Research Outputs
An a-priori error analysis for discontinuous Lagrangian finite elements applied to nonconforming dual-mixed formulations: Poisson and stokes problems
In this paper, we discuss the well-posedness of a mixed discontinuous Galerkin (DG) scheme for the Poisson and Stokes problems in 2D, considering only piecewise Lagrangian finite elements. The complication here lies in the fact that the classical Babuška-Brezzi theory is difficult to verify for low-order finite elements, so we proceed in a non-standard way. First, we prove uniqueness, and then we apply a discrete version of Fredholm's alternative theorem to ensure existence. The a-priori error analysis is done by introducing suitable projections of the exact solution. As a result, we prove that the method is convergent, and, under standard additional regularity assumptions on the exact solution, the optimal rate of convergence of the method is guaranteed.
New a posteriori error estimator for an stabilized mixed method applied to incompressible fluid flows
We consider an augmented mixed finite element method for incompressible fluid flows and develop a simple a posteriori error analysis. We obtain an a posteriori error estimator that is reliable and locally efficient. We provide numerical experiments that illustrate the performance of the corresponding adaptive algorithm and support its use in practice.
Augmented mixed finite element method for the Oseen problem: A priori and a posteriori error analyses
We propose a new augmented dual-mixed method for the Oseen problem based on the pseudostress–velocity formulation. The stabilized formulation is obtained by adding to the dual-mixed approach suitable least squares terms that arise from the constitutive and equilibrium equations. We prove that for appropriate values of the stabilization parameters, the new 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 or Brezzi–Douglas–Marini elements and the velocity is approximated by continuous piecewise polynomials. Moreover, we derive a simple a posteriori error estimator of residual type that consists of two residual terms and prove that it is reliable and locally efficient. Finally, we include several numerical experiments that support the theoretical results.
Stabilised finite element methods for a bending moment formulation of the Reissner-Mindlin plate model
This work presents new stabilised finite element methods for a bending moments formulation of the Reissner-Mindlin plate model. The introduction of the bending moment as an extra unknown leads to a new weak formulation, where the symmetry of this variable is imposed strongly in the space. This weak problem is proved to be well-posed, and stabilised Galerkin schemes for its discretisation are presented and analysed. The finite element methods are such that the bending moment tensor is sought in a finite element space constituted of piecewise linear continuos and symmetric tensors. Optimal error estimates are proved, and these findings are illustrated by representative numerical experiments.
A stabilized mixed method applied to Stokes system with nonhomogeneous source terms: The stationary case Dedicated to Prof. R. Rodríguez, on the occasion of his 65th birthday
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.
A posteriori error analysis of an augmented dual-mixed method in linear elasticity with mixed boundary conditions
We consider the augmented mixed finite element method introduced in [7] for the equations of plane linear elasticity with mixed boundary conditions. We develop an a posteriori error analysis based on the Ritz projection of the error and obtain an a posteriori error estimator that is reliable and efficient, but that involves a non-local term. Then, introducing an auxiliary function, we derive fully local reliable a posteriori error estimates that are locally efficient up to the elements that touch the Neumann boundary. We provide numerical experiments that illustrate the performance of the corresponding adaptive algorithm and support its use in practice.
Adaptive numerical solution of a discontinuous Galerkin method for a Helmholtz problem in low-frequency regime
We develop an a posteriori error analysis for Helmholtz problem using the local discontinuous Galerkin (LDG for short) approach. For the sake of completeness, we give a description of the main a priori results of this method. Indeed, under some assumptions on regularity of the solution of an adjoint problem, we prove that: (a) the corresponding indefinite discrete scheme is well posed; (b) the approach is convergent, with the expected convergence rates as long as the meshsize h is small enough. We give precise information on how small h has to be in term soft he size of the wave number and its distance to the set of eigenvalues for the same boundary value problem for the Laplacian. After that, we present a reliable and efficient a posteriori error estimator with detailed information on the dependence of the constants on the wave number. We finish presenting extensive numerical experiments which illustrate the theoretical results proven in this paper and suggest that stability and convergence may occur under less restrictive assumptions than those taken in the present work.
On an adaptive stabilized mixed finite element method for the Oseen problem with mixed boundary conditions
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.
Analysis of DG approximations for Stokes problem based on velocity-pseudostress formulation
In this article, we first discuss the well posedness of a modified LDG scheme of Stokes problem, considering a velocity-pseudostress formulation. The difficulty here relies on the fact that the application of classical Babuška-Brezzi theory is not easy, so we proceed in a nonstandard way. For uniqueness, we apply a discrete version of Fredholm's alternative theorem, while the a priori error analysis is done introducing suitable projections of exact solution. As a result, we prove that the method is convergent, and under suitable regularity assumptions on the exact solution, the optimal rate of convergence is guaranteed. Next, we explore two stabilizations to the previous scheme, by adding least squares type terms. For these cases, well posedness and the a priori error estimates are proved by the application of standard theory. We end this work with some numerical experiments considering our third scheme, whose results are in agreement with the theoretical properties we deduce.
A note on a priori error estimates for augmented mixed methods
In this note we describe a strategy that improves the a priori error bounds for augmented mixed methods under appropriate hypotheses. This means that we can derive a priori error estimates for each one of the involved unknowns. Usually, the standard a priori error estimate is for the total error. Finally, a numerical example is included, that illustrates the theoretical results proven in this paper.