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Explicit T -coercivity for the Stokes problem: a coercive finite element discretization

Patrick Ciarlet, Erell Jamelot

Abstract

Using the T-coercivity theory as advocated in [Chesnel, Ciarlet, T -coercivity and continuous Galerkin methods: application to transmission problems with sign changing coefficients (2013)], we propose a new variational formulation of the Stokes problem which does not involve nonlocal operators. With this new formulation, unstable finite element pairs are stabilized. In addition, the numerical scheme is easy to implement, and a better approximation of the velocity and the pressure is observed numerically when the viscosity is small.

Explicit T -coercivity for the Stokes problem: a coercive finite element discretization

Abstract

Using the T-coercivity theory as advocated in [Chesnel, Ciarlet, T -coercivity and continuous Galerkin methods: application to transmission problems with sign changing coefficients (2013)], we propose a new variational formulation of the Stokes problem which does not involve nonlocal operators. With this new formulation, unstable finite element pairs are stabilized. In addition, the numerical scheme is easy to implement, and a better approximation of the velocity and the pressure is observed numerically when the viscosity is small.

Paper Structure

This paper contains 21 sections, 14 theorems, 81 equations, 14 figures, 7 tables.

Table of Contents

  1. Introduction
  2. $T$-coercivity
  3. The Stokes problem
  4. New variational formulations
  5. Explicit $T$-coercivity
  6. New variational formulation using orthogonality
  7. Numerical algorithms
  8. Discretization and stability estimates
  9. Discretizations
  10. Error estimate when $z_{\mathbf{f}}$ is known
  11. Error estimate when $z_{\mathbf{f}}$ is not known
  12. Discussion on the error indicators
  13. Numerical results
  14. Resolution algorithm
  15. Settings
  16. ...and 6 more sections

Key Result

Theorem 1

Let $a(\cdot,\cdot)$ be a continuous bilinear form on $V\times W$. The Problem eq:var-pb is well-posed if, and only if, the form $a(\cdot,\cdot)$ is $T$-coercive in the sense of definition def:Tcoer.

Figures (14)

  • Figure 1: Linear velocity, $\nu=1$. Plots of $\varepsilon_0^\nu (\mathbf{u}_h)$ (left) and $\varepsilon_0^\nu(p_h)$ (right).
  • Figure 2: Linear velocity, $\nu=10^{-6}$. Plots of $\varepsilon_0^\nu (\mathbf{u}_h)$ (left) and $\varepsilon_0^\nu(p_h)$ (right).
  • Figure 3: Linear velocity, $\nu=10^{-6}$. Plots of $\varepsilon_1^\nu$ (left) and $\varepsilon_D^\nu$ (right).
  • Figure 4: Linear velocity, $\nu=10^{-6}$. Plots of $\varepsilon_0^\nu (\mathbf{u}_h)$ (left) and $\varepsilon_0^\nu(p_h)$ (right) against CPU time (s).
  • Figure 5: Sinusoidal velocity, $\nu=1$. Plots of $\varepsilon_0^\nu(\mathbf{u}_h)$ (left) and $\varepsilon_0^\nu(p_h)$ (right).
  • ...and 9 more figures

Theorems & Definitions (25)

  • Definition 1: basic $T$-coercivity
  • Theorem 1: well-posedness Ciar12ChCi13
  • Proposition 1
  • Proposition 2
  • proof
  • Theorem 2
  • proof
  • Proposition 3
  • Proposition 4
  • proof
  • ...and 15 more