ESAIM. Mathematical modeling and numerical analysis. Modelisation mathématique

Vol. 51, num. 4, p. 1407

DOI: 10.1051/m2an/2016068

Date of publication: 2017-07

Abstract:

In this paper we present the numerical analysis of a three-field stabilized finite element formulation recently proposed to approximate viscoelastic flows. The three-field viscoelastic fluid flow problem may suffer from two types of numerical instabilities: on the one hand we have the two inf-sup conditions related to the mixed nature problem and, on the other, the convective nature of the momentum and constitutive equations may produce global and local oscillations in the numerical approximation. Both can be overcome by resorting from the standard Galerkin method to a stabilized formulation. The one presented here is based on the subgrid scale concept, in which unresolvable scales of the continuous solution are approximately accounted for. In particular, the approach developed herein is based on the decomposition into their finite element component and a subscale, which is approximated properly to yield a stable formulation. The analyzed problem corresponds to a linearized version of the Navier-Stokes/Oldroyd-B case where the advection velocity of the momentum equation and the non-linear terms in the constitutive equation are treated using a fixed point strategy for the velocity and the velocity gradient. The proposed method permits the resolution of the problem using arbitrary interpolations for all the unknowns. We describe some important ingredients related to the design of the formulation and present the results of its numerical analysis. It is shown that the formulation is stable and optimally convergent for small Weissenberg numbers, independently of the interpolation used.

The original publication is available at www.esaimm2an.org.]]>

ESAIM. Mathematical modeling and numerical analysis. Modelisation mathématique

Vol. 46, num. 5, p. 1003-1028

DOI: 10.1051/m2an/2011046

Date of publication: 2012

Abstract:

We introduce a family of mixed discontinuous Galerkin (DG) finite element methods for nearly and perfectly incompressible linear elasticity. These mixed methods allow the choice of polynomials of any order k 2: 1 for the approximation of the displacement field, and of order k or k - 1 for the pressure space, and are stable for any positive value of the stabilization parameter. We prove the optimal convergence of the displacement and stress fields in both cases, with error estimates that are independent of the value of the Poisson's ratio. These estimates demonstrate that these methods are locking-free. To this end, we prove the corresponding inf-sup condition, which for the equal-order case, requires a construction to establish the surjectivity of the space of discrete divergences on the pressure space. In the particular case of near incompressibility and equal-order approximation of the displacement and pressure fields, the mixed method is equivalent to a displacement method proposed earlier by Lew et al. [29]. The absence of locking of this displacement method then follows directly from that of the mixed method, including the uniform error estimate for the stress with respect to the Poisson's ratio. We showcase the performance of these methods through numerical examples, which show that locking may appear if Dirichlet boundary conditions are imposed strongly rather than weakly, as we do here.]]>

ESAIM. Mathematical modeling and numerical analysis. Modelisation mathématique

Vol. 36, num. 6, p. 1027-1042

DOI: 10.1051/m2an:2003004

Date of publication: 2002-11

Abstract:

In the framework of meshless methods the interpolation is based on a distribution of particles : it is not necessary to define connectivities. In this methods the interpolation can be easily enriched, increasing the number of particles (as in h-refinement of finite elements) or increasing the order of consistency (as in p-refinement of finite elements). However, comparing with finite elements, particle methods suffers from an increase in the computational cost, mainly due to the computation of the shape functions. In this paper, a mixed interpolation that combines finite elements and particles is presented. The objective is to take advantage of both methods. In order to define h-p refinement strategies an a priori error estimate is needed, and thus, some convergence results are presented and proved for this mixed interpolation.]]>