Juho Könnö

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

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.

Numerical computations with H (div)-finite elements for the Brinkman problem

The H(div)-conforming approach for the Brinkman equation is studied numerically, verifying the theoretical a priori and a posteriori analysis in Könnö and Stenberg (2010; Math Models Methods Appl Sci, 2011). Furthermore, the results are extended to cover a non-constant permeability. A hybridization technique for the problem is presented, complete with a convergence analysis and numerical verification. Finally, the numerical convergence studies are complemented with numerical examples of applications to domain decomposition and adaptive mesh refinement.

Non-Conforming Finite Element Method for the Brinkman Problem

The Brinkman equations describe the flow of a viscous fluid in a porous matrix. Mathematically the Brinkman model is a parameter-dependent combination of the Darcy and Stokes models. A dual mixed framework is introduced for the problem, and H(div)-conforming finite elements are used with Nitsche’s method to obtain a stable formulation. We show the formulation to be stable in a mesh-dependent norm for all values of the parameter and introduce a postprocessing scheme for the pressure, which gives optimal convergence for the pressure.

Mixed Finite Element Methods for Problems with Robin Boundary Conditions

We derive new a priori and a posteriori error estimates for mixed finite element discretizations of second-order elliptic problems with general Robin boundary conditions, parameterized by

Finite element methods for flow in porous media

\varepsilon\geq0

Analysis of H(div)-conforming finite elements for the Brinkman problem

. The estimates are robust in

Finite element analysis of composite plates with an application to the paper cockling problem

\varepsilon

22nd Nordic Seminar on Computational Mechanics, Aalborg, Denmark, Oct 22-23, 2009

, ranging from pure Dirichlet conditions to pure Neumann conditions. We also show that hybridization leads to a well-conditioned linear system. A series of numerical experiments is presented that verify our theoretical results.