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On the construction of a family of well-posed approximate formulations for the stationary Stokes problem using an extended system

Cătălin Liviu Bichir

TL;DR

This work addresses stable, inf-sup-free approximations for the stationary Stokes problem by formulating an exact parameterized extended system that couples the velocity, pressure, and auxiliary variables through a momentum equation, a Glowinski–Pironneau pressure decomposition, and a parameterized pressure Poisson equation. The authors prove well-posedness of the exact extended system via density-based isomorphism arguments and develop a general framework for well-posed approximations that do not rely on the discrete inf-sup condition. They construct an abstract approximation and then implement a finite-element discretization, including equal-order FE spaces and a discrete Laplacian, leading to a robust matrix form with suitable preconditioners. Convergence and error estimates follow from a Kantorovich–Akilov–type analysis and Céa’s lemma, with detailed treatment of density arguments and isomorphisms; the approach extends boundary-pressure treatment and incompressibility considerations to the extended formulation and its discretization. Overall, the paper provides a novel, inf-sup-independent pathway to accurate stationary Stokes approximations and clarifies the interplay between pressure on the boundary and velocity incompressibility in extended-system frameworks.

Abstract

We introduce an exact parameterized extended system such that, under adequate data, between the components of its solution, there is the solution of the weak formulation of the homogeneous Dirichlet problem for the stationary Stokes equations. In the extended system, we introduce the momentum equation together with two other forms of this one. This allows us to reformulate, for the stationary case, the consistent pressure Poisson equation of Sani, Shen, Pironneau, Gresho [\textit{Int. J. Numer. Meth. Fluids}, \textbf{50} (2006), pp. 673-682], from the unsteady case. In this way, we can retain the information we need for the approximate pressure on the boundary. We obtain a parameterized perturbed pressure Poisson equation for the stationary Stokes problem. We prove that to solve the stationary Stokes problem is equivalent to solve a problem for the momentum equation, the parameterized equation and the equation that defines the Laplace operator acting on velocity. The approximation of this last problem give a family of well - posed approximate formulations. The solution of each element of this family approximates the solution of the stationary Stokes problem. Some necessary variants of existing results on general Banach spaces are also developed. These concern the density of subspaces related to isomorphisms and the connection between the exact and the approximate problems. First, the well - posedness of the exact extended system is proved in some dense subspaces. The well posed approximate problem does not necessitate the discrete inf-sup condition. The paper is also related to the existing discussions on the boundary conditions for the pressure and on the imposing of the incompressibility constraint on the boundary.

On the construction of a family of well-posed approximate formulations for the stationary Stokes problem using an extended system

TL;DR

This work addresses stable, inf-sup-free approximations for the stationary Stokes problem by formulating an exact parameterized extended system that couples the velocity, pressure, and auxiliary variables through a momentum equation, a Glowinski–Pironneau pressure decomposition, and a parameterized pressure Poisson equation. The authors prove well-posedness of the exact extended system via density-based isomorphism arguments and develop a general framework for well-posed approximations that do not rely on the discrete inf-sup condition. They construct an abstract approximation and then implement a finite-element discretization, including equal-order FE spaces and a discrete Laplacian, leading to a robust matrix form with suitable preconditioners. Convergence and error estimates follow from a Kantorovich–Akilov–type analysis and Céa’s lemma, with detailed treatment of density arguments and isomorphisms; the approach extends boundary-pressure treatment and incompressibility considerations to the extended formulation and its discretization. Overall, the paper provides a novel, inf-sup-independent pathway to accurate stationary Stokes approximations and clarifies the interplay between pressure on the boundary and velocity incompressibility in extended-system frameworks.

Abstract

We introduce an exact parameterized extended system such that, under adequate data, between the components of its solution, there is the solution of the weak formulation of the homogeneous Dirichlet problem for the stationary Stokes equations. In the extended system, we introduce the momentum equation together with two other forms of this one. This allows us to reformulate, for the stationary case, the consistent pressure Poisson equation of Sani, Shen, Pironneau, Gresho [\textit{Int. J. Numer. Meth. Fluids}, \textbf{50} (2006), pp. 673-682], from the unsteady case. In this way, we can retain the information we need for the approximate pressure on the boundary. We obtain a parameterized perturbed pressure Poisson equation for the stationary Stokes problem. We prove that to solve the stationary Stokes problem is equivalent to solve a problem for the momentum equation, the parameterized equation and the equation that defines the Laplace operator acting on velocity. The approximation of this last problem give a family of well - posed approximate formulations. The solution of each element of this family approximates the solution of the stationary Stokes problem. Some necessary variants of existing results on general Banach spaces are also developed. These concern the density of subspaces related to isomorphisms and the connection between the exact and the approximate problems. First, the well - posedness of the exact extended system is proved in some dense subspaces. The well posed approximate problem does not necessitate the discrete inf-sup condition. The paper is also related to the existing discussions on the boundary conditions for the pressure and on the imposing of the incompressibility constraint on the boundary.
Paper Structure (18 sections, 44 theorems, 125 equations)

This paper contains 18 sections, 44 theorems, 125 equations.

Table of Contents

  1. Introduction
  2. Preliminaries and the Stokes problem
  3. Notations
  4. Some results of functional analysis
  5. The Stokes problem
  6. A lemma that reduces the proofs of well-posedness to dense subspaces
  7. A theorem on the well-posedness of a class of approximate linear problems
  8. Preliminary remarks for the formulation of the problem we study
  9. Some remarks on some operators and subspaces
  10. Some definitions introduced in order to apply Lemma \ref{['well_posed_lema_deschisa']} and Theorem \ref{['well_posed_teorema5_1_Stokes_math_B_L2']}
  11. An extended system based on the Stokes problem
  12. The formulation of the extended system
  13. The well-posedness of the exact extended system
  14. Construction of another formulation of the isomorphism for the exact problem
  15. An abstract approximation. Construction of the isomorphism for the approximate problem
  16. ...and 3 more sections

Key Result

Lemma 2.1

(CLBichir_bib_Atkinson_Han2009, Theorem 2.4.1; CLBichir_bib_Kreyszig1978, Theorem 2.7-11; CLBichir_bib_Sir_AF1982, Theorem 5.3.2, Corollary 1; CLBichir_bib_E_Zeidler_NFA_IIA, Proposition 18.29 (a)) Let $E$ be a normed space, let $E_{0} \subseteq E$ be a linear subspace dense in $E$, and let $F$ be a

Theorems & Definitions (89)

  • Lemma 2.1
  • Lemma 2.2
  • Corollary 2.3
  • proof
  • Corollary 2.4
  • proof
  • Corollary 2.5
  • proof
  • Lemma 2.6
  • proof
  • ...and 79 more