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Mixed finite element projection methods for the unsteady Stokes equations

Costanza Aricò, Rainer Helmig, Ivan Yotov

Abstract

We develop $H$(div)-conforming mixed finite element methods for the unsteady Stokes equations modeling single-phase incompressible fluid flow. A projection method in the framework of the incremental pressure correction methodology is applied, where a predictor and a corrector problems are sequentially solved, accounting for the viscous effects and incompressibility, respectively. The predictor problem is based on a stress-velocity mixed formulation, while the corrector projection problem uses a velocity-pressure mixed formulation. The scheme results in pointwise divergence-free velocity computed at the end of each time step. We establish unconditional stability and first order in time accuracy. In the implementation we focus on generally unstructured triangular grids. We employ a second order multipoint flux mixed finite element method based on the next-to-the-lowest order Raviart-Thomas space $RT_1$ and a suitable quadrature rule. In the predictor problem this approach allows for a local stress elimination, resulting in element-based systems for each velocity component with three degrees of freedom per element. Similarly, in the corrector problem, the velocity is locally eliminated and an element-based system for the pressure is solved. At the end of each time step we obtain a second order accurate $H$(div)-conforming piecewise linear velocity, which is pointwise divergence free. We present a series of numerical tests to illustrate the performance of the method.

Mixed finite element projection methods for the unsteady Stokes equations

Abstract

We develop (div)-conforming mixed finite element methods for the unsteady Stokes equations modeling single-phase incompressible fluid flow. A projection method in the framework of the incremental pressure correction methodology is applied, where a predictor and a corrector problems are sequentially solved, accounting for the viscous effects and incompressibility, respectively. The predictor problem is based on a stress-velocity mixed formulation, while the corrector projection problem uses a velocity-pressure mixed formulation. The scheme results in pointwise divergence-free velocity computed at the end of each time step. We establish unconditional stability and first order in time accuracy. In the implementation we focus on generally unstructured triangular grids. We employ a second order multipoint flux mixed finite element method based on the next-to-the-lowest order Raviart-Thomas space and a suitable quadrature rule. In the predictor problem this approach allows for a local stress elimination, resulting in element-based systems for each velocity component with three degrees of freedom per element. Similarly, in the corrector problem, the velocity is locally eliminated and an element-based system for the pressure is solved. At the end of each time step we obtain a second order accurate (div)-conforming piecewise linear velocity, which is pointwise divergence free. We present a series of numerical tests to illustrate the performance of the method.

Paper Structure

This paper contains 25 sections, 3 theorems, 80 equations, 21 figures, 11 tables.

Table of Contents

  1. Introduction
  2. Governing equations and numerical method
  3. Time discretization
  4. Space discretization
  5. The mixed finite element projection method
  6. Stability and time discretization error analysis
  7. Stability analysis
  8. Time discretization error analysis
  9. A second order multipoint flux mixed finite element method
  10. Mixed finite element spaces
  11. The quadrature rule
  12. Predictor problem
  13. Local stress elimination in the predictor problem
  14. The case of any vertex.
  15. The case of any center of mass.
  16. ...and 10 more sections

Key Result

Lemma 3.1

For $\mathbf{q}_h^{n+1}$ computed in eq:up_q-0, it holds that

Figures (21)

  • Figure 1: Degrees of freedom of $\mathbf{V}(\hat{E})$ and the associated basis functions.
  • Figure 2: Mapping from the reference triangle $\hat{E}$ to a computational triangle E with DOFs of $RT_1$ in $\hat{E}$ and E.
  • Figure 3: Interaction of the DOFs of $\boldsymbol{\sigma}^{n+1}_{x,h}$ for a vertex (left) and a center of mass (center). Stencil of the reduced system for $\tilde{\mathbf{u}}_h^{n+1}$ (right).
  • Figure 4: Test 1. Coarse computational grids and BCs: MDG and BCs scenario 1 (left), BDG and BCs scenario 2 (center). Right: Zoom of the BDG at the 5$^{th}$ refinement level.
  • Figure 5: Test 1. Number of triangles of the coarse MDG and BDG falling in each sub-range of the interval $\left[0, 1\right]$.
  • ...and 16 more figures

Theorems & Definitions (10)

  • Remark 2.1
  • Remark 3.1
  • Lemma 3.1
  • proof
  • Theorem 3.1
  • proof
  • Theorem 3.2
  • proof
  • Remark 4.1
  • Remark 4.2