arXiv subjects

Antonio Márquez

Publications and source records attributed to Antonio Márquez.

5 recordsLinked to original sources

arXiv2020

Mixed-hybrid and mixed-discontinuous Galerkin methods for linear dynamical elastic-viscoelastic composite structures

We introduce and analyze a stress-based formulation for Zener's model in linear viscoelasticity. The method is aimed to tackle efficiently heterogeneous materials that admit purely elastic and viscoelastic parts in their composition. We write the mixed variational formulation of the problem in terms of a class of tensorial wave equation and obtain an energy estimate that guaranties the well-posedness of the problem through a standard Galerkin procedure. We propose and analyze mixed continuous and discontinuous Galerkin space discretizations of the problem and derive optimal error bounds for each semidiscrete solution in the corresponding energy norm.

math.NA↗

arXiv2020

A mixed finite element method with reduced symmetry for the standard model in linear viscoelasticity

We introduce and analyze a new mixed finite element method with reduced symmetry for the standard linear model in viscoelasticity. Following a previous approach employed for linear elastodynamics, the present problem is formulated as a second-order hyperbolic partial differential equation in which, after using the motion equation to eliminate the displacement unknown, the stress tensor remains as the main variable to be found. The resulting variational formulation is shown to be well-posed, and a class of $\text{H}(\text{div})$-conforming semi-discrete schemes is proved to be convergent. Then, we use the Newmark trapezoidal rule to obtain an associated fully discrete scheme, whose main convergence results are also established. Finally, numerical examples illustrating the performance of the method are reported.

math.NA↗

arXiv2014

A mixed discontinuous Galerkin method for the time harmonic elasticity problem with reduced symmetry

The aim of this paper is to analyze a mixed discontinuous Galerkin discretization of the time-harmonic elasticity problem. The symmetry of the Cauchy stress tensor is imposed weakly, as in the traditional dual-mixed setting. We show that the discontinuous Galerkin scheme is well-posed and uniformly stable with respect to the mesh parameter $h$ and the Lamé coefficient $λ$. We also derive optimal a-priori error bounds in the energy norm. Several numerical tests are presented in order to illustrate the performance of the method and confirm the theoretical results.

math.NA↗

arXiv2012

A decoupled preconditioning technique for a mixed Stokes-Darcy model

We propose an efficient iterative method to solve the mixed Stokes-Dracy model for coupling fluid and porous media flow. The weak formulation of this problem leads to a coupled, indefinite, ill-conditioned and symmetric linear system of equations. We apply a decoupled preconditioning technique requiring only good solvers for the local mixed-Darcy and Stokes subproblems. We prove that the method is asymptotically optimal and confirm, with numerical experiments, that the performance of the preconditioners does not deteriorate on arbitrarily fine meshes.

math.NA↗

arXiv2012

Strong coupling of finite element methods for the Stokes-Darcy problem

The aim of this paper is to propose a systematic way to obtain convergent finite element schemes for the Darcy-Stokes flow problem by combining well-known mixed finite elements that are separately convergent for Darcy and Stokes problems. In the approach in which the Darcy problem is set in its natural $\mathbf{H}(\text{div})$ formulation and the Stokes problem is expressed in velocity-pressure form, the transmission condition ensuring global mass conservation becomes essential. As opposed to the strategy that handles weakly this transmission condition through a Lagrange multiplier, we impose here this restriction exactly in the space of global velocity field. Our analysis of the Galerkin discretization of the resulting problem reveals that, if the mixed finite element space used in the Darcy domain admits an $\mathbf{H}(\text{div})$-stable discrete lifting of the normal trace, then it can be combined with any stable Stokes mixed finite element of the same order to deliver a stable global method with quasi-optimal convergence rate. Finally, we present a series of numerical tests confirming our theoretical convergence estimates.

math.NA↗