We propose an efficient iterative method to solve the mixed Stokes-Dracy model for coupling fluid and porous media flow. The weak formulation of this problem leads to a coupled, indefinite, ill-conditioned and symmetric linear system of equations. We apply a decoupled preconditioning technique requiring only good solvers for the local mixed-Darcy and Stokes subproblems. We prove that the method is asymptotically optimal and confirm, with numerical experiments, that the performance of the preconditioners does not deteriorate on arbitrarily fine meshes.
In this paper, we propose a new finite element solution approach to the multi-compartmental Darcy equations describing flow and interactions in a porous medium with multiple fluid compartments. We introduce a new numerical formulation and a block-diagonal preconditioner. The robustness with respect to variations in material parameters is demonstrated by theoretical considerations and numerical examples.
The coupled Darcy-Stokes problem is widely used for modeling fluid transport in physical systems consisting of a porous part and a free part. In this work we consider preconditioners for monolitic solution algorithms of the coupled Darcy-Stokes problem, where the Darcy problem is in primal form. We employ the operator preconditioning framework and utilize a fractional solver at the interface between the problems to obtain order optimal schemes that are robust with respect to the material parameters, i.e. the permeability, viscosity and Beavers-Joseph-Saffman parameter. Our approach is similar …
Robust stability and preconditioning of Darcy-Forchheimer equations
Indian Institute of Technology Bombay（印度理工学院孟买分校）；BITS-Pilani, KK Birla Goa Campus（巴季学院-皮拉尼，KK维尔戈阿校区）；Monash University（蒙纳士大学）；Universidad Adventista de Chile（智利基督复临安息日会大学）
We consider the time-dependent Stokes-Darcy problem as a model case for the challenges involved in solving coupled systems. Keeping the model, its discretization, and the underlying numerics for the subproblems in the free-flow domain and the porous medium constant, we focus on different solver approaches for the coupled problem. We compare a partitioned coupling approach using the coupling library preCICE with a monolithic block-preconditioned one that is tailored to different formulations of the problem. Both approaches enable the reuse of already available iterative solvers and precondition…
We analyze a fully discrete finite element numerical scheme for the Cahn-Hilliard-Stokes-Darcy system that models two-phase flows in coupled free flow and porous media. To avoid a well-known difficulty associated with the coupling between the Cahn-Hilliard equation and the fluid motion, we make use of the operator-splitting in the numerical scheme, so that these two solvers are decoupled, which in turn would greatly improve the computational efficiency. The unique solvability and the energy stability have been proved in~\cite{CHW2017}. In this work, we carry out a detailed convergence analysis…
High-order, long-time stable and parallel decoupled GBDF$k$ SAV ensemble schemes for the Navier--Stokes--Darcy flow with random hydraulic conductivity tensors
Eastern Institute of Technology（宁波东方理工大学）；Ningbo University（宁波大学）
In this report, we propose a new adaptive time filter algorithm for the unsteady Stokes/Darcy model. First we present a first order $θ$-scheme with the variable time step which is one parameter family of Linear Multi-step methods and use a time filter algorithm to increase the convergence order to second order with almost no increasing the amount of computation. Furthermore, we construct coupled and decoupled adaptive algorithms. Then we analyze stabilities and the second-order accuracy of variable time-stepping algorithm for Linear Multi-step methods plus time filter, respectively. Finally, w…
When solving a multi-physics problem one often decomposes a monolithic system into simpler, frequently single-physics, subproblems. A comprehensive solution strategy may commonly be attempted, then, by means of combining strategies devised for the constituent subproblems. When decomposing the monolithic problem, however, it may be that requiring a particular scaling for one subproblem enforces an undesired scaling on another. In this manuscript we consider the H(div)-based mixed formulation of the Darcy problem as a single-physics subproblem; the hydraulic conductivity, K, is considered intrin…
In this paper, we develop the numerical theory of decoupled modified characteristic finite element method with different subdomain time steps for the mixed stabilized formulation of nonstationary dual-porosity-Navier-Stokes model. Based on partitioned time-stepping methods, the system is decoupled, which means that the Navier-Stokes equations and two different Darcy equations are solved independently at each time step of subdomain. In particular, the Navier-Stokes equations are solved by the modified characteristic finite element method, which overcome the computational difficulties caused by …
In this article we consider the numerical modeling and simulation via the phase field approach of two-phase flows of different densities and viscosities in superposed fluid and porous layers. The model consists of the Cahn-Hilliard-Navier-Stokes equations in the free flow region and the Cahn-Hilliard-Darcy equations in porous media that are coupled by seven domain interface boundary conditions. We show that the coupled model satisfies an energy law. Based on the ideas of pressure stabilization and artificial compressibility, we propose an unconditionally stable time stepping method that decoup…
