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

An adaptive algorithm for the Crank–Nicolson scheme applied to a time-dependent convection–diffusion problem

AbstractAn a posteriori upper bound is derived for the nonstationary convection–diffusion problem using the Crank–Nicolson scheme and continuous, piecewise linear stabilized finite elements with large aspect ratio. Following Lozinski et al. (2009) [13], a quadratic time reconstruction is used.A space and time adaptive algorithm is developed to ensure the control of the relative error in the L2(H1) norm. Numerical experiments illustrating the efficiency of this approach are reported; it is shown that the error indicator is of optimal order with respect to both the mesh size and the time step, even in the convection dominated regime and in the presence of boundary layers

Mathematical analysis of a simplified Hookean dumbbells model arising from viscoelastic flows

Abstract.: A stochastic model corresponding to a simplified Hookean dumbbells viscoelastic fluid is considered, the convective terms being disregarded. Existence on a fixed time interval is proved provided the data are small enough, using the implicit function theorem and a maximum regularity property for a three fields Stokes proble

Mathematical and numerical analysis of a simplified time-dependent viscoelastic flow

A time-dependent model corresponding to an Oldroyd-B viscoelastic fluid is considered, the convective terms being disregarded. Global existence in time is proved in Banach spaces provided the data are small enough, using the implicit function theorem and a maximum regularity property for a three fields Stokes problem. A finite element discretization in space is then proposed. Existence of the numerical solution is proved for small data, so as a priori error estimates, using again an implicit function theore

An adaptive finite element method for the wave equation based on anisotropic a posteriori error estimates in the L2(H1) norm

An adaptive finite element algorithm is presented for the wave equation in two space dimensions. The goal of the adaptive algorithm is to control the error in the same norm as for parabolic problems, namely the L2(0,T;H1(\Omega)) norm, where T denotes the final time and Omega the computational domain. The mesh aspect ratio can be large whenever needed, thus allowing a given level of accuracy to be reached with fewer vertices than with classical isotropic meshes. The refinement and coarsening criteria are based on anisotropic, a posteriori error estimates and on an elliptic reconstruction. A numerical study of the effectivity index on non-adapted meshes confirms the sharpness of the error estimator. Numerical results on adapted meshes indicate that the error indicator slightly underestimates the true error. We conjecture that the missing information corresponds to the interpolation error between successive meshes. It is observed that the error indicator becomes sharp again when considering the damped wave equation with a large damping coefficient, thus when the parabolic character of the PDE becomes predominant

Simulation numérique des traitements de surface par laser

This work deals with the numerical modeling of laser surface treatments, particularly laser remelting (a laser beam melts part of a moving work piece) and laser cladding (injection of powder in the melt pool produces a thin metallic layer on the work piece). A two-dimensional stationary model is presented for laser cladding. This model takes into account the solid-liquid phase change process and the important velocity field in the melt pool. A pure thermal problem is studied, which corresponds to a laser remelting model when the liquid particles movements are neglected. The enthalpy variable is introduced and thus the model reduces to solving the so-called stationary Stefan problem (a diffusion-convection equation, degenerated on the phase change interface). We present a Finite Element discretization for the corresponding regularized problem and give some a priori and a posteriori estimates. An efficient adaptive mesh algorithm is then presented. Finally we solve the complete laser cladding model (the two phase change and hydrodynamic problems) using a Finite Element Method and discuss some numerical results

Anisotropic, Adaptive Finite Elements for a Thin 3D Plate

International audienceAn adaptive, anisotropic finite element algorithm is proposed to solve the 3D linear elasticity equations in a thin 3D plate. Numerical experiments show that adaptive computations can be performed in thin 3D domains having geometrical aspect ratio 1:1000