H(div)-CONFORMING FINITE ELEMENTS FOR THE BRINKMAN PROBLEM

2011 ◽  
Vol 21 (11) ◽  
pp. 2227-2248 ◽  
Author(s):  
JUHO KÖNNÖ ◽  
ROLF STENBERG
Keyword(s):  
Finite Elements ◽  
Viscous Fluid ◽  
Galerkin Method ◽  
Porous Matrix ◽  
Brinkman Model ◽  
A Posteriori ◽  
Brinkman Equations ◽  
Parameter Dependent ◽  
Parameter Values ◽  
Mixed Framework

The Brinkman equations describe the flow of a viscous fluid in a porous matrix. Mathematically the Brinkman model is a parameter-dependent combination of both the Darcy and Stokes models. We introduce a dual mixed framework for the problem, and use H( div )-conforming finite elements with the symmetric interior penalty Galerkin method to obtain a stable formulation. We show that the formulation is stable in a mesh-dependent norm for all values of the parameter. We also introduce a postprocessing scheme for the pressure along with a residual-based a posteriori estimator, which is shown to be efficient and reliable for all parameter values.

A New Weak Galerkin Finite Element Scheme for the Brinkman Model

2016 ◽  
Vol 19 (5) ◽  
pp. 1409-1434 ◽  
Author(s):  
Qilong Zhai ◽  
Ran Zhang ◽  
Lin Mu
Keyword(s):  
Finite Element ◽  
Three Dimensional ◽  
Optimal Order ◽  
Brinkman Model ◽  
Weak Galerkin ◽  
Order Error ◽  
Brinkman Equations ◽  
Generalized Distributions ◽  
Variational Form ◽  
Weak Derivatives

AbstractThe Brinkman model describes flow of fluid in complex porous media with a high-contrast permeability coefficient such that the flow is dominated by Darcy in some regions and by Stokes in others. A weak Galerkin (WG) finite element method for solving the Brinkman equations in two or three dimensional spaces by using polynomials is developed and analyzed. The WG method is designed by using the generalized functions and their weak derivatives which are defined as generalized distributions. The variational form we considered in this paper is based on two gradient operators which is different from the usual gradient-divergence operators for Brinkman equations. The WG method is highly flexible by allowing the use of discontinuous functions on arbitrary polygons or polyhedra with certain shape regularity. Optimal-order error estimates are established for the corresponding WG finite element solutions in various norms. Some computational results are presented to demonstrate the robustness, reliability, accuracy, and flexibility of the WG method for the Brinkman equations.

The Sinc-Galerkin method for parameter-dependent self-adjoint problems

1992 ◽  
Vol 50 (2-3) ◽  
pp. 175-202 ◽  
Author(s):  
Kelly M. McArthur ◽  
Ralph C. Smith ◽  
John Lund ◽  
Kenneth L. Bowers
Keyword(s):  
Galerkin Method ◽  
Parameter Dependent

Averaging of gradient in the space of linear triangular and bilinear rectangular finite elements

2013 ◽  
Vol 11 (4) ◽  
Author(s):  
Josef Dalík ◽  
Václav Valenta
Keyword(s):  
Finite Element ◽  
Finite Elements ◽  
Averaging Method ◽  
Order Approximation ◽  
A Posteriori Error Estimates ◽  
Posteriori Error ◽  
Second Order ◽  
Complete Analysis ◽  
A Posteriori ◽  
A Posteriori Error

AbstractAn averaging method for the second-order approximation of the values of the gradient of an arbitrary smooth function u = u(x 1, x 2) at the vertices of a regular triangulation T h composed both of rectangles and triangles is presented. The method assumes that only the interpolant Πh[u] of u in the finite element space of the linear triangular and bilinear rectangular finite elements from T h is known. A complete analysis of this method is an extension of the complete analysis concerning the finite element spaces of linear triangular elements from [Dalík J., Averaging of directional derivatives in vertices of nonobtuse regular triangulations, Numer. Math., 2010, 116(4), 619–644]. The second-order approximation of the gradient is extended from the vertices to the whole domain and applied to the a posteriori error estimates of the finite element solutions of the planar elliptic boundary-value problems of second order. Numerical illustrations of the accuracy of the averaging method and of the quality of the a posteriori error estimates are also presented.

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