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A mortar method for the coupled Stokes-Darcy problem using the MAC scheme for Stokes and mixed finite elements for Darcy

Wietse M. Boon, Dennis Gläser, Rainer Helmig, Kilian Weishaupt, Ivan Yotov

TL;DR

The paper addresses coupling of Stokes flow and Darcy flow on non-matching grids using a mortar interface. It develops a mortar MAC--MFE discretization that combines the MAC scheme for the Stokes region with a $RT_0$ mixed finite element discretization for the Darcy region, linked by a mortar variable that encodes the Darcy pressure and Stokes normal stress. The resulting system is shown to be well-posed with first-order convergence in all variables, and it can be reduced to a symmetric positive definite interface problem solved by CG, requiring only decoupled Stokes and Darcy solves per iteration. Numerical experiments demonstrate convergence, effectiveness on non-matching interfaces, and the ability to locally adapt grids for efficiency and accuracy in challenging geometries.

Abstract

A discretization method with non-matching grids is proposed for the coupled Stokes-Darcy problem that uses a mortar variable at the interface to couple the marker and cell (MAC) method in the Stokes domain with the Raviart-Thomas mixed finite element pair in the Darcy domain. Due to this choice, the method conserves linear momentum and mass locally in the Stokes domain and exhibits local mass conservation in the Darcy domain. The MAC scheme is reformulated as a mixed finite element method on a staggered grid, which allows for the proposed scheme to be analyzed as a mortar mixed finite element method. We show that the discrete system is well-posed and derive a priori error estimates that indicate first order convergence in all variables. The system can be reduced to an interface problem concerning only the mortar variables, leading to a non-overlapping domain decomposition method. Numerical examples are presented to illustrate the theoretical results and the applicability of the method.

A mortar method for the coupled Stokes-Darcy problem using the MAC scheme for Stokes and mixed finite elements for Darcy

TL;DR

The paper addresses coupling of Stokes flow and Darcy flow on non-matching grids using a mortar interface. It develops a mortar MAC--MFE discretization that combines the MAC scheme for the Stokes region with a mixed finite element discretization for the Darcy region, linked by a mortar variable that encodes the Darcy pressure and Stokes normal stress. The resulting system is shown to be well-posed with first-order convergence in all variables, and it can be reduced to a symmetric positive definite interface problem solved by CG, requiring only decoupled Stokes and Darcy solves per iteration. Numerical experiments demonstrate convergence, effectiveness on non-matching interfaces, and the ability to locally adapt grids for efficiency and accuracy in challenging geometries.

Abstract

A discretization method with non-matching grids is proposed for the coupled Stokes-Darcy problem that uses a mortar variable at the interface to couple the marker and cell (MAC) method in the Stokes domain with the Raviart-Thomas mixed finite element pair in the Darcy domain. Due to this choice, the method conserves linear momentum and mass locally in the Stokes domain and exhibits local mass conservation in the Darcy domain. The MAC scheme is reformulated as a mixed finite element method on a staggered grid, which allows for the proposed scheme to be analyzed as a mortar mixed finite element method. We show that the discrete system is well-posed and derive a priori error estimates that indicate first order convergence in all variables. The system can be reduced to an interface problem concerning only the mortar variables, leading to a non-overlapping domain decomposition method. Numerical examples are presented to illustrate the theoretical results and the applicability of the method.
Paper Structure (12 sections, 10 theorems, 85 equations, 8 figures, 4 tables)

This paper contains 12 sections, 10 theorems, 85 equations, 8 figures, 4 tables.

Table of Contents

  1. Introduction
  2. The model problem and its variational formulation
  3. Mortar MAC--MFE method
  4. Well posedness
  5. Error estimates
  6. Domain decomposition algorithm
  7. Implementation
  8. Numerical results
  9. Case 1: Convergence test
  10. Case 2: Porous obstacle
  11. Case 3: Locally adapted grids
  12. Conclusions

Key Result

Lemma 1

The MAC scheme MAC-scheme is equivalent to the following mixed finite element method: find $(\bm{u}_{S,h},p_{S,h}) \in \bm{V}_{S,h}\times W_{S,h}$ such that

Figures (8)

  • Figure 1: The Stokes-Darcy domain.
  • Figure 2: Grids and velocity degrees of freedom for the MAC scheme.
  • Figure 3: Finite difference stencils for the MAC momentum balance equations on volumes $G_1$ (left) and $G_2$ (right).
  • Figure 4: Primal MAC grid with velocity degrees of freedom (left) and staggered grids with degrees of freedom for the horizontal (center) and vertical (right) velocities.
  • Figure 5: Velocity and pressure distributions after the first refinement for Case 1.
  • ...and 3 more figures

Theorems & Definitions (19)

  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • Lemma 3
  • proof
  • Lemma 4
  • proof
  • Lemma 5
  • proof
  • ...and 9 more