In this paper we develop an a priori error analysis of a new unified mixed finite element method for the coupling of fluid flow with porous media flow in $\mathbb{R}^N$, $N\in\{2,3\}$ on isotropic meshes. Flows are governed by the Stokes and Darcy equations, respectively, and the corresponding transmission conditions are given by mass conservation, balance of normal forces, and the Beavers-Joseph-Saffman law. The approach utilizes a modification of the Darcy problem which allows us to apply a variant nonconforming Crouzeix-Raviart finite element to the whole coupled Stokes-Darcy problem. The well-posedness of the finite element scheme and its convergence analysis are derived. Finally, the numerical experiments are presented, which confirm the excellent stability and accuracy of our method.
In this manuscript we focus on the question: what is the correct notion of Stokes-Biot stability? Stokes-Biot stable discretizations have been introduced, independently by several authors, as a means of discretizing Biot's equations of poroelasticity; such schemes retain their stability and convergence properties, with respect to appropriately defined norms, in the context of a vanishing storage coefficient and a vanishing hydraulic conductivity. The basic premise of a Stokes-Biot stable discretization is: one part Stokes stability and one part mixed Darcy stability. In this manuscript we rema…
In this paper, we develop a discretization for the non-linear coupled model of classical Darcy-Forchheimer flow in deformable porous media, an extension of the quasi-static Biot equations. The continuous model exhibits a generalized gradient flow structure, identifying the dissipative character of the physical system. The considered mixed finite element discretization is compatible with this structure, which gives access to a simple proof for the existence, uniqueness, and stability of discrete approximations. Moreover, still within the framework, the discretization allows for the development …
We study the existence of solutions for Darcy's problem coupled with the heat equation under singular forcing; the right-hand side of the heat equation corresponds to a Dirac measure. The studied model allows thermal diffusion and viscosity depending on the temperature. We propose a finite element solution technique and analyze its convergence properties. In the case that the thermal diffusion is constant, we propose an a posteriori error estimator and investigate reliability and efficiency properties. We illustrate the theory with numerical examples.
In this paper we develop an a posteriori error analysis for the stationary Stokes-Darcy coupled problem approximated by conforming finite element method on isotropic meshes in $\mathbb{R}^d$, $d\in\{2,3\}$. The approach utilizes a new robust stabilized fully mixed discretization developed by Jiaping Yu et al. (Advances in Difference Equations, SpringerOpen Journal, 2018). The a posteriori error estimate is based on a suitable evaluation on the residual of the finite element solution plus the stabilization terms. It is proven that the a posteriori error estimate provided in this paper is both r…
A priori analysis for a generalized local projection stabilized conforming finite element approximation of Darcy flow and Stokes problems is presented in this paper. A first-order conforming P1 finite element space is used to approximate both the velocity and the pressure. It is shown that the stabilized discrete bilinear form satisfy the inf-sup condition with respect to a generalized local projection norm. Moreover, a priori error estimates are derived for both problems. Finally, the validation of the proposed stabilization scheme is demonstrated with appropriate numerical examples.
Gradient discretization of Hybrid Dimensional Darcy Flows in Fractured Porous Media with discontinuous pressures at the matrix fracture interfaces
Laboratoire de Mathématiques J.A. Dieudonné, UMR 7351 CNRS, University Nice Sophia Antipolis（尼斯索菲亚安提波利斯大学）；INRIA Sophia Antipolis Méditerranée（法国国家信息与自动化研究所索菲亚安提波利斯地中海中心）；CSTJF, TOTAL S.A.（道达尔公司）
We consider the problem of iteratively solving large and sparse double saddle-point systems arising from the stationary Stokes-Darcy equations in two dimensions, discretized by the Marker-and-Cell (MAC) finite difference method. We analyze the eigenvalue distribution of a few ideal block preconditioners. We then derive practical preconditioners that are based on approximations of Schur complements that arise in a block decomposition of the double saddle-point matrix. We show that including the interface conditions in the preconditioners is key in the pursuit of scalability. Numerical results s…
We construct mesh-independent and parameter-robust monolithic solvers for the coupled primal Stokes-Darcy problem. Three different formulations and their discretizations in terms of conforming and non-conforming finite element methods and finite volume methods are considered. In each case, robust preconditioners are derived using a unified theoretical framework. In particular, the suggested preconditioners utilize operators in fractional Sobolev spaces. Numerical experiments demonstrate the parameter-robustness of the proposed solvers.
