Abstract
Three Galerkin methods using discontinuous approximation spaces are introduced to solve elliptic problems. The underlying bilinear form for all three methods is the same and is nonsymmetric. In one case, a penalty is added to the form and in another, a constraint on jumps on each face of the triangulation. All three methods are locally conservative and the third one is not restricted. Optimal a priori hp error estimates are derived for all three procedures.
Similar content being viewed by others
References
D.N. Arnold, An interior penalty finite element method with discontinuous elements, SIAM J. Numer. Anal. 19(4) (1982) 742–760.
I. Babuška and M. Suri, The h-p version of the finite element method with quasiuniform meshes, Math. Modeling Numer. Anal. 21(2) (1987) 199–238.
I. Babuška and M. Suri, The optimal convergence rates of the p version of the finite element method, SIAM J. Numer. Anal. 24(4) (1987).
C.E. Baumann, An h-p adaptive discontinuous finite element method for computational fluid dynamics, Ph.D. thesis, The University of Texas at Austin (1997).
J. Douglas and T. Dupont, Interior penalty procedures for elliptic and parabolic Galerkin methods, Lecture Notes in Physics, Vol. 58 (1976) pp. 207–216.
J.T. Oden, I. Babuška and C.E. Baumann, A discontinous hp finite element method for diffusion problems, J. Comput. Phys. 146 (1998) 491–519.
P. Percell and M.F. Wheeler, A local residual finite element procedure for elliptic equations, SIAM J. Numer. Anal. 15(4) (1978) 705–714.
M.F. Wheeler, An elliptic collocation-finite element method with interior penalties, SIAM J. Numer. Anal. 15(1) (1978) 152–161.
M.F. Wheeler and B.L. Darlow, Interior penalty Galerkin procedures for miscible displacement problems in porous media, Comput. Methods Nonlinear Mech. (1980) 485–506.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Rivière, B., Wheeler, M.F. & Girault, V. Improved energy estimates for interior penalty, constrained and discontinuous Galerkin methods for elliptic problems. Part I. Computational Geosciences 3, 337–360 (1999). https://doi.org/10.1023/A:1011591328604
Issue Date:
DOI: https://doi.org/10.1023/A:1011591328604