Multigrid methods for nonconforming finite element methods
SIAM Journal on Numerical Analysis
Mixed and hybrid finite element methods
Mixed and hybrid finite element methods
A parallel multigrid algorithm for solving the Navier-Stokes equations
IMPACT of Computing in Science and Engineering
Error estimators for nonconforming finite element approximations of the Stokes problem
Mathematics of Computation
Multigrid and multilevel methods for nonconforming Q1 elements
Mathematics of Computation
Multigrid Algorithms for Nonconforming and Mixed Methods for Nonsymmetric and Indefinite Problems
SIAM Journal on Scientific Computing
Coupling Fluid Flow with Porous Media Flow
SIAM Journal on Numerical Analysis
High-resolution FEM-TVD schemes based on a fully multidimensional flux limiter
Journal of Computational Physics
Robust A Posteriori Error Estimation for Nonconforming Finite Element Approximation
SIAM Journal on Numerical Analysis
Inf-sup stable non-conforming finite elements of arbitrary order on triangles
Numerische Mathematik
SIAM Journal on Numerical Analysis
A unifying theory of a posteriori error control for nonconforming finite element methods
Numerische Mathematik
Framework for the A Posteriori Error Analysis of Nonconforming Finite Elements
SIAM Journal on Numerical Analysis
Multilevel preconditioning of rotated bilinear non-conforming FEM problems
Computers & Mathematics with Applications
Nonconforming, Anisotropic, Rectangular Finite Elements of Arbitrary Order for the Stokes Problem
SIAM Journal on Numerical Analysis
A monolithic FEM-multigrid solver for non-isothermal incompressible flow on general meshes
Journal of Computational Physics
Applied Numerical Mathematics - Selected papers from the 16th Chemnitz finite element symposium 2003
Newton multigrid least-squares FEM for the V-V-P formulation of the Navier-Stokes equations
Journal of Computational Physics
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We show that existing quadrilateral nonconforming finite elements of higher order exhibit a reduction in the order of approximation if the sequence of meshes is still shape-regular but consists no longer of asymptotically affine equivalent mesh cells. We study second order nonconforming finite elements as members of a new family of higher order approaches which prevent this order reduction. We present a new approach based on the enrichment of the original polynomial space on the reference element by means of nonconforming cell bubble functions which can be removed at the end by static condensation. Optimal estimates of the approximation and consistency error are shown in the case of a Poisson problem which imply an optimal order of the discretization error. Moreover, we discuss the known nonparametric approach to prevent the order reduction in the case of higher order elements, where the basis functions are defined as polynomials on the original mesh cell. Regarding the efficient treatment of the resulting linear discrete systems, we analyze numerically the convergence of the corresponding geometrical multigrid solvers which are based on the canonical full order grid transfer operators. Based on several benchmark configurations, for scalar Poisson problems as well as for the incompressible Navier-Stokes equations (representing the desired application field of these nonconforming finite elements), we demonstrate the high numerical accuracy, flexibility and efficiency of the discussed new approaches which have been successfully implemented in the FeatFlow software (www.featflow.de). The presented results show that the proposed FEM-multigrid combinations (together with discontinuous pressure approximations) appear to be very advantageous candidates for efficient simulation tools, particularly for incompressible flow problems.