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

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.

### 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

Zbl 0875.65090
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