In this paper, we study the "a posteriori" error estimate corresponding to the Brinkman-Darcy-Forchheimer problem. We introduce the variational formulation discretised by using the finite element method. Then, we establish an "a posteriori" error estimation with two types of error indicators related to the discretization and to the linearization. Finally, numerical investigations are shown and discussed.
Residual-based a posteriori error estimates for a conforming finite element discretization of the Navier-Stokes/Darcy coupled problem
Université d’Abomey-Calavi (UAC)（阿波美-卡拉维大学）；African Institute for Mathematical Sciences (AIMS)（非洲数学科学研究所）；Institut de Mathématiques et de Sciences Physiques (IMSP)（数学与物理科学研究所）
We propose two different discrete formulations for the weak imposition of the Neumann boundary conditions of the Darcy flow. The Raviart-Thomas mixed finite element on both triangular and quadrilateral meshes is considered for both methods. One is a consistent discretization depending on a weighting parameter scaling as $\mathcal O(h^{-1})$, while the other is a penalty-type formulation obtained as the discretization of a perturbation of the original problem and relies on a parameter scaling as $\mathcal O(h^{-k-1})$, $k$ being the order of the Raviart-Thomas space. We rigorously prove that bo…
In this paper, we present block preconditioners for a stabilized discretization of the poroelastic equations developed in [45]. The discretization is proved to be well-posed with respect to the physical and discretization parameters, and thus provides a framework to develop preconditioners that are robust with respect to such parameters as well. We construct both norm-equivalent (diagonal) and field-of-value-equivalent (triangular) preconditioners for both the stabilized discretization and a perturbation of the stabilized discretization that leads to a smaller overall problem after static cond…
In this work, we consider a hybrid mixed finite element method for Biot's model. The hybrid P1-RT0-P0 discretization of the displacement-pressure-Darcy's velocity system of Biot's model presented in \cite{C. Niu} is not uniformly stable with respect to the physical parameters, resulting in some issues in numerical simulations. To alleviate such problems, following \cite{V. Girault}, we stabilize the hybrid scheme with face bubble functions and show the well-posedness with respect to physical and discretization parameters, which provide optimal error estimates of the stabilized method. We intro…
This report considers a variable step time discretization algorithm proposed by Dahlquist, Liniger and Nevanlinna and applies the algorithm to the unsteady Stokes/Darcy model. Although long-time forgotten and little explored, the algorithm performs advantages in variable timestep analysis of various fluid flow systems, including the coupled Stokes/Darcy model. The paper proves that the approximate solutions to the unsteady Stokes/Darcy model are unconditionally stable due to the G-stability of the algorithm. Also variable time stepping error analysis follows from the combination of G-stability…
This paper proposes a parallel numerical algorithm to simulate the flow and the transport in a discrete fracture network taking into account the mass exchanges with the surrounding matrix. The discretization of the Darcy fluxes is based on the Vertex Approximate Gradient finite volume scheme adapted to polyhedral meshes and to heterogeneous anisotropic media, and the transport equation is discretized by a first order upwind scheme combined with an Euler explicit integration in time. The parallelization is based on the SPMD (Single Program, Multiple Data) paradigm and relies on a distribution o…
We consider a coupled model of free-flow and porous medium flow, governed by stationary Stokes and Darcy flow, respectively. The coupling between the two systems is enforced by introducing a single variable representing the normal flux across the interface. The problem is reduced to a system concerning only the interface flux variable, which is shown to be well-posed in appropriately weighted norms. An iterative solution scheme is then proposed to solve the reduced problem such that mass is conserved at each iteration. By introducing a preconditioner based on the weighted norms from the analys…
We present and analyze a cut finite element method for the weak imposition of the Neumann boundary conditions of the Darcy problem. The Raviart-Thomas mixed element on both triangular and quadrilateral meshes is considered. Our method is based on the Nitsche formulation studied in [10.1515/jnma-2021-0042] and can be considered as a first attempt at extension in the unfitted case. The key feature is to add two ghost penalty operators to stabilize both the velocity and pressure fields. We rigorously prove our stabilized formulation to be well-posed and derive a priori error estimates for the vel…
A solution technique is proposed for flows in porous media that guarantees local conservation of mass. We first compute a flux field to balance the mass source and then exploit exact co-chain complexes to generate a solenoidal correction. A reduced basis method based on proper orthogonal decomposition is employed to construct the correction and we show that mass balance is ensured regardless of the quality of the reduced basis approximation. The method is directly applicable to mixed finite and virtual element methods, among other structure-preserving discretization techniques, and we present …
We consider the flow of a viscous incompressible fluid through a porous medium. We allow the permeability of the medium to depend exponentially on the pressure and provide an analysis for this model. We study a splitting formulation where a convection diffusion problem is used to define the permeability, which is then used in a linear Darcy equation. We also study a discretization of this problem, and provide an error analysis for it.
