Monolithic multigrid method for the coupled stokes flow and deformable porous medium system

The interaction between fluid flow and a deformable porous medium is a complicated multi-physics problem, which can be described by a coupled model based on the Stokes and poroelastic equations. A monolithic multigrid method together with either a coupled Vanka smoother or a decoupled Uzawa smoother is employed as an efficient numerical technique for the linear discrete system obtained by finite volumes on staggered grids. A specialty in our modeling approach is that at the interface of the fluid and poroelastic medium, two unknowns from the different subsystems are defined at the same grid point. We propose a special discretization at and near the points on the interface, which combines the approximation of the governing equations and the considered interface conditions. In the decoupled Uzawa smoother, Local Fourier Analysis (LFA) helps us to select optimal values of the relaxation parameter appearing. To implement the monolithic multigrid method, grid partitioning is used to deal with the interface updates when communication is required between two subdomains. Numerical experiments show that the proposed numerical method has an excellent convergence rate. The efficiency and robustness of the method are confirmed in numerical experiments with typically small realistic values of the physical coefficients

Poroelasticity problem: numerical difficulties and efficient multigrid solution

This work contains some of the more relevant results obtained by the author regarding the numerical solution of the Biot’s consolidation problem. The emphasis here is on the stable discretization and the highly efficient solution of the resulting algebraic system of equations, which is of saddle point type. On the one hand, a stabilized linear finite element scheme providing oscillation-free solutions for this model is proposed and theoretically analyzed. On the other hand, a monolithic multigrid method is considered for the solution of the resulting system of equations after discretization by using the stabilized scheme. Since this system is of saddle point type, special smoothers of “Vanka”-type have to be considered. This multigrid method is designed with the help of an special local Fourier analysis that takes into account the specific characteristics of the considered block-relaxations. Results from this analysis are presented and compared with those experimentally computed

On an Uzawa smoother in multigrid for poroelasticity equations

A poroelastic saturated medium can be modeled by means of Biot''s theory of consolidation. It describes the time-dependent interaction between the deformation of porous material and the fluid flow inside of it. Here, for the efficient solution of the poroelastic equations, a multigrid method is employed with an Uzawa-type iteration as the smoother. The Uzawa smoother is an equation-wise procedure. It shall be interpreted as a combination of the symmetric Gauss-Seidel smoothing for displacements, together with a Richardson iteration for the Schur complement in the pressure field. The Richardson iteration involves a relaxation parameter which affects the convergence speed, and has to be carefully determined. The analysis of the smoother is based on the framework of local Fourier analysis (LFA) and it allows us to provide an analytic bound of the smoothing factor of the Uzawa smoother as well as an optimal value of the relaxation parameter. Numerical experiments show that our upper bound provides a satisfactory estimate of the exact smoothing factor, and the selected relaxation parameter is optimal. In order to improve the convergence performance, the acceleration of multigrid by iterant recombination is taken into account. Numerical results confirm the efficiency and robustness of the acceleration scheme

Smoothing analysis of two-color distributive relaxation for solving 2D Stokes flow by multigrid method

Smoothing properties of two-color distributive relaxation for solving a two-dimensional (2D) Stokes flow by multigrid method are theoretically investigated by using the local Fourier analysis (LFA) method. The governing equation of the 2D Stokes flow in consideration is discretized with the non-staggered grid and an added pressure stabilization term with stabilized parameters to be determined is introduced into the discretization system in order to enhance the smoothing effectiveness in the analysis. So, an important problem caused by the added pressure stabilization term is how to determine a suitable zone of parameters in the added term. To that end, theoretically, a two-color distributive relaxation, developed on the two-color Jacobi point relaxation, is established for the 2D Stokes flow. Firstly, a mathematical constitution based on the Fourier modes with various frequency components is constructed as a base of the two-color smoothing analysis, in which the related Fourier representation is presented by the form of two-color Jacobi point relaxation. Then, an optimal one-stage relaxation parameter and related smoothing factor for the two-color distributive relaxation are applied to the discretization system, and an analytical expression of the parameter zone on the added pressure stabilization term is established by LFA. The obtained analytical results show that numerical schemes for solving 2D Stokes flow by multigrid method on the two-color distributive relaxation have a specific convergence zone on the parameters of the added pressure stabilization term, and the property of convergence is independent of mesh size, but depends on the parameters of the pressure stabilization term

Multigrid method for nonlinear poroelasticity equations

<p>In this study, a nonlinear multigrid method is applied for solving the system of incompressible poroelasticity equations considering nonlinear hydraulic conductivity. For the unsteady problem, an additional artificial term is utilized to stabilize the solutions when the equations are discretized on collocated grids. We employ two nonlinear multigrid methods, i.e. the “full approximation scheme” and “Newton multigrid” for solving the corresponding system of equations arising after discretization. For the steady case, both homogeneous and heterogeneous cases are solved and two different smoothers are examined to search for an efficient multigrid method. Numerical results show a good convergence performance for all the strategies.</p