Dr. Gatica-Simpertigue, Luis

In this paper we introduce and analyze a hybridizable discontinuous Galerkin (HDG) method for the linear Brinkman model of porous media flow in two and three dimensions and with non-homogeneous Dirichlet boundary conditions. We consider a fully-mixed formulation in which the main unknowns are given by the pseudostress, the velocity and the trace of the velocity, whereas the pressure is easily recovered through a simple postprocessing. We show that the corresponding continuous and discrete schemes are well-posed. In particular, we use the projection-based error analysis in order to derive a priori error estimates. Furthermore, we develop a reliable and efficient residual-based a posteriori error estimator, and propose the associated adaptive algorithm for our HDG approximation. Finally, several numerical results illustrating the performance of the method, confirming the theoretical properties of the estimator and showing the expected behavior of the adaptive refinements are presented.

The numerical approximation of hyperelasticity must address nonlinear constitutive laws, geometric nonlinearities associated with large strains and deformations, the imposition of the incompressibility of the solid, and the solution of large linear systems arising from the discretisation of 3D problems in complex geometries. We adapt the three-field formulation for nearly incompressible hyperelasticity introduced in Chavan et al. (2007) to the fully incompressible case. The mixed formulation is of Hu–Washizu type and it differs from other approaches in that we use the Kirchhoff stress, displacement, and pressure as principal unknowns. We also discuss the solvability of the linearised problem restricted to neo-Hookean materials, illustrating the interplay between the coupling blocks. We construct a family of mixed finite element schemes (with different polynomial degrees) for simplicial meshes and verify its error decay through computational tests. We also propose a new augmented Lagrangian preconditioner that improves convergence properties of iterative solvers. The numerical performance of the family of mixed methods is assessed with benchmark solutions, and the applicability of the formulation is further tested in a model of cardiac biomechanics using orthotropic strain energy densities. The proposed methods are advantageous in terms of physical fidelity (as the Kirchhoff stress can be approximated with arbitrary accuracy and no locking is observed) and convergence (the discretisation and the preconditioners are robust and computationally efficient, and they compare favourably at least with respect to classical displacement–pressure schemes).

In this paper we present the apriori and aposteriori error analyses of a non-standard mixed finite element method for the linear elasticity problem with non-homogeneous Dirichlet boundary conditions. More precisely, the approach introduced here is based on a simplified interpretation of the pseudostress–displacement formulation originally proposed in Arnold and Falk (1988), which does not require symmetric tensor spaces in the finite element discretization. In addition, physical quantities such as the stress, the strain tensor of small deformations, and the rotation, are computed through a simple postprocessing in terms of the pseudostress variable. Furthermore, we also introduce a second elementby-element postprocessing formula for the stress, which yields an optimally convergent approximation of this unknown with respect to the broken H(div)-norm. We apply the classical Babuška–Brezzi theory to prove that the corresponding continuous and discrete schemes are well-posed. In particular, Raviart–Thomas spaces of order k ≥ 0 for the pseudostress and piece wise polynomials of degree≤ k for the displacement can be utilized. Moreover, were mark that in the 3D case the number of unknowns behaves approximately as 9 times the number of elements (tetrahedra) of the triangulation when k = 0. This factor increases to 12.5 when one uses the classical PEERS. Next, we derive a reliable and efficient residual-based a posteriori error estimator for the mixed finite element scheme. Finally, several numerical results illustrating the performance of the method, confirming the theoretical properties of the estimator, and showing the expected behaviour of the associated adaptive algorithm, are provided.

In this paper we introduce and analyze an augmented mixed finite element method for the twodimensional nonlinear Brinkman model of porous media flow with mixed boundary conditions. More precisely, we extend a previous approach for the respective linear model to the present nonlinear case, and employ a dual-mixed formulation in which the main unknowns are given by the gradient of the velocity and the pseudostress. In this way, and similarly as before, the original velocity and pressure unknowns are easily recovered through a simple postprocessing. In addition, since the Neumann boundary condition becomes essential, we impose it in a weak sense, which yields the introduction of the trace of the fluid velocity over the Neumann boundary as the associated Lagrange multiplier. We apply known results from nonlinear functional analysis to prove that the corresponding continuous and discrete schemes are well-posed. In particular, a feasible choice of finite element subspaces is given by Raviart-Thomas elements of order k ≥ 0 for the pseudostress, piecewise polynomials of degree ≤ k for the gradient, and continuous piecewise polynomials of degree ≤ k + 1 for the Lagrange multiplier. We also derive a reliable and efficient residual-based a posteriori error estimator for this problem. Finally, several numerical results illustrating the performance and the robustness of the method, confirming the theoretical properties of the estimator, and showing the behaviour of the associated adaptive algorithm, are provided.