We study a decoupling iterative algorithm based on domain decomposition for the time-dependent nonlinear Stokes-Darcy model, in which different time steps can be used in the flow region and in the porous medium. The coupled system is formulated as a space-time interface problem based on the interface condition for mass conservation. The nonlinear interface problem is then solved by a nested iteration approach which involves, at each Newton iteration, the solution of a linearized interface problem and, at each Krylov iteration, parallel solution of time-dependent linearized Stokes and Darcy pro…
In this work we derive a posteriori error estimates for the convection-diffusion-reaction equation coupled with the Darcy-Forchheimer problem by a nonlinear external source depending on the concentration of the fluid. We introduce the variational formulation associated to the problem, and discretize it by using the finite element method. We prove optimal a posteriori errors with two types of calculable error indicators. The first one is linked to the linearization and the second one to the discretization. Then we find upper and lower error bounds under additional regularity assumptions on the …
This work deals with the a posteriori error estimates for the Darcy-Forchheimer problem. We introduce the corresponding variational formulation and discretize it by using the finite-element method. A posteriori error estimate with two types of computable error indicators is showed. The first one is linked to the linearization and the second one to the discretization. Finally, numerical computations are performed to show the effectiveness of the error indicators.
This paper develops the a priori analysis of a mixed finite element method for the filtration of an incompressible fluid through a non-deformable saturated porous medium with heterogeneous permeability. Flows are governed by the Brinkman--Forchheimer and Darcy equations in the more and less permeable regions, respectively, and the corresponding transmission conditions are given by mass conservation and continuity of momentum. We consider the standard mixed formulation in the Brinkman--Forchheimer domain and the dual-mixed one in the Darcy region, and we impose the continuity of the normal velo…
We consider the interaction between a free flowing fluid and a porous medium flow, where the free flowing fluid is described using the time dependent Stokes equations, and the porous medium flow is described using Darcy's law in the primal formulation. To solve this problem numerically, we use a diffuse interface approach, where the weak form of the coupled problem is written on an extended domain which contains both Stokes and Darcy regions. This is achieved using a phase-field function which equals one in the Stokes region and zero in the Darcy region, and smoothly transitions between these …
An abstract framework is developed that enables the analysis of algebraically stabilized discretizations in a unified way. This framework is applied to a discretization of this kind for convection-diffusion-reaction equations. The definition of this scheme contains a new limiter that improves a standard one in such a way that local and global discrete maximum principles are satisfied on arbitrary simplicial meshes.
Postprocessing and Higher Order Convergence of Stabilized Finite Element Discretizations of the Stokes Eigenvalue Problem
LSEC, Academy of Mathematics and Systems Science, Chinese Academy of Sciences（中国科学院数学与系统科学研究院计算数学与科学工程计算实验室）；Institute for Analysis and Computational Mathematics, Otto-von-Guericke-University Magdeburg（马格德堡奥托·冯·格里克大学分析与计算数学研究所）
We derive a numerical method for Darcy flow, hence also for Poisson's equation in mixed (first order) form, based on discrete exterior calculus (DEC). Exterior calculus is a generalization of vector calculus to smooth manifolds and DEC is one of its discretizations on simplicial complexes such as triangle and tetrahedral meshes. DEC is a coordinate invariant discretization, in that it does not depend on the embedding of the simplices or the whole mesh. We start by rewriting the governing equations of Darcy flow using the language of exterior calculus. This yields a formulation in terms of flux…
In this paper, an efficient ensemble domain decomposition algorithm is proposed for fast solving the fully-mixed random Stokes-Darcy model with the physically realistic Beavers-Joseph (BJ) interface conditions. We utilize the Monte Carlo method for the coupled model with random inputs to derive some deterministic Stokes-Darcy numerical models and use the idea of the ensemble to realize the fast computation of multiple problems. One remarkable feature of the algorithm is that multiple linear systems share a common coefficient matrix in each deterministic numerical model, which significantly red…
The paper develops an unfitted finite element method for solving the Darcy system of equations posed in a network of fractures embedded in a porous matrix. The approach builds on the Hughes--Masud stabilized formulation of the Darcy problem and the trace finite element method. The system of fractures is allowed to cut through the background mesh in an arbitrary way. Moreover, the fractures are not triangulated in the common sense and the junctions of fractures are not fitted by the mesh. To couple the flow variables at multiple fracture junctions, we extend the Hughes--Masud formulation by inc…
