Discontinuous Galerkin methods for Friedrichs’ systems. II: Second-order elliptic PDEs. (English) Zbl 1133.65098
The aim to give an unified analysis of discontinuous Galerkin methods based on Friedrichs’ system [cf. part I, A. Ern and J.-L. Guermond, SIAM J. Numer. Anal. 44, No. 2, 753–778 (2006; Zbl 1122.65111)] is continued in this second part specializing to two-field Friedrichs’ systems such that after partitioning of the dependent variable one variables can be eliminated at the continuous level as well as at the dicretized level of their approximation by means of discontinuous Galerkin methods.
To get these results new design criteria for the boundary and interface operators of the obtained system of second-order elliptic-like partial differential equations are developed. Convergence and error analysis are performed. Three important examples of two-field Friedrichs’ systems are discussed: advection-diffusion-reaction equations, linear continuum mechanics equations, and time-discretized magnetohydrodynamic equations. Finally it is shown that in the literature well-known discontinuous Galerkin methods for the Poisson equation are covered by the presented results.
To get these results new design criteria for the boundary and interface operators of the obtained system of second-order elliptic-like partial differential equations are developed. Convergence and error analysis are performed. Three important examples of two-field Friedrichs’ systems are discussed: advection-diffusion-reaction equations, linear continuum mechanics equations, and time-discretized magnetohydrodynamic equations. Finally it is shown that in the literature well-known discontinuous Galerkin methods for the Poisson equation are covered by the presented results.
Reviewer: Georg Hebermehl (Berlin)
MSC:
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 |
35F15 | Boundary value problems for linear first-order PDEs |
35J05 | Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation |
76W05 | Magnetohydrodynamics and electrohydrodynamics |
76M10 | Finite element methods applied to problems in fluid mechanics |