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Fully consistent lowest-order finite element methods for generalised Stokes flows with variable viscosity

Felipe Galarce, Douglas R. Q. Pacheco

Abstract

Variable viscosity arises in many flow scenarios, often imposing numerical challenges. Yet, discretisation methods designed specifically for non-constant viscosity are few, and their analysis is even scarcer. In finite element methods for incompressible flows, the most popular approach to allow equal-order velocity-pressure interpolation are residual-based stabilisations. For low-order elements, however, the viscous part of that residual cannot be approximated, often compromising accuracy. Assuming slightly more regularity on the viscosity field, we can construct stabilisation methods that fully approximate the residual, regardless of the polynomial order of the finite element spaces. This work analyses two variants of this fully consistent approach, with the generalised Stokes system as a model problem. We prove unique solvability and derive expressions for the stabilisation parameter, generalising some classical results for constant viscosity. Numerical results illustrate how our method completely eliminates the spurious pressure boundary layers typically induced by low-order PSPG-like stabilisations.

Fully consistent lowest-order finite element methods for generalised Stokes flows with variable viscosity

Abstract

Variable viscosity arises in many flow scenarios, often imposing numerical challenges. Yet, discretisation methods designed specifically for non-constant viscosity are few, and their analysis is even scarcer. In finite element methods for incompressible flows, the most popular approach to allow equal-order velocity-pressure interpolation are residual-based stabilisations. For low-order elements, however, the viscous part of that residual cannot be approximated, often compromising accuracy. Assuming slightly more regularity on the viscosity field, we can construct stabilisation methods that fully approximate the residual, regardless of the polynomial order of the finite element spaces. This work analyses two variants of this fully consistent approach, with the generalised Stokes system as a model problem. We prove unique solvability and derive expressions for the stabilisation parameter, generalising some classical results for constant viscosity. Numerical results illustrate how our method completely eliminates the spurious pressure boundary layers typically induced by low-order PSPG-like stabilisations.

Paper Structure

This paper contains 14 sections, 3 theorems, 44 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Residual-based finite element stabilisations
  4. The classical PSPG method
  5. The boundary vorticity stabilisation
  6. Stability analysis
  7. Stress-divergence formulation
  8. On the reaction term in the residual
  9. Generalised Laplacian formulation
  10. Convergence
  11. Numerical experiments
  12. A reaction-free experiment
  13. Generalised Stokes experiment
  14. Concluding remarks

Key Result

Lemma 4.1

For $\hbox{\boldmath $f$}\in [L^2(\Omega)]^d$ and $\nu\in W^{1,\infty}(\Omega)$ bounded as in viscosityBounds, problem weakMomentumSD,BVS has a unique solution if the stabilisation parameter $\delta$ satisfies with $C$ being a positive constant depending only on the geometry and the discretisation.

Figures (5)

  • Figure 1: 4 coarser structured mesh refinements used for the computational domain $\Omega = (0,5) \times (0,1)$.
  • Figure 2: Centreline pressure for the channel flow with BVS and PSPG. The pressure boundary layer, a numerical artefact, can clearly be seen when using PSPG. The BVS results, on the other hand, show that a careful treatment of the viscous term in the stabilisation leads to a consistent approach, even for over-regularised setups.
  • Figure 3: Numerical solution using BVS for the generalised Stokes experiment
  • Figure 4: Solution convergence for $\hbox{\boldmath $u$}$ and $p$ in generalized Stokes experiment and its sensitivity with respect to the parameter $\gamma$ in \ref{['eq:stab_gamma']}.
  • Figure 5: Spurious pressure boundary layers induced by PSGP and corrected by BVS.

Theorems & Definitions (3)

  • Lemma 4.1: Unique solvability of the BVS formulation in SD form
  • Lemma 4.2: Unique solvability of the BVS formulation in GL form
  • Lemma 5.1: Galerkin orthogonality