A note on a posteriori error analysis for dual mixed methods with mixed boundary conditions

Abstract

In this article, we give a description of a technique to develop an a posteriori error estimator for the dual mixed methods, when applied to elliptic partial differential equations with non homogeneous mixed boundary conditions. The approach considers conforming finite elements for the discrete scheme, and a quasi‐Helmholtz decomposition result to obtain a residual a posteriori error estimator. After applying first a homogenization technique (for the Neumann boundary condition), we derive an a posteriori error estimator, which looks to be expensive to compute. This motivates the derivation of another a posteriori error estimator, that is fully computable. As a consequence, we establish the equivalence between the latter a posteriori error estimator and the natural norm of the error, that is, we prove the reliability and local efficiency of the aforementioned estimator. Finally, we report numerical examples showing the good properties of the estimator, in agreement with the theoretical results of this work.