## Preprint 2024-17

## Tomás Barrios, Edwin Behrens, Rommel Bustinza, Jose M. Cascon:

### An a posteriori error estimator for an augmented variational formulation of the Brinkman problem with mixed boundary conditions and non-null source terms

### Abstract:

The aim of this work is the development of an a posteriori error analysis for the Brinkman problem with non-homogeneous mixed boundary conditions. In order to clarify the analysis, we first study, for simplicity, the model problem with null mixed boundary conditions. Then, we derive a suitable augmented variational formulation, based on the pseudo-stress and the velocity unknowns. This process involves the elimination of the pressure, which can be recovered once the system is solved. Applying known arguments, we can prove the unique solvability of the referred formulation, as well as of the corresponding Galerkin scheme. Moreover, we can establish the convergence of the method, when consider row-wise Raviart-Thomas elements to approximate the pseudo-stress in $H(\mathbf{div};\Omega)$, and continuous piecewise polynomials for the velocity. Then, we proceed to deduce an a posteriori error estimator, which results to be reliable and local efficient. To obtain this, we basically take into account the ellipticity of the bilinear form that defines our scheme. Since we do not require any type of Helmholtz decomposition of functions living in $H(\mathbf{div};\Omega)$, the analysis is valid for 2D and 3D. The novelty of the current work relies on how we deal with the case we have non homogeneous mixed boundary conditions. The strategy is to perform first a suitable lifting for the Neumann and Dirichlet data, respectively, in order to homogenize them, and apply the procedure introduced at the beginning. Then, we also can establish the well posedness of the augmented variational formulation, at continuous and discrete levels, as well as the convergence of the method and the derivation of an a posteriori error estimator. We point out that in this case, the corresponding estimator consists of two residual terms, and two oscillation terms related to the boundary data, which are not present when the boundary data are piecewise polynomials. In that situation, the a posteriori error estimator results to be reliable, and locally efficient. We include some numerical experiments, which are in agreement with the theoretical results we have obtained here.