Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Multigrid solvers for multipoint flux approximations of the Darcy problem on rough quadrilateral grids

Andrés Arrarás, Francisco J. Gaspar, Laura Portero, Carmen Rodrigo

TL;DR

The paper addresses efficiently solving multipoint flux approximations for the Darcy problem on rough quadrilateral grids by introducing a blackbox, cell-centered multigrid method tailored to MPFA discretizations (MFMFE) on logically rectangular grids. It leverages velocity elimination via a BD MFE framework and performs coarse-grid Galerkin corrections with transfer operators that promote robustness, aided by local Fourier analysis on Cartesian grids to predict convergence and guide smoothing choices. Numerical experiments across smooth, perturbed, and random permeability configurations demonstrate robust convergence with low iteration counts, validating the method’s effectiveness for anisotropic and discontinuous coefficients. The work also outlines potential extensions to three dimensions and to higher-order MFMFE schemes, highlighting the practical impact for scalable groundwater and porous media simulations.

Abstract

In this work, an efficient blackbox-type multigrid method is proposed for solving multipoint flux approximations of the Darcy problem on logically rectangular grids. The approach is based on a cell-centered multigrid algorithm, which combines a piecewise constant interpolation and the restriction operator by Wesseling/Khalil with a line-wise relaxation procedure. A local Fourier analysis is performed for the case of a Cartesian uniform grid. The method shows a robust convergence for different full tensor coefficient problems and several rough quadrilateral grids.

Multigrid solvers for multipoint flux approximations of the Darcy problem on rough quadrilateral grids

TL;DR

The paper addresses efficiently solving multipoint flux approximations for the Darcy problem on rough quadrilateral grids by introducing a blackbox, cell-centered multigrid method tailored to MPFA discretizations (MFMFE) on logically rectangular grids. It leverages velocity elimination via a BD MFE framework and performs coarse-grid Galerkin corrections with transfer operators that promote robustness, aided by local Fourier analysis on Cartesian grids to predict convergence and guide smoothing choices. Numerical experiments across smooth, perturbed, and random permeability configurations demonstrate robust convergence with low iteration counts, validating the method’s effectiveness for anisotropic and discontinuous coefficients. The work also outlines potential extensions to three dimensions and to higher-order MFMFE schemes, highlighting the practical impact for scalable groundwater and porous media simulations.

Abstract

In this work, an efficient blackbox-type multigrid method is proposed for solving multipoint flux approximations of the Darcy problem on logically rectangular grids. The approach is based on a cell-centered multigrid algorithm, which combines a piecewise constant interpolation and the restriction operator by Wesseling/Khalil with a line-wise relaxation procedure. A local Fourier analysis is performed for the case of a Cartesian uniform grid. The method shows a robust convergence for different full tensor coefficient problems and several rough quadrilateral grids.
Paper Structure (11 sections, 49 equations, 10 figures, 7 tables)

This paper contains 11 sections, 49 equations, 10 figures, 7 tables.

Table of Contents

  1. Introduction
  2. Multipoint flux approximations of the Darcy system
  3. The stencil coefficients for homogeneous media and Cartesian uniform grids
  4. Cell-centered multigrid and local Fourier analysis
  5. Cell-centered multigrid
  6. Local Fourier analysis
  7. Numerical experiments
  8. A test with a known analytical solution
  9. Several tests with discontinuous permeability tensors
  10. A test with a random permeability tensor
  11. Conclusions

Figures (10)

  • Figure 1: Degrees of freedom for the $\mathcal{BDM}_1$ spaces on quadrilaterals; crosses and circles denote pressure and velocity degrees of freedom, respectively. For the sake of clarity, the velocity degrees of freedom are drawn at a distance from the corresponding vertex.
  • Figure 2: Interaction of degrees of freedom in the MFMFE method.
  • Figure 3: Local notations for the 9-point stencil.
  • Figure 4: Some logically rectangular grids considered in the numerical examples.
  • Figure 5: Logically rectangular mesh (left), numerical pressure (center) and logarithm of the norm of the numerical velocity (right) for the flow problem that contains an impermeable streak.
  • ...and 5 more figures

Theorems & Definitions (1)

  • remark thmcounterremark