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A posteriori error estimation for the Poisson equation with mixed Dirichlet/Neumann boundary conditions. (English) Zbl 1038.65114

Summary: The present work is devoted to the a posteriori error estimation for the Poisson equation with mixed Dirichlet/Neumann boundary conditions. Using the duality technique we derive a reliable and efficient a posteriori error estimator that measures the error in the energy norm. The estimator can be used in assessing the error of any approximate solution which belongs to the Sobolev space ${H}^{1}$, independently of the discretization method chosen.

Only two global constants appear in the definition of the estimator; both constants depend solely on the domain geometry, and the estimator is quite nonsensitive to the error in the constants evaluation. It is also shown how accurately the estimator captures the local error distribution, thus, creating a base for a justified adaptivity of an approximation.

##### MSC:
 65N15 Error bounds (BVP of PDE) 65N30 Finite elements, Rayleigh-Ritz and Galerkin methods, finite methods (BVP of PDE) 35J05 Laplacian operator, reduced wave equation (Helmholtz equation), Poisson equation