Volume 28, Issue 2

The aim of this paper is to provide new perspectives on the relative finite elements accuracy. Starting from a geometrical interpretation of the error estimate which can be deduced from Bramble–Hilbert lemma, we derive a probability law that evaluates the relative accuracy, considered as a random variable, between two finite elements P k and P m , k < m . We extend this probability law to get a cumulated probabilistic law for two main applications. The first one concerns a family of meshes, the second one is dedicated to a sequence of simplexes constituting a given mesh. Both of these applications could be considered as a first step toward application for adaptive mesh refinement with probabilistic methods.

In [Nkemzi and Jung, 2013] explicit extraction formulas for the computation of the edge flux intensity functions for the Laplacian at axisymmetric edges are presented. The present paper proposes a new adaptation for the Fourier-finite-element method for efficient numerical treatment of boundary value problems for the Poisson equation in axisymmetric domains Ω̂ ⊂ ℝ 3 with edges. The novelty of the method is the use of the explicit extraction formulas for the edge flux intensity functions to define a postprocessing procedure of the finite element solutions of the reduced boundary value problems on the two-dimensional meridian of Ω̂ . A priori error estimates show that the postprocessing finite element strategy exhibits optimal rate of convergence on regular meshes. Numerical experiments that validate the theoretical results are presented.

The paper is concerned with a diffuse-interface model that describes two-phase flow of dilute polymeric solutions with a variable particle density. The additional stresses, which arise by elongations of the polymers caused by deformations of the fluid, are described by Kramers stress tensor. The evolution of Kramers stress tensor is modeled by an Oldroyd-B type equation that is coupled to a Navier–Stokes type equation, a Cahn–Hilliard type equation, and a parabolic equation for the particle density. We present a regularized finite element approximation of this model, prove that our scheme is energy stable and that there exist discrete solutions to it. Furthermore, in the case of equal mass densities and two space dimensions, we are able to pass to the limit rigorously as the regularization parameters and the spatial and temporal discretization parameters tend towards zero and prove that a subsequence of discrete solutions converges to a global-in-time weak solution to the unregularized coupled system. To the best of our knowledge, this is the first existence result for a two-phase flow model of viscoelastic fluids with an Oldroyd-B type equation. Additionally, we show that our finite element scheme is fully practical and we present numerical simulations.