Analysis of a family of discontinuous Galerkin methods for elliptic problems: the one dimensional case. (English) Zbl 1064.65082
This paper deals with the properties of discontinuous Galerkin methods for elliptic problems in one spatial dimension. The analysis is based on a splitting of the discrete space into a direct sun of continuous piecewise polynomials and a space representing the discontinuous part of the functions also satisfying a special orthogonality relation. The authors prove stability results and \(L^2\) error estimates. Finally, some remarks on the elementwise conservative nature of the discontinuous Galerkin method are presented.
Reviewer: Pavol Chocholatý (Bratislava)
MSC:
| 65L60 | Finite element, Rayleigh-Ritz, Galerkin and collocation methods for ordinary differential equations |
| 65L10 | Numerical solution of boundary value problems involving ordinary differential equations |
| 65L20 | Stability and convergence of numerical methods for ordinary differential equations |
| 65L70 | Error bounds for numerical methods for ordinary differential equations |
| 34B05 | Linear boundary value problems for ordinary differential equations |
References:
| [1] | Arnold, D.N.: An interior penalty finite element method with discontinuous elements. SIAM J. Numer. Anal. 19, 742 (1982) · Zbl 0482.65060 · doi:10.1137/0719052 |
| [2] | Arnold, D.N., Brezzi, F., Cockburn, B., Marini, L.D.: Unified analysis of discontinuous Galerkin methods for elliptic problems. SIAM J. Numer. Anal. 39(5), 1749-1779 (2001) · Zbl 1008.65080 · doi:10.1137/S0036142901384162 |
| [3] | Babu?ka, I., Baumann, C.E., Oden, J.T.: A discontinuous hp finite element method for diffusion problems:1-D. Anal. Comput. Math. Appl. 37(9), 103-122 (1999) · Zbl 0940.65076 |
| [4] | Brenner, S.C., Scott, L.R.: The mathematical theory of finite element methods. Berlin, Springer-Verlag, 1994 · Zbl 0804.65101 |
| [5] | Cockburn, B., Karniadakis, K.E. Shu, C.-W.: Discontinuous Galerkin methods: theory computation and applications, volume 11 of Lecture Notes in Computational science and engineering, Springer Verlag, 2000. Papers from the 1st International Symposium held in Newport RI, May 24-26, 1999 |
| [6] | Nitsche, J.: ?ber ein Variationzprinzip zur L?sung von Dirichlet Problemen bei Verwendung von Teilr?umen. Abh. Math. Sem. Univ. Hamburg 36(9), (1971) |
| [7] | Oden, J.T., Babu?ka, I., Baumann, C.E.: A discontinuous hp finite element method for diffusion problems. J. Comput. Phys. 146, 491-519 (1998) · Zbl 0926.65109 · doi:10.1006/jcph.1998.6032 |
| [8] | Oden, J.T., Baumann, C.E.: A conservative DGM for convection-diffusion and Navier-Stokes problems. In: Discontinuous Galerkin methods (Newport RI 1999) volume 11 of Lect. Notes Comput. Sci. Eng. Springer, 2000 · Zbl 0988.76053 |
| [9] | Rivi?re, B., Wheeler, M.F., Girault, V.: Improved energy estimates for interior penalty constrained and discontinuous Galerkin methods for elliptic problems I. Comput. Geosci. 3(3-4), 337-360 (2000) · Zbl 0951.65108 |
| [10] | Rivi?re, B., Wheeler, M.F., Girault, V.: A priori error estimates for finite element methods based on discontinuous approximation spaces for elliptic problems. SIAM J. Numer. Anal. 39(3), 902-931 (2001) (electronic) · Zbl 1010.65045 · doi:10.1137/S003614290037174X |
| [11] | S?li, E., Schwab, Ch., Houston, P.: hp-DGFEM for partial differential equations with nonnegative characteristic form. In: Discontinuous Galerkin methods (Newport RI 1999), volume 11 of Lect. Notes Comput. Sci. Eng. Springer, 2000, pp. 221-230 · Zbl 0946.65102 |
| [12] | Wheeler, M.F.: An elliptic collocation-finite element method with interior penalties. SIAM J. Numer. Anal. 15, 152 (1978) · Zbl 0384.65058 · doi:10.1137/0715010 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
