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Locally energy-stable finite element schemes for incompressible flow problems: Design and analysis for equal-order interpolations

Hennes Hajduk, Dmitri Kuzmin, Gert Lube, Philipp Öffner

Abstract

We show that finite element discretizations of incompressible flow problems can be designed to ensure preservation/dissipation of kinetic energy not only globally but also locally. In the context of equal-order (piecewise-linear) interpolations, we prove the validity of a semi-discrete energy inequality for a quadrature-based approximation to the nonlinear convective term, which we combine with the Becker--Hansbo pressure stabilization. An analogy with entropy-stable algebraic flux correction schemes for the compressible Euler equations and the shallow water equations yields a weak `bounded variation' estimate from which we deduce the semi-discrete Lax--Wendroff consistency and convergence towards dissipative weak solutions. The results of our numerical experiments for standard test problems confirm that the method under investigation is non-oscillatory and exhibits optimal convergence behavior.

Locally energy-stable finite element schemes for incompressible flow problems: Design and analysis for equal-order interpolations

Abstract

We show that finite element discretizations of incompressible flow problems can be designed to ensure preservation/dissipation of kinetic energy not only globally but also locally. In the context of equal-order (piecewise-linear) interpolations, we prove the validity of a semi-discrete energy inequality for a quadrature-based approximation to the nonlinear convective term, which we combine with the Becker--Hansbo pressure stabilization. An analogy with entropy-stable algebraic flux correction schemes for the compressible Euler equations and the shallow water equations yields a weak `bounded variation' estimate from which we deduce the semi-discrete Lax--Wendroff consistency and convergence towards dissipative weak solutions. The results of our numerical experiments for standard test problems confirm that the method under investigation is non-oscillatory and exhibits optimal convergence behavior.
Paper Structure (13 sections, 9 theorems, 71 equations, 7 figures, 8 tables)

This paper contains 13 sections, 9 theorems, 71 equations, 7 figures, 8 tables.

Table of Contents

  1. Introduction
  2. Stabilized finite element method
  3. Galerkin space discretization
  4. Energy and pressure stabilization
  5. Crank--Nicolson time discretization
  6. Analysis of stability and consistency
  7. Energy stability
  8. Lax--Wendroff consistency
  9. Convergence analysis
  10. Numerical results
  11. Taylor--Green vortex
  12. Gresho vortex
  13. Conclusions

Key Result

Theorem 1

Assume that the boundary conditions are periodic and Then the validity of galerkin-stab implies where $\mathbf n_{ij}=\frac{\mathbf{c}_{ij}}{|\mathbf{c}_{ij}|}$ and $\mathcal{Q}({\mathbf u}_L,p_L,{\mathbf u}_R,p_R;\mathbf n)$ is a numerical flux consistent with $\mathbf{q}({\mathbf u},p)\cdot\mathbf{n}$.

Figures (7)

  • Figure 1: Energy evolutions for the Taylor--Green vortex, exact profile and numerical results obtained on refinement levels ref for three types of meshes.
  • Figure 2: Snapshots of finite element approximations to $|\mathbf u|$ for the Taylor--Green vortex obtained on a uniform quadrilateral mesh with $256\times 256$ elements using the lumped mass matrix and the time step $\Delta t = 1/512$.
  • Figure 3: Snapshots of finite element approximations to $p$ for the Taylor--Green vortex obtained on a uniform quadrilateral mesh with $256\times 256$ elements using the lumped mass matrix and $\Delta t = 1/512$.
  • Figure 4: Long-term energy evolutions for the Taylor--Green vortex on finest meshes.
  • Figure 5: Energy evolutions for the Gresho vortex, exact profile and numerical results obtained on refinement levels ref for three types of meshes.
  • ...and 2 more figures

Theorems & Definitions (26)

  • remark 1
  • remark 2
  • remark 3
  • remark 4
  • Theorem 1
  • proof
  • remark 5
  • remark 6
  • Lemma 1
  • proof
  • ...and 16 more