A Mortar Method Using Nonconforming and Mixed Finite Elements for the Coupled Stokes-Darcy Model

2017 ◽  
Vol 9 (3) ◽  
pp. 596-620 ◽  
Author(s):  
Peiqi Huang ◽  
Jinru Chen ◽  
Mingchao Cai
Keyword(s):  
Finite Element ◽  
Porous Media ◽  
Stokes Equations ◽  
Coupled Model ◽  
Weak Form ◽  
Porous Media Flow ◽  
Optimal Error Estimate ◽  
Discrete Problem ◽  
Interface Conditions ◽  
Element Space

AbstractIn this work, we study numerical methods for a coupled fluid-porous media flow model. The model consists of Stokes equations and Darcy's equations in two neighboring subdomains, coupling together through certain interface conditions. The weak form for the coupled model is of saddle point type. A mortar finite element method is proposed to approximate the weak form of the coupled problem. In our method, nonconforming Crouzeix-Raviart elements are applied in the fluid subdomain and the lowest order Raviart-Thomas elements are applied in the porous media subdomain; Meshes in different subdomains are allowed to be nonmatching on the common interface; Interface conditions are weakly imposed via adding constraint in the definition of the finite element space. The well-posedness of the discrete problem and the optimal error estimate for the proposed method are established. Numerical experiments are also given to confirm the theoretical results.

A fully-mixed finite element method for the Navier–Stokes/Darcy coupled problem with nonlinear viscosity

2017 ◽  
Vol 25 (2) ◽  
Author(s):  
Sergio Caucao ◽  
Gabriel N. Gatica ◽  
Ricardo Oyarzúa ◽  
Ivana Šebestová
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Porous Media ◽  
Stokes Equations ◽  
Mixed Finite Element ◽  
Mixed Finite Element Method ◽  
Transmission Conditions ◽  
Navier Stokes ◽  
Porous Media Flow ◽  
Element Method

AbstractWe propose and analyze an augmented mixed finite element method for the coupling of fluid flow with porous media flow. The flows are governed by a class of nonlinear Navier–Stokes and linear Darcy equations, respectively, and the transmission conditions are given by mass conservation, balance of normal forces, and the Beavers–Joseph–Saffman law. We apply dual-mixed formulations in both domains, and the nonlinearity involved in the Navier–Stokes region is handled by setting the strain and vorticity tensors as auxiliary unknowns. In turn, since the transmission conditions become essential, they are imposed weakly, which yields the introduction of the traces of the porous media pressure and the fluid velocity as the associated Lagrange multipliers. Furthermore, since the convective term in the fluid forces the velocity to live in a smaller space than usual, we augment the variational formulation with suitable Galerkin type terms arising from the constitutive and equilibrium equations of the Navier–Stokes equations, and the relation defining the strain and vorticity tensors. The resulting augmented scheme is then written equivalently as a fixed point equation, so that the well-known Schauder and Banach theorems, combined with classical results on bijective monotone operators, are applied to prove the unique solvability of the continuous and discrete systems. In particular, given an integer

Interface‐capturing finite element technique for transient two‐phase flow

2003 ◽  
Vol 20 (5/6) ◽  
pp. 725-740 ◽  
Author(s):  
Nahidh Hamid Sharif ◽  
Nils‐Erik Wiberg
Keyword(s):  
Finite Element ◽  
Porous Media ◽  
Stokes Equations ◽  
Time Dependent ◽  
Navier Stokes ◽  
Porous Media Flow ◽  
Advection Equation ◽  
Fluid Interfaces ◽  
Navier Stokes Equations ◽  
Two Fluid

A numerical model is presented for the computation of unsteady two‐fluid interfaces in nonlinear porous media flow. The nonlinear Forchheimer equation is included in the Navier‐Stokes equations for porous media flow. The model is based on capturing the interface on a fixed mesh domain. The zero level set of a pseudo‐concentration function, which defines the interface between the two fluids, is governed by a time‐dependent advection equation. The time‐dependent Navier‐Stokes equations and the advection equation are spatially discretized by the finite element (FE) method. The fully coupled implicit time integration scheme and the explicit forward Eulerian scheme are implemented for the advancement in time. The trapezoidal rule is applied to the fully implicit scheme, while the operator‐splitting algorithm is used for the velocity‐pressure segregation in the explicit scheme. The spatial and time discretizations are stabilized using FE stabilization techniques. Numerical examples of unsteady flow of two‐fluid interfaces in an earth dam are investigated.

scholarly journals On the Stream Function-Vorticity Finite Element Formulation for Incompressible Flow in Porous Media

2014 ◽  
Vol 2014 ◽  
pp. 1-11
Author(s):  
Abdellatif Agouzal ◽  
Karam Allali ◽  
Siham Binna
Keyword(s):  
Finite Element ◽  
Porous Media ◽  
Stream Function ◽  
Incompressible Flow ◽  
Equations Of Motion ◽  
Finite Element Formulation ◽  
Flow In Porous Media ◽  
Darcy Law ◽  
Element Formulation ◽  
Discrete Problem

Stream function-vorticity finite element formulation for incompressible flow in porous media is presented. The model consists of the heat equation, the equation for the concentration, and the equations of motion under the Darcy law. The existence of solution for the discrete problem is established. Optimal a priori error estimates are given.

scholarly journals Finite element methods for compressible Stokes equations with inflow boundary condition

1997 ◽  
Vol 56 (2) ◽  
pp. 217-225
Author(s):  
Jae Ryong Kweon
Keyword(s):  
Finite Element Method ◽  
Boundary Condition ◽  
Finite Element ◽  
Error Analysis ◽  
Finite Element Methods ◽  
Stokes Equations ◽  
Discrete Problem ◽  
Unique Existence ◽  
Inflow Boundary Condition ◽  
The One

A finite element method for solving the compressible viscous Stokes equation with an inflow boundary condition is presented. The unique existence of the solution of the discrete problem is established, and an error analysis is given. It is shown that the error in pressure is dominated by the one in velocity and an error at the inflow portion of the boundary.

scholarly journals Fully Discrete Finite Element Approximation for the Stabilized Gauge-Uzawa Method to Solve the Boussinesq Equations

2013 ◽  
Vol 2013 ◽  
pp. 1-21
Author(s):  
Jae-Hong Pyo
Keyword(s):  
Finite Element ◽  
Stokes Equations ◽  
Boussinesq Equations ◽  
Finite Element Approximation ◽  
Navier Stokes ◽  
Element Approximation ◽  
Uzawa Method ◽  
Element Space ◽  
Fully Discrete ◽  
Discrete Finite Element

The stabilized Gauge-Uzawa method (SGUM), which is a 2nd-order projection type algorithm used to solve Navier-Stokes equations, has been newly constructed in the work of Pyo, 2013. In this paper, we apply the SGUM to the evolution Boussinesq equations, which model the thermal driven motion of incompressible fluids. We prove that SGUM is unconditionally stable, and we perform error estimations on the fully discrete finite element space via variational approach for the velocity, pressure, and temperature, the three physical unknowns. We conclude with numerical tests to check accuracy and physically relevant numerical simulations, the Bénard convection problem and the thermal driven cavity flow.

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