Results 1  10
of
74
A stabilized mixed finite element method for Darcy flow
 Comput. Methods Appl. Mech. Engrg
, 2002
"... This paper presents a new stabilized finite element method for the Darcy–Stokes equations also known as the Brinkman model of lubrication theory. These equations also govern the flow of incompressible viscous fluids through permeable media. The proposed method arises from a decomposition of the velo ..."
Abstract

Cited by 53 (5 self)
 Add to MetaCart
(Show Context)
This paper presents a new stabilized finite element method for the Darcy–Stokes equations also known as the Brinkman model of lubrication theory. These equations also govern the flow of incompressible viscous fluids through permeable media. The proposed method arises from a decomposition of the velocity field into coarse/resolved scales and fine/unresolved scales. Modelling of the unresolved scales corrects the lack of stability of the standard Galerkin formulation for the Darcy–Stokes equations. A significant feature of the present method is that the structure of the stabilization tensor s appears naturally via the solution of the finescale problem. The issue of arbitrary combinations of pressure–velocity interpolation functions is addressed, and equalorder combinations of C ◦ interpolations are shown to be stable and convergent.
Stabilization mechanisms in discontinuous Galerkin finite element methods
 Comput. Methods Appl. Mech. Engrg
, 2006
"... In this paper we propose a new general framework for the construction and the analysis of Discontinuous Galerkin (DG) methods which reveals a basic mechanism, responsible for certain distinctive stability properties of DG methods. We show that this mechanism is common to apparently unrelated stabili ..."
Abstract

Cited by 44 (6 self)
 Add to MetaCart
(Show Context)
In this paper we propose a new general framework for the construction and the analysis of Discontinuous Galerkin (DG) methods which reveals a basic mechanism, responsible for certain distinctive stability properties of DG methods. We show that this mechanism is common to apparently unrelated stabilizations, including jump penalty, upwinding, and Hughes–Franca type residualbased stabilizations.
Continuous interior penalty finite element method for Oseen’s equations
 SIAM J. Numer. Anal
"... Abstract. In this paper we present an extension of the continuous interior penalty method of Douglas and Dupont [Interior penalty procedures for elliptic and parabolic Galerkin methods, in Computing Methods in Applied Sciences, Lecture Notes in Phys. 58, SpringerVerlag, Berlin, 1976, pp. 207–216] t ..."
Abstract

Cited by 36 (8 self)
 Add to MetaCart
(Show Context)
Abstract. In this paper we present an extension of the continuous interior penalty method of Douglas and Dupont [Interior penalty procedures for elliptic and parabolic Galerkin methods, in Computing Methods in Applied Sciences, Lecture Notes in Phys. 58, SpringerVerlag, Berlin, 1976, pp. 207–216] to Oseen’s equations. The method consists of a stabilized Galerkin formulation using equal order interpolation for pressure and velocity. To counter instabilities due to the pressure/velocity coupling, or due to a high local Reynolds number, we add a stabilization term giving L2control of the jump of the gradient over element faces (edges in two dimensions) to the standard Galerkin formulation. Boundary conditions are imposed in a weak sense using a consistent penalty formulation due to Nitsche. We prove energytype a priori error estimates independent of the local Reynolds number and give some numerical examples recovering the theoretical results.
CONTINUOUS INTERIOR PENALTY hpFINITE ELEMENT METHODS FOR ADVECTION AND ADVECTIONDIFFUSION EQUATIONS
"... Abstract. A continuous interior penalty hpfinite element method that penalizes the jump of the gradient of the discrete solution across mesh interfaces is introduced. Error estimates are obtained for advection and advectiondiffusion equations. The analysis relies on three technical results that ar ..."
Abstract

Cited by 29 (10 self)
 Add to MetaCart
(Show Context)
Abstract. A continuous interior penalty hpfinite element method that penalizes the jump of the gradient of the discrete solution across mesh interfaces is introduced. Error estimates are obtained for advection and advectiondiffusion equations. The analysis relies on three technical results that are of independent interest: an hpinverse trace inequality, a local discontinuous to continuous hpinterpolation result, and hperrorestimatesforcontinuousL 2orthogonal projections. 1.
A unified analysis for conforming and nonconforming stabilized finite element methods using interior penalty
"... We discuss stabilized Galerkin approximations in a new framework, widening the scope from the usual dichotomy of the discontinuous Galerkin method on the one hand and Petrov– Galerkin methods such as the SUPG method on the other. The idea is to use interior penalty terms as a means of stabilizing th ..."
Abstract

Cited by 28 (14 self)
 Add to MetaCart
(Show Context)
We discuss stabilized Galerkin approximations in a new framework, widening the scope from the usual dichotomy of the discontinuous Galerkin method on the one hand and Petrov– Galerkin methods such as the SUPG method on the other. The idea is to use interior penalty terms as a means of stabilizing the finite element method using conforming or nonconforming approximation, thus circumventing the need of a Petrov–Galerkintype choice of spaces. This is made possible by adding a higherorder penalty term giving L²control of the jumps in the gradients between adjacent elements. We consider convectiondiffusionreaction problems using piecewise linear approximations and prove optimal order a priori error estimates for two different finite element spaces, the standard H¹conforming space of piecewise linears and the nonconforming space of piecewise linear elements where the nodes are situated at the midpoint of the element sides (the Crouzeix–Raviart element). Moreover, we show how the formulation extends to discontinuous Galerkin interior penalty methods in a natural way by domain decomposition using Nitsche’s method.
Optimal control of the convectiondiffusion equation using stabilized finite element methods
, 2007
"... ..."
(Show Context)
Constrained Dirichlet boundary control in L² for a class of evolution equations
"... Optimal Dirichlet boundary control based on the very weak solution of a parabolic state equation is analysed. This approach allows to consider the boundary controls in L2 which has advantages over approaches which consider control in Sobolev involving (fractional) derivatives. Pointwise constrain ..."
Abstract

Cited by 15 (3 self)
 Add to MetaCart
Optimal Dirichlet boundary control based on the very weak solution of a parabolic state equation is analysed. This approach allows to consider the boundary controls in L2 which has advantages over approaches which consider control in Sobolev involving (fractional) derivatives. Pointwise constraints on the boundary are incorporated by the primaldual active set strategy. Its global and local superlinear convergence are shown. A discretization based on spacetime finite elements is proposed and numerical examples are included.
A Stabilized Nonconforming Finite Element Method For Incompressible Flow
, 2004
"... In this paper we extend the recently introduced edge stabilization method to the case of nonconforming finite element approximations of the linearized NavierStokes equation. To get stability also in the convective dominated regime we add a term giving control of the jump in the gradient over el ..."
Abstract

Cited by 11 (3 self)
 Add to MetaCart
(Show Context)
In this paper we extend the recently introduced edge stabilization method to the case of nonconforming finite element approximations of the linearized NavierStokes equation. To get stability also in the convective dominated regime we add a term giving control of the jump in the gradient over element boundaries. An a priori error estimate that is uniform in the Reynolds number is proved and some numerical examples are presented. 1.