non-Newtonian flow. Numer. Math., 68, 437-456.
5. BAUMANN, C. (1997) An hp-adaptive discontinuous Galerkin FEM for computational fluid dynamics. Doctoral Dissertation, TICAM, UT Austin, Texas.
6. BRAESS, D. (1997) Finite Elements. Theory, Fast Solvers, and Applications in Solid Mechanics. Cambridge University Press.
7. BUSTINZA, R. & GATICA, G. N. (2004) A local discontinuous Galerkin method for nonlinear diffusion problems with mixed boundary conditions. SIAM J. Sci. Comp., 26, 152-177.
8. CASTILLO, P., COCKBURN, B., PERUGIA, I. & SCHÖTZAU, D. (2000) An a priori error analysis of the local discontinuous Galerkin method for elliptic problems. SIAM J. Numer. Anal., 38, 1676-1706.
9. COCKBURN, B., HOU, S. & SHU, C.-W. (1990) TVB Runge-Kutta local projection discontinuous Galerkin finite elements for hyperbolic conservation laws. Math. Comp., 54, 545-581.
10. COCKBURN, B., KANSCHAT, G., PERUGIA, I. & SCHÖTZAU, D. (2001) Superconvergence of the local discontinuous Galerkin method for elliptic problems on Cartesian Grids. SIAM J. Numer. Anal., 39, 264-285.
11. COCKBURN, B., KARNIADAKIS, G. E. & SHU, C.-W. (2000) The development of discontinuous Galerkin methods. Discontinuous Galerkin Finite Element Methods. (B. Cockburn, G. E. Karniadakis & C.-W. Shu eds). Lecture Notes in Computational Science and Engineering, Vol. 11. Springer.
12. COCKBURN, B. & 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.
13. 2AN Math. Model. Numer. Anal., 25, 337-361.
14. COCKBURN, B. & SHU, C.-W. (1998a) The local discontinuous Galerkin method for time-dependent reaction-diffusion systems. SIAM J. Numer. Anal., 35, 2440-2463.
15. COCKBURN, B. & SHU, C.-W. (1998b) The Runge-Kutta discontinuous Galerkin method for conservation laws: multidimensional systems. J. Comput. Phys., 141, 199-244.
16. DAWSON, C. N., SUN, S. & WHEELER, M. F. (2004) Compatible algorithms for coupled flow and transport. Comput. Meth. Appl. Mech. Engng., 193, 2565-2580.
17. DOLEIŠÍ, V., FEISTAUER, M. & SOBOTÍKOVÁ, V. Analysis of the discontinuous Galerkin method for nonlinear convection-diffusion problems. Comput. Meth. Appl. Mech. Engng., (to appear).
18. FLAHERTY, J. E., LOY, R. M., SHEPHARD, M. S. & TERESCO, J. D. (2000) Software for parallel adaptive solution of conservation laws by discontinuous Galerkin methods. Discontinuous Galerkin Methods: Theory, Computation and Applications (B. Cockburn, G. E. Karniadakis & C.-W. Shu eds). Lecture Notes in Computational Science and Engineering, Vol. 11. Berlin: Springer, pp. 113-123.
19. GATICA, G. N., GONZÁLEZ, M. & MEDDAHI, S. (2004) A low-order mixed finite element method for a class of quasi-Newtonian Stokes flows. Part I: a-priori error analysis. Comput. Meth. Appl. Mech. Engng., 193, 881-892.
20. HARRIMAN, K., HOUSTON, P., SENIOR, B. & SÜLI, E. (2003) hp-version discontinuous Galerkin methods with interior penalty for partial differential equations with nonnegative characteristic form. Recent Advances in Scientific Computing and Partial Differential Equations. Contemporary Mathematics, vol. 330. (C.-W. Shu, T. Tang & S.-Y. Cheng eds). AMS, pp. 89-119.
21. HOUSTON, P., ROBSON, J. & SÜLI, E. Discontinuous Galerkin finite element approximation of quasilinear elliptic boundary value problems II: non-Newtonian flows. (In preparation).
22. HOUSTON, P., SCHWAB, CH. & SÜLI, E. (2002) Discontinuous hp-finite element methods for advection-diffusionreaction problems. SIAM J. Numer. Anal., 39, 2133-2163.
23. JOHNSON, C., NÄVERT, U. & PITKÄRANTA, J. (1984) Finite element methods for linear hyperbolic problems. Comp. Meth. Appl. Mech. Engng., 45, 285-312.
24. JOHNSON, C. & PITKÄRANTA, J. (1986) An analysis of the discontinuous Galerkin method for a scalar hyperbolic equation. Math. Comp., 46, 1-26.
25. KARNIADAKIS, G. E. & SHERWIN, S. (1999) Spectral/hp Finite Element Methods in CFD. Oxford University Press.
26. LESAINT, P. & RAVIART, P. A. (1974) On a finite element method for solving the neutron transport equation. Mathematical Aspects of Finite Elements in Partial Differential Equations (C. A. deBoor ed.). Academic Press, pp. 89-123.
27. NEČAS, J. (1986) Introduction to the Theory of Nonlinear Elliptic Equations. John Wiley and Sons.
28. NITSCHE, J. (1971) Über ein variationsprinzip zur lösung von dirichlet problemen bei verwendung von teilräumen, die keinen randbedingungen unterworfen sind. Abh. Math. Sem. Univ. Hamburg, 36, 9-15.
29. ODEN, J. T., BABUŠKA, I. & BAUMANN, C. (1998) A discontinuous hp-FEM for diffusion problems. J. Comput. Phys., 146, 491-519.
30. REED, W. H. & HILL, T. R. (1973) Triangular mesh methods for the neutron transport equation. Technical Report LA-UR-73-479 Los Alamos Scientific Laboratory.
31. RIVIÈRE, B., WHEELER, M. F. & GIRAULT, V. (1999) Improved energy estimates for interior penalty, constrained and discontinuous Galerkin methods for elliptic problems. Part I. Computational Geosciences 3 (1999) vol. 3-4. pp. 337-360.
32. SUN, S. (2003) Discontinuous Galerkin methods for reactive transport in porous media. Ph.D. Thesis, The University of Texas, Austin.
33. WHEELER, M. F. (1978) An elliptic collocation finite element method with interior penalties. SIAM J. Numer. Anal., 15, 152-161.