This paper presents two new finite element methods for two-dimensional Stokes problems. These methods are developed by relaxing the constraints of the Crouzeix–Raviart nonconforming P1 finite elements. Penalty terms are introduced to compensate for lack of continuity or the divergence-free property. However, there is no need for choosing penalty factors, and the formulations are symmetric. Thes...

In this talk we will discuss a nonoverlapping domain decomposition pre-conditioner for the symmetric interior penalty Galerkin method [1, 2, 3]. Thepreconditioner is based on balancing domain decomposition by constraints [4].Theoretical results on the condition number estimate of the preconditioned sys-tem will be presented along with numerical results. References[1] J. ...

In this paper we consider discontinuous Galerkin (DG) finite element approximations of a model scalar linear hyperbolic equation. We show that in order to ensure continuous stabilization of the method it suffices to add a jump-penalty-term to the discretized equation. In particular, the method does not require upwinding in the usual sense. For a specific value of the penalty parameter we recove...

The two-level penalty finite element methods for Navier-Stokes equations with nonlinear slip boundary conditions are investigated in this paper, whose variational formulation is the Navier-Stokes type variational inequality problem of the second kind. The basic idea is to solve the Navier-Stokes type variational inequality problem on a coarse mesh with mesh size H in combining with solving a St...

Analysis of the fictitious domain method with L 2-penalty for elliptic and parabolic problems Abstract. The fictitious domain method with L 2-penalty for elliptic and parabolic problems are considered, respectively. The regularity theorems and a priori estimates for L 2-penalty problems are given. We derive error estimates for penalization and finite element interpolation with P 1-element. Nume...

A review of recent work and new developments are presented for the penalty-function, finite element formulation of incompressible viscous flows. Basic features of the penalty method are described in the context of the steady and unsteady Navier-Stokes equations. Galerkin and “upwind” treatments of convection terms are discussed. Numerical results indicate the versatility and effectiveness of th...

Finite element approximations based on a penalty formulation of the elliptic obstacle problem are analyzed in the maximum norm. A posteriori error estimates, which involve a residual of the approximation and a spatially variable penalty parameter, are derived in the cases of both smooth and rough obstacles. An adaptive algorithm is suggested and implemented in one dimension.

We analyze discontinuous Galerkin methods with penalty terms, namely, symmetric interior penalty Galerkin methods, to solve nonlinear Sobolev equations. We construct finite element spaces on which we develop fully discrete approximations using extrapolated Crank-Nicolson method. We adopt an appropriate elliptic-type projection, which leads to optimal ∞ L2 error estimates of discontinuous Galerk...

A study of a class of finite element methods for the analysis of Stokes’ problem based on the use of exterior penalty formulations is described. The effects of selective reduced integration (i.e., the use of quadrature rules for integrating the penalty terms which are of lower order than that required to integrate polynomial approximations of these terms exactly) are investigated. Error estimat...