We propose and analyze a new mixed finite element method for the nonlinear problem given by the stationary convective Brinkman–Forchheimer equations. In addition to the original fluid variables, the pseudostress is introduced as an auxiliary unknown, and then the incompressibility condition is used to eliminate the pressure, which is computed afterwards by a postprocessing formula depending on the aforementioned tensor and the velocity. As a consequence, we obtain a mixed variational formulation consisting of a nonlinear perturbation of, in turn, a perturbed saddle point problem in a Banach spaces framework. In this way, and differently from the techniques previously developed for this model, no augmentation procedure needs to be incorporated into the formulation nor into the solvability analysis. The resulting non-augmented scheme is then written equivalently as a fixed-point equation, so that recently established solvability results for perturbed saddle-point problems in Banach spaces, along with the well-known Banach–Nečas–Babuška and Banach theorems, are applied to prove the well-posedness of the continuous and discrete systems. The finite element discretization involves Raviart–Thomas elements of order for the pseudostress tensor and discontinuous piecewise polynomial elements of degree for the velocity. Stability, convergence, and optimal a priori error estimates for the associated Galerkin scheme are obtained. Numerical examples confirm the theoretical rates of convergence and illustrate the performance and flexibility of the method. In particular, the case of flow through a 2D porous media with fracture networks is considered.

We introduce and analyze an augmented mixed finite element method for the Navier–Stokes–Brinkman problem with nonsolenoidal velocity. We employ a technique previously applied to the stationary Navier–Stokes equation, which consists of the introduction of a modified pseudostress tensor relating the gradient of the velocity and the pressure with the convective term, and propose an augmented pseudostress–velocity formulation for the model problem. The resulting augmented scheme is then written equivalently as a fixed point equation, so that the well-known Banach fixed point theorem, combined with the Lax–Milgram lemma, are applied to prove the unique solvability of the continuous and discrete systems. We point out that no discrete inf–sup conditions are required for the solvability analysis, and hence, in particular for the Galerkin scheme, arbitrary finite element subspaces of the respective continuous spaces can be utilized. For instance, given an integer k≥0, the Raviart–Thomas spaces of order k and continuous piecewise polynomials of degree ≤k+1 constitute feasible choices of discrete spaces for the pseudostress and the velocity, respectively, yielding optimal convergence. We also emphasize that, since the Dirichlet boundary condition becomes a natural condition, the analysis for both the continuous an discrete problems can be derived without introducing any lifting of the velocity boundary datum. In addition, we derive a reliable and efficient residual-based a posteriori error estimator for the augmented mixed method. The proof of reliability makes use of a global inf–sup condition, a Helmholtz decomposition, and local approximation properties of the Clément interpolant and Raviart–Thomas operator. On the other hand, inverse inequalities, the localization technique based on element-bubble and edge-bubble functions, approximation properties of the L2-orthogonal projector, and known results from previous works, are the main tools for proving the efficiency of the estimator. Finally, some numerical results illustrating the performance of the augmented mixed method, confirming the theoretical rate of convergence and properties of the estimator, and showing the behavior of the associated adaptive algorithms, are reported.

We propose and analyze an augmented mixed formulation for the time-dependent Brinkman–Forchheimer equations written in terms of vorticity, velocity and pressure. The weak formulation is based on the introduction of suitable least squares terms arising from the incompressibility condition and the constitutive equation relating the vorticity and velocity. We establish existence and uniqueness of a solution to the weak formulation, and derive the corresponding stability bounds, employing classical results on nonlinear monotone operators. We then propose a semidiscrete continuous-in-time approximation based on stable Stokes elements for the velocity and pressure, and continuous or discontinuous piecewise polynomial spaces for the vorticity. In addition, by means of the backward Euler time discretization, we introduce a fully discrete finite element scheme. We prove well-posedness and derive the stability bounds for both schemes, and establish the corresponding error estimates. We provide several numerical results verifying the theoretical rates of convergence and illustrating the performance and flexibility of the method for a range of domain configurations and model parameters.

We present a non-standard mixed finite element method for the linear elasticity problem in Rn with non-homogeneous Dirichlet boundary conditions. More precisely, our approach his based on a simplified interpretation of the pseudo stress–displacement formulation originally proposed in Arnold and Falk (1988), which does not require symmetric tensor spaces in the finite element discretization. We apply the classical Babuˇ ska–Brezzi theory to prove that the corresponding continuous and discrete schemes are well-posed. In particular, Raviart–Thomas spaces of orderk≥0 for the pseudo stress and piece wise polynomials of degree ≤k for the displacement can be utilized. In addition, complementing the results in the afore mentioned reference, we introduceanewpostprocessingformulaforthestressrecoveringtheoptimally convergent approximation of the broken H(div)-norm. Numerical results confirm our theoretical findings.

In this paper we provide some Matlab tools for efficient vectorized coding of the Hybridizable Discontinuous Galerkin for linear variable coencient reaction-discusion problems in polyhedral domains. The resulting tools are modular and include enhanced structures to deal with convection-discusion problems, plus several projections and a superconvergent postprocess of the solution. Loops over the elements are exclusively local and, as such, have been parallelized.