Crouzeix-Raviart Element Research Articles

A variationally consistent finite element approach to the two-fluid internal contact-line problem

A new method for simulating incompressible viscous fluid flow involving moving internal contact lines is presented. The steady state interface shape is determined by a variationally consistent formulation of the surface tension contribution to the equations of motion adapted to the case of internal contact lines through the application of a global force balance compatibility condition that consistently removes the pressure indeterminacy. The Crouzeix-Raviart element is chosen to capture the pressure discontinuity at the two-fluid interface. The resulting discrete equations are solved by an iterative procedure which displays fast convergence characteristics for small capillary numbers. Numerical results for the case of the steady movement of a fluid meniscus in a two-dimensional channel are presented

A residual based error estimator for mortar finite element discretizations

A residual based error estimator for the approximation of linear elliptic boundary value problems by nonconforming finite element methods is introduced and analyzed. In particular, we consider mortar finite element techniques restricting ourselves to geometrically conforming domain decomposition methods using P1 approximations in each subdomain. Additionally, a residual based error estimator for Crouzeix-Raviart elements of lowest order is presented and compared with the error estimator obtained in the more general mortar situation. It is shown that the computational effort of the error estimator can be considerably reduced if the special structure of the Lagrange multiplier is taken into account.

A posteriori error estimates¶for nonconforming finite element schemes

We derive a posteriori error estimates for nonconforming discretizations of Poisson's and Stokes' equations. The estimates are residual based and make use of weight factors obtained by a duality argument. Crouzeix-Raviart elements on triangles and rotated bilinear elements are considered. The quadrilateral case involves the introduction of additional local trial functions. We show that their influence is of higher order and that they can be neglected. The validity of the estimate is demonstrated by computations for the Laplacian and for Stokes' equations.

A finite volume method based on the Crouzeix-Raviart element for elliptic PDE's in two dimensions

We introduce and analyse a finite volume method for the discretization of elliptic boundary value problems in $\mathbb{R}^2$ . The method is based on nonuniform triangulations with piecewise linear nonconforming spaces. We prove optimal order error estimates in the $L^2$ –norm and a mesh dependent $H^1$ –norm.

THE MORTAR ELEMENT METHOD WITH LOCALLY NONCONFORMING ELEMENTS

We consider a discretization of linear elliptic boundary value problems in 2-D by the new version of the mortar finite element method which uses locally nonconforming Crouzeix-Raviart elements. We show that if a solution of the original differential problem belongs to the space H2(Ω), then an error is of the same order as in the standard nonconforming finite element method. We also propose an additive Schwarz method of solving the discrete problem and show that its rate of convergence is almost optimal.

An a posteriori error estimator for the Navier-Stokes equations

ABSTRACT This paper presents an a posteriori error for the incompressible Navier-Stokes equations. Two different solutions are computed, one using the Crouzeix-Raviart element and the second one with the non-conforming Fortin-Soulie element. The error estimator is computed as the difference between these solutions. This estimate is used in an adaptative remeshing strategy for steady state solution of the Navier-Stokes equations.