Existence, a priori and a posteriori error estimates for a nonlinear three-field problem arising from Oldroyd-B viscoelastic flows

In this paper, a nonlinear problem corresponding to a simplified Oldroyd-B model without convective terms is considered. Assuming the domain to be a convex polygon, existence of a solution is proved for small relaxation times. Continuous piecewise linear finite elements together with a Galerkin Least Square (GLS) method are studied for solving this problem. Existence and a priori error estimates are established using a Newton-chord fixed point theorem, a posteriori error estimates are also derived. An Elastic Viscous Split Stress (EVSS) scheme related to the GLS method is introduced. Numerical results confirm the theoretical predictions

Modelling and mathematical results arising from ferromagnetic problems

In this article, we investigate the equations of magnetostatics for a configuration where a ferromagnetic material occupies a bounded domain and is surrounded by vacuum. Furthermore, the ferromagnetic law takes the form $$B = \mu _0 \mu _r \left( {\left| H \right|} \right)H,$$ i.e., the magnetizing field H and the magnetic induction B are collinear, but the relative permeability µr is allowed to depend on the modulus of H. We prove the well-posedness of the magnetostatic problem under suitable convexity assumptions, and the convergence of several iterative methods, both for the original problem set in the Beppo-Levi space W 1(ℝ3), and for a finite-dimensional approximation. The theoretical results are illustrated by numerical examples, which capture the known physical phenomen

On Some Weighted Stokes Problems. Application on Smagorinsky Models

In this paper we study existence and uniqueness of weak solutions for some non-linear weighted Stokes problems using convex analysis. The characteri- zation of these considered equations is that the viscosity depends on the strain rate of the velocity field with a weight being a positive power of the distance to the boundary of the domain. These non-linear relations can be seen as a first approach of mixing-length eddy viscosity from turbulent modeling. A well known model is von Karman’s on which the viscosity depends on the square of the distance to the boundary of the domain. Numerical experiments conclude the work and show prop- erties from the theory

On a two-dimensional magnetohydrodynamic problem. II. Numerical analysis

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Finite element approximation of multi-scale elliptic problems using patches of elements

In this paper we present a method for the numerical solution of elliptic problems with multi-scale data using multiple levels of not necessarily nested grids. The method consists in calculating successive corrections to the solution in patches whose discretizations are not necessarily conforming. This paper provides proofs of the results published earlier (see C. R. Acad. Sci. Paris, Ser. I 337 (2003) 679-684), gives a generalization of the latter to more than two domains and contains extensive numerical illustrations. New results including the spectral analysis of the iteration operator and a numerical method to evaluate the constant of the strengthened Cauchy-Buniakowski-Schwarz inequality are presente

Modelling and mathematical results arising from ferromagnetic problems

In this article, we investigate the equations of magnetostatics for a configuration where a ferromagnetic material occupies a bounded domain and is surrounded by vacuum. Furthermore, the ferromagnetic law takes the for