GLS and EVSS methods for a three-field Stokes problem arising from viscoelastic flows

Marco Picasso
2001
Article

In this paper, order one finite elements together with Galerkin least-squares (GLS) methods are used for solving a three-held Stokes problem arising from the numerical study of viscoelastic hows. Stability and convergence results are established, even when the solvent viscosity is small compared to the viscosity due to the polymer chains. An iterative algorithm decoupling velocity-pressure and stress calculations is proposed. The link with the modified elastic viscous split stress (EVSS) method studied in (M. Fortin, R. Guenette, R. Pierre, Comput. Methods Appl. Mech. Engrg. 143 (1997) 79-95; R. Guenette, M. Fortin, J. Non-Newtonian Fluid Mech. 60 (1995) 27-52) is presented. Numerical results are in agreement with theoretical predictions, and with those presented in (M. Fortin, R. Guenette, R. Pierre, Comput. Methods Appl. Mech. Engrg. 143 (1997) 79-95). (C) 2001 Elsevier Science B.V. All rights reserved.

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Marco Picasso
2001
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https://infoscience.epfl.ch/record/89270?ln=fr

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An algebraic least squares reduced basis method for the solution of nonaffinely parametrized Stokes equations

Niccolò Dal Santo, Simone Deparis, Andrea Manzoni, Alfio Quarteroni

In this paper we propose a new, purely algebraic, Petrov-Galerkin reduced basis (RB) method to solve the parametrized Stokes equations, where parameters serve to identify the (variable) domain geometry. Our method is obtained as an algebraic least squares reduced basis (aLS-RB) method, and improves the existing RB methods for Stokes equations in several directions. First of all, it does not require to enrich the velocity space, as often done when dealing with a velocity-pressure formulation, relying on a Petrov- Galerkin RB method rather than on a Galerkin RB (G-RB) method. Then, it exploits a suitable approximation of the matrix-norm in the definition of the (global) supremizing operator. The proposed method also provides a fully automated procedure to assemble and solve the RB problem, able to treat any kind of parametrization, and we rigorously prove the stability of the resulting aLS-RB problem (in the sense of a suitable inf-sup condition). Next, we introduce a coarse aLSRB (caLSRB) method, which is obtained by employing an approximated RB test space, and further improves the efficiency of the aLSRB method both offline and online. We provide numerical comparisons between the proposed methods and the current state-of-art G-RB methods. The new approach results in a more convenient option both during the offline and the online stage of computation, as shown by the numerical results. (C) 2018 Elsevier B.Y. All rights reserved.

ELSEVIER SCIENCE SA2019

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Stability of time-splitting schemes for the Stokes problem with stabilized finite elements

Marco Picasso, Jacques Rappaz

The time-dependent Stokes problem is solved using continuous, piecewise linear finite elements and a classical stabilization procedure. Four order-one methods are proposed for the time discretization. The first one is nothing but the Euler backward scheme and requires a large linear system involving the velocity and pressure unknowns to be solved. The other three schemes allow velocity and pressure computations to be decoupled, namely the pressure-matrix method, a method based on an inexact LU factorization, and an operator splitting method. Stability and condition number estimates are derived. Numerical experiments in two space dimensions confirm the theoretical predictions. (C) 2001 John Wiley & Sons, Inc.

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Anisotropic Error Estimates And Space Adaptivity For A Semidiscrete Finite Element Approximation Of The Transient Transport Equation

Marco Picasso

A stabilized semidiscrete finite element discretization of the transient transport equation is studied in the framework of anisotropic meshes. A priori and a posteriori error estimates are derived, the involved constants being independent of the mesh aspect ratio, only space discretization being considered. Numerical results on nonadapted, anisotropic meshes and small time steps confirm the sharpness of the theoretical predictions. An anisotropic, adaptive finite element algorithm is then proposed with the goal to control the L-2 error in space at final time, the time step being kept constant. Numerical results are then presented on anisotropic, adapted meshes and small time steps. Three different methods are proposed to interpolate the solution between two adapted meshes. Numerical results indicate that both the conservative interpolation method of [F. Alauzet and M. Mehrenberger, Internat. J. Numer. Methods Engrg., 84 (2010), pp. 1552-1588] or the L-2 projection advocated in [P. E. Farrell et al., Comput. Methods Appl. Mech. Engrg., 198 (2009), pp. 2632-2642] yield accurate results.

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