zbMATH — the first resource for mathematics

Estimation of penalty parameters for symmetric interior penalty Galerkin methods. (English) Zbl 1141.65078
The numerical approximation of a linear elliptic problem by the discontinuous Galerkin method is considered. Computable lower bounds for the penalty parameters are presented.

65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
65N12 Stability and convergence of numerical methods for boundary value problems involving PDEs
35J25 Boundary value problems for second-order elliptic equations
Full Text: DOI
[1] D.N. Arnold, An interior penalty finite element method with discontinuous elements, Ph.D. Thesis, The University of Chicago, 1979.
[2] Arnold, D.N., An interior penalty finite element method with discontinuous elements, SIAM J. numer. anal., 19, 742-760, (1982) · Zbl 0482.65060
[3] 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, (2002) · Zbl 1008.65080
[4] Babus˘ka, I., The finite element method with penalty, Math. comput., 27, 221-228, (1973) · Zbl 0299.65057
[5] Babus˘ka, I.; Zlamal, M., Nonconforming elements in the finite element method with penalty, SIAM J. numer. anal., 10, 863-875, (1973) · Zbl 0237.65066
[6] Baker, G.A., Finite element methods for elliptic equations using nonconforming elements, Math. comput., 31, 137, 45-59, (1977) · Zbl 0364.65085
[7] Baker, G.A.; Jureidini, W.N.; Karakashian, O.A., Piecewise solenoidal vector fields and the Stokes problem, SIAM J. numer. anal., 27, 1466-1485, (1990) · Zbl 0719.76047
[8] Becker, R.; Hansbo, P.; Stenberg, R., A finite element method for domain decomposition with non-matching grids, M2AN math. model. numer. anal., 37, 209-225, (2003) · Zbl 1047.65099
[9] Ciarlet, P., The finite element method for elliptic problems, (1978), North-Holland Amsterdam
[10] B. Cockburn, G.E. Karniadakis, C.-W. Shu (Eds.), First International Symposium on Discontinuous Galerkin Methods, Lecture Notes in Computational Science and Engineering, vol. 11, Springer, Berlin, 2000.
[11] Dawson, C.; Sun, S.; Wheeler, M.F., Compatible algorithms for coupled flow and transport, Comput. meth. appl. mech. eng., 193, 2565-2580, (2004) · Zbl 1067.76565
[12] J. Douglas, T. Dupont, Interior penalty procedures for elliptic and parabolic Galerkin methods, Lecture Notes in Physics, vol. 58, Springer, Berlin, 1976.
[13] A. Ern, J.-L. Guermond, Theory and practice of finite elements, Applied Mathematical Sciences, vol. 159, Springer, Berlin, 2003.
[14] Girault, V.; Rivière, B.; Wheeler, M.F., A discontinuous Galerkin method with non-overlapping domain decomposition for the Stokes and navier – stokes problems, Math. comput., 74, 53-84, (2005) · Zbl 1057.35029
[15] Houston, P.; Schwab, C.; Süli, E., Discontinuous hp-finite element methods for advection – diffusion reaction problems, SIAM J. numer. anal., 39, 6, 2133-2163, (2002) · Zbl 1015.65067
[16] Karakashian, O.A.; Jureidini, W., A nonconforming finite element method for the stationary navier – stokes equations, SIAM J. numer. anal., 35, 93-120, (1998) · Zbl 0933.76047
[17] Kaya, S.; Rivière, B., A discontinuous subgrid eddy viscosity method for the time-dependent navier – stokes equations, SIAM J. numer. anal., 43, 4, 1572-1595, (2005) · Zbl 1096.76026
[18] Lee, J.R., The law of cosines in a tetrahedron, J. Korea soc. math. ed. ser. B: pure appl. math., 4, 1-6, (1997)
[19] Nitsche, J.A., Uber ein variationsprinzip zur losung von Dirichlet-problemen bei verwendung von teilraumen, die keinen randbedingungen unteworfen sind, Abh. math. sem. univ. Hamburg, 36, 9-15, (1971) · Zbl 0229.65079
[20] Rivière, B.; Girault, V., Discontinuous finite element methods for incompressible flows on subdomains with non-matching interfaces, Comput. meth. appl. mech. eng., 195, 3274-3292, (2006) · Zbl 1121.76038
[21] Rivière, 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) · Zbl 1010.65045
[22] Rivière, B.; Yotov, I., Locally conservative coupling of Stokes and Darcy flow, SIAM J. numer. anal., 42, 5, 1959-1977, (2005) · Zbl 1084.35063
[23] Sun, S.; Wheeler, M.F., Symmetric and nonsymmetric discontinuous Galerkin methods for reactive transport in porous media, SIAM J. numer. anal., 43, 1, 195-219, (2005) · Zbl 1086.76043
[24] Warburton, T.; Hesthaven, J.S., On the constants in hp-finite element trace inverse inequalities, Comput. meth. appl. mech. eng., 192, 2765-2773, (2003) · Zbl 1038.65116
[25] Wheeler, M.F., An elliptic collocation-finite element method with interior penalties, SIAM J. numer. anal., 15, 1, 152-161, (1978) · Zbl 0384.65058
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.