Dr. Barrios-Faundez, Tomas

In this work, we focus our attention in the Stokes flow with nonhomogeneous source terms, formulated in dual mixed form. For the sake of completeness, we begin recalling the corresponding well-posedness at continuous and discrete levels. After that, and with the help of a kind of a quasi-Helmholtz decomposition of functions in H (div), we develop a residual type a posteriori error analysis, deducing an estimator that is reliable and locally efficient. Finally, we provide numerical experiments, which confirm our theoretical results on the a posteriori error estimator and illustrate the performance of the corresponding adaptive algorithm, supporting its use in practice.

This article is concerned with the Stokes system with nonhomogeneous source terms and nonhomogeneous Dirichlet boundary condition. First, we reformulate the problem in its dual mixed form, and then, we study its corresponding well‐posedness. Next, in order to circumvent the well‐known Babuška‐Brezzi condition, we analyze a stabilized formulation of the resulting approach. Additionally, we endow the scheme with an a posteriori error estimator that is reliable and efficient. Finally, we provide numerical experiments that illustrate the performance of the corresponding adaptive algorithm and support its use in practice.

We consider the Oseen problem with nonhomogeneous Dirichlet boundary conditions on a part of the boundary and a Neumann type boundary condition on the remaining part. Suitable least squares terms that arise from the constitutive law, the momentum equation and the Dirichlet boundary condition are added to a dual-mixed formulation based on the pseudostress-velocity variables. We prove that the new augmented variational formulation and the corresponding Galerkin scheme are well-posed, and a Céa estimate holds for any finite element subspaces. We also provide the rate of convergence when each row of the pseudostress is approximated by Raviart–Thomas elements and the velocity is approximated by continuous piecewise polynomials. We develop an a posteriori error analysis based on a Helmholtz-type decomposition, and derive a posteriori error indicators that consist of two residual terms per element except on those elements with a side on the Dirichlet boundary, where they both have two additional terms. We prove that these a posteriori error indicators are reliable and locally efficient. Finally, we provide several numerical experiments that support the theoretical results.

We analyze a new stabilized dual-mixed method applied to incompressible linear elasticity problems, considering two kinds of data on the boundary of the domain: non homogeneous Dirichlet and mixed boundary conditions. In this approach, we circumvent the standard use of the rotation to impose weakly the symmetry of stress tensor. We prove that the new variational formulation and the corresponding Galerkin scheme are well-posed. We also provide the rate of convergence when each row of the stress is approximated by Raviart-Thomas elements and the displacement is approximated by continuous piecewise polynomials. Moreover, we derive a residual a posteriori error estimator for each situation. The corresponding analysis is quite different, depending on the type of boundary conditions. For known displacement on the whole boundary, we based our analysis on Ritz projection of the error, which requires a suitable quasi-Helmholtz decomposition of functions living in H (div; Ω). As a result, we obtain a simple a posteriori error estimator, which consists of five residual terms, and results to be reliable and locally efficient. On the other hand, when we consider mixed boundary conditions, these tools are not necessary. Then, we are able to develop an a posteriori error analysis, which provides us of an estimator consisting of three residual terms. In addition, we prove that in general this estimator is reliable, and when the traction datum is piecewise polynomial, locally efficient. In the second situation, we propose a numerical procedure to compute the numerical approximation, at a reasonable cost. Finally, we include several numerical experiments that illustrate the performance of the corresponding adaptive algorithm for each problem, and support its use in practice.