A fully mixed finite element method for the coupling of the Stokes and Darcy–Forchheimer problems

2019 ◽  
Vol 40 (2) ◽  
pp. 1454-1502
Author(s):  
Javier A Almonacid ◽  
Hugo S Díaz ◽  
Gabriel N Gatica ◽  
Antonio Márquez
Keyword(s):  
Finite Element ◽  
Piecewise Linear ◽  
Mixed Finite Element ◽  
A Priori ◽  
Mixed Finite Element Method ◽  
Rates Of Convergence ◽  
Normed Spaces ◽  
New Approach ◽  
Darcy Velocity ◽  
Linear Elements

Abstract In this paper we introduce and analyze a fully mixed formulation for the nonlinear problem given by the coupling of the Stokes and Darcy–Forchheimer equations with the Beavers–Joseph–Saffman condition on the interface. This new approach yields non-Hilbert normed spaces and a twofold saddle point structure for the corresponding operator equation, whose continuous and discrete solvabilities are analyzed by means of a suitable abstract theory developed for this purpose. In particular, feasible choices of finite element subspaces include PEERS of the lowest order for the stress of the fluid, Raviart–Thomas of the lowest order for the Darcy velocity, piecewise constants for the pressures and continuous piecewise linear elements for the vorticity. An a priori error estimates and associated rates of convergence are derived, and several numerical results illustrating the good performance of the method are reported.

The Numerical Analysis and Simulation of a Linearized Crank-Nicolson H1-GMFEM for Nonlinear Coupled BBM Equations

2014 ◽  
Vol 513-517 ◽  
pp. 1919-1926 ◽  
Author(s):  
Min Zhang ◽  
Zu Deng Yu ◽  
Yang Liu ◽  
Hong Li
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Numerical Scheme ◽  
Mixed Finite Element ◽  
A Priori ◽  
Numerical Test ◽  
Mixed Finite Element Method ◽  
Spatial Direction ◽  
Time Direction ◽  
Crank Nicolson

In this article, the numerical scheme of a linearized Crank-Nicolson (C-N) method based on H1-Galerkin mixed finite element method (H1-GMFEM) is studied and analyzed for nonlinear coupled BBM equations. In this method, the spatial direction is approximated by an H1-GMFEM and the time direction is discretized by a linearized Crank-Nicolson method. Some optimal a priori error results are derived for four important variables. For conforming the theoretical analysis, a numerical test is presented.

A mixed finite-element method for elliptic operators with Wentzell boundary condition

2018 ◽  
Vol 40 (1) ◽  
pp. 87-108
Author(s):  
Eberhard Bänsch ◽  
Markus Gahn
Keyword(s):  
Boundary Condition ◽  
Finite Element ◽  
Mixed Finite Element ◽  
A Priori ◽  
Mixed Finite Element Method ◽  
Finite Element Approximation ◽  
Element Approximation ◽  
Weak Formulation ◽  
Wentzell Boundary Condition ◽  
Polyhedral Domain

Abstract In this paper we introduce and analyze a mixed finite-element approach for a coupled bulk-surface problem of second order with a Wentzell boundary condition. The problem is formulated on a domain with a curved smooth boundary. We introduce a mixed formulation that is equivalent to the usual weak formulation. Furthermore, optimal a priori error estimates between the exact solution and the finite-element approximation are derived. To this end, the curved domain is approximated by a polyhedral domain introducing an additional geometrical error that has to be bounded. A computational result confirms the theoretical findings.

A Stabilized Mixed Finite Element Method for Nearly Incompressible Elasticity

2005 ◽  
Vol 72 (5) ◽  
pp. 711-720 ◽  
Author(s):  
Arif Masud ◽  
Kaiming Xia
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Mixed Finite Element ◽  
A Priori ◽  
Mixed Finite Element Method ◽  
Element Formulation ◽  
Patch Tests ◽  
Stabilized Finite Element Method ◽  
Incompressible Elasticity ◽  
Element Method

