Mixed finite element methods for problems with Robin boundary conditions. (English) Zbl 1226.65094

Könnö, Juho; Schötzau, Dominik; Stenberg, Rolf

SIAM J. Numer. Anal. 49, No. 1, 285-308 (2011).

The authors derive new a priori and a posteriori error estimates for mixed finite element discretization of second-order elliptic problems with general Robin boundary conditions, parametrized by \(\varepsilon \geq 0\). It is observed that Robin conditions have been treated by J. E. Roberts and J.-M. Thomas [P. G. Ciarlet (ed.) et al., Handbook of numerical analysis. Volume II: Finite element methods (Part 1). Amsterdam: North-Holland. 523–639 (1991; Zbl 0875.65090)], but the robustness with respect to the parameter has not been studied there. The authors prove both a priori and a posteriori error estimates that are uniformly valid, independently of the value of the parameter \(epsilon\). A solution approach based on hybridization is also discussed. Numerical results presented here show the optimality and sharpness of the estimates obtained and confirm their robustness.

Reviewer: H. P. Dikshit (Bhopal)

MSC:

65N30	Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
65N15	Error bounds for boundary value problems involving PDEs
35J25	Boundary value problems for second-order elliptic equations

Keywords:

mixed finite element methods; Robin boundary conditions; parametrized boundary conditions; a posteriori error estimates; post processing; second-order elliptic problems; numerical results; robustness

Citations:

Zbl 0875.65090