We present a new multiscale/stabilized finite element method for compressible and incompressible elasticity. The multiscale method arises from a decomposition of the displacement field into coarse (resolved) and fine (unresolved) scales. The resulting stabilized-mixed form consistently represents the fine computational scales in the solution and thus possesses higher coarse mesh accuracy. The ensuing finite element formulation allows arbitrary combinations of interpolation functions for the displacement and stress fields. Specifically, equal order interpolations that are easy to implement but violate the celebrated Babushka-Brezzi inf-sup condition, become stable and convergent. Since the proposed framework is based on sound variational foundations, it provides a basis for a priori error analysis of the system. Numerical simulations pass various element patch tests and confirm optimal convergence in the norms considered.

scholarly journals MIXED FINITE ELEMENT METHODS FOR LINEAR VISCOELASTICITY USING WEAK SYMMETRY

2010 ◽  
Vol 20 (06) ◽  
pp. 955-985 ◽  
Author(s):  
MARIE E. ROGNES ◽  
RAGNAR WINTHER
Keyword(s):  
Finite Element ◽  
Linear Viscoelasticity ◽  
Mixed Finite Element ◽  
A Priori ◽  
Viscoelastic Body ◽  
Mixed Finite Element Method ◽  
Three Dimensions ◽  
Unified Framework ◽  
Weak Symmetry ◽  
Small Deformations

Small deformations of a viscoelastic body are considered through the linear Maxwell and Kelvin–Voigt models in the quasi-static equilibrium. A robust mixed finite element method, enforcing the symmetry of the stress tensor weakly, is proposed for these equations on simplicial tessellations in two and three dimensions. A priori error estimates are derived and numerical experiments presented. The approach can be applied to general models for linear viscoelasticity and thus offers a unified framework.

scholarly journals NONCONFORMING TETRAHEDRAL MIXED FINITE ELEMENTS FOR ELASTICITY

2014 ◽  
Vol 24 (04) ◽  
pp. 783-796 ◽  
Author(s):  
DOUGLAS N. ARNOLD ◽  
GERARD AWANOU ◽  
RAGNAR WINTHER
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Piecewise Linear ◽  
Mixed Finite Element ◽  
Linear Convergence ◽  
Mixed Finite Element Method ◽  
Two Dimensions ◽  
Element Approximation ◽  
Symmetric Tensors ◽  
Element Method

This paper presents a nonconforming finite element approximation of the space of symmetric tensors with square integrable divergence, on tetrahedral meshes. Used for stress approximation together with the full space of piecewise linear vector fields for displacement, this gives a stable mixed finite element method which is shown to be linearly convergent for both the stress and displacement, and which is significantly simpler than any stable conforming mixed finite element method. The method may be viewed as the three-dimensional analogue of a previously developed element in two dimensions. As in that case, a variant of the method is proposed as well, in which the displacement approximation is reduced to piecewise rigid motions and the stress space is reduced accordingly, but the linear convergence is retained.

scholarly journals A New Linearized Crank-Nicolson Mixed Element Scheme for the Extended Fisher-Kolmogorov Equation

2013 ◽  
Vol 2013 ◽  
pp. 1-11 ◽  
Author(s):  
Jinfeng Wang ◽  
Hong Li ◽  
Siriguleng He ◽  
Wei Gao ◽  
Yang Liu
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Mixed Finite Element ◽  
A Priori ◽  
Mixed Finite Element Method ◽  
Order Equation ◽  
Second Order ◽  
Fully Discrete ◽  
Priori Error Estimates ◽  
Element Method

We present a new mixed finite element method for solving the extended Fisher-Kolmogorov (EFK) equation. We first decompose the EFK equation as the two second-order equations, then deal with a second-order equation employing finite element method, and handle the other second-order equation using a new mixed finite element method. In the new mixed finite element method, the gradient∇ubelongs to the weaker(L2(Ω))2space taking the place of the classicalH(div;Ω)space. We prove some a priori bounds for the solution for semidiscrete scheme and derive a fully discrete mixed scheme based on a linearized Crank-Nicolson method. At the same time, we get the optimal a priori error estimates inL2andH1-norm for both the scalar unknownuand the diffusion termw=−Δuand a priori error estimates in(L2)2-norm for its gradientχ=∇ufor both semi-discrete and fully discrete schemes.

Sign in / Sign up

Export Citation Format

Share Document