A lowest order divergence-free finite element on rectangular grids

It is shown that the conforming Q _2,1;1,2-Q′₁ mixed element is stable, and provides optimal order of approximation for the Stokes equations on rectangular grids. Here, Q _2,1;1,2 = Q _2,1 × Q _1,2, and Q _2,1 denotes the space of continuous piecewise-polynomials of degree 2 or less in the x direction but of degree 1 in the y direction. Q′₁ is the space of discontinuous bilinear polynomials, with spurious modes filtered. To be precise, Q′₁ is the divergence of the discrete velocity space Q _2,1;1,2. Therefore, the resulting finite element solution for the velocity is divergence-free pointwise, when solving the Stokes equations. This element is the lowest order one in a family of divergence-free element, similar to the families of the Bernardi-Raugel element and the Raviart-Thomas element.     

On transport equations driven by a non‐divergence‐free force field

The paper is devoted to initial boundary value problems for transport equations with non‐divergence‐free external field. The crucial role is played by integration along characteristics and associated Green's formula for which we provide a new proof which generalizes and clarifies previous versions. The paper concludes with an application of general theory to the Spencer–Lewis equation. Copyright © 2007 John Wiley & Sons, Ltd.     

Low‐order stabilized finite element methods for the unsteady Stokes/Navier‐Stokes equations with friction boundary conditions

In this paper, we consider low‐order stabilized finite element methods for the unsteady Stokes/Navier‐Stokes equations with friction boundary conditions. The time discretization is based on the Euler implicit scheme, and the spatial discretization is based on the low‐order element (P₁−P₁ or P₁−P₀) for the approximation of the velocity and pressure. Moreover, some error estimates for the numerical solution of fully discrete stabilized finite element scheme are obtained. Finally, numerical experiments are performed to confirm our theoretical results.     

WEB‐spline based mesh‐free finite element analysis for the heat equation and the time‐dependent Navier–Stokes equation: A survey

In this article, we give some numerical techniques and error estimates using web‐spline based mesh‐free finite element method for the heat equation and the time‐dependent Navier–Stokes equations on bounded domains. The web‐spline method uses weighted extended B‐splines on a regular grid as basis functions and does not require any grid generation. We demonstrate the method by providing numerical results for the Poisson's and stationary Stokes equation. © 2012 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2013     

Superconvergence of a 3D finite element method for stationary Stokes and Navier‐Stokes problems

For the Poisson equation on rectangular and brick meshes it is well known that the piecewise linear conforming finite element solution approximates the interpolant to a higher order than the solution itself. In this article, this type of supercloseness property is established for a special interpolant of the Q₂ ? P element applied to the 3D stationary Stokes and Navier‐Stokes problem, respectively. Moreover, applying a Q₃ ? P postprocessing technique, we can also state a superconvergence property for the discretization error of the postprocessed discrete solution to the solution itself. Finally, we show that inhomogeneous boundary values can be approximated by the Lagrange Q₂‐interpolation without influencing the superconvergence property. Numerical experiments verify the predicted convergence rates. Moreover, a cost‐benefit analysis between the two third‐order methods, the post‐processed Q₂ ? P discretization, and the Q₃ ? P discretization is carried out. © 2004 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2005     

A nonconforming quadrilateral element with maximal inf‐sup constant

A new nonconforming element is introduced for quadrilateral meshes. The element is designed to maximize the inf‐sup constant for a Stokes element pair. Numerical results are presented and we observe that the maximizing inf‐sup constant results in efficiency of computing time. © 2013 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 30: 12‐132, 2014     

Numerical analysis of a locking‐free mixed finite element method for a bending moment formulation of Reissner‐Mindlin plate model

This article deals with the approximation of the bending of a clamped plate, modeled by Reissner‐Mindlin equations. It is known that standard finite element methods applied to this model lead to wrong results when the thickness t is small. Here, we propose a mixed formulation based on the Hellinger‐Reissner principle which is written in terms of the bending moments, the shear stress, the rotations and the transverse displacement. To prove that the resulting variational formulation is well posed, we use the Babu?ka‐Brezzi theory with appropriate t ‐dependent norms. The problem is discretized by standard mixed finite elements without the need of any reduction operator. Error estimates are proved. These estimates have an optimal dependence on the mesh size h and a mild dependence on the plate thickness t. This allows us to conclude that the method is locking‐free. The proposed method yields direct approximation of the bending moments and the shear stress. A local postprocessing leading to H¹ ‐type approximations of transverse displacement and rotations is introduced. Moreover, we propose a hybridization procedure, which leads to solving a significantly smaller positive definite system. Finally, we report numerical experiments which allow us to assess the performance of the method. © 2012 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2013     

A finite element method for the axisymmetric three‐field Stokes system with a discontinuous pressure

A quadrilateral based velocity‐pressure‐extrastress tensor mixed finite element method for solving the three‐field Stokes system in the axisymmetric case is studied. The method derived from Fortin's Q₂ − P₁ velocity‐pressure element is to be used in connection with the standard Galerkin formulation. This makes it particularly suitable for the numerical simulation of viscoelastic flow. It is proven to be second‐order convergent in the natural weighted Sobolev norms, for the system under consideration. The crucial result that the method is uniformly stable is proven for the case of rectangular meshes. © 1999 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 15: 739–763, 1999     

Two‐level Fourier analysis of multigrid for higher‐order finite‐element discretizations of the Laplacian

In this paper, we employ local Fourier analysis (LFA) to analyze the convergence properties of multigrid methods for higher‐order finite‐element approximations to the Laplacian problem. We find that the classical LFA smoothing factor, where the coarse‐grid correction is assumed to be an ideal operator that annihilates the low‐frequency error components and leaves the high‐frequency components unchanged, fails to accurately predict the observed multigrid performance and, consequently, cannot be a reliable analysis tool to give good performance estimates of the two‐grid convergence factor. While two‐grid LFA still offers a reliable prediction, it leads to more complex symbols that are cumbersome to use to optimize parameters of the relaxation scheme, as is often needed for complex problems. For the purposes of this analytical optimization as well as to have simple predictive analysis, we propose a modification that is “between” two‐grid LFA and smoothing analysis, which yields reasonable predictions to help choose correct damping parameters for relaxation. This exploration may help us better understand multigrid performance for higher‐order finite element discretizations, including for Q₂‐Q₁ (Taylor‐Hood) elements for the Stokes equations. Finally, we present two‐grid and multigrid experiments, where the corrected parameter choice is shown to yield significant improvements in the resulting two‐grid and multigrid convergence factors.     

A multilevel finite element method in space‐time for the Navier‐Stokes problem

A multilevel finite element method in space‐time for the two‐dimensional nonstationary Navier‐Stokes problem is considered. The method is a multi‐scale method in which the fully nonlinear Navier‐Stokes problem is only solved on a single coarsest space‐time mesh; subsequent approximations are generated on a succession of refined space‐time meshes by solving a linearized Navier‐Stokes problem about the solution on the previous level. The a priori estimates and error analysis are also presented for the J‐level finite element method. We demonstrate theoretically that for an appropriate choice of space and time mesh widths: h_j ～ h, k_j ～ k, j = 2, …, J, the J‐level finite element method in space‐time provides the same accuracy as the one‐level method in space‐time in which the fully nonlinear Navier‐Stokes problem is solved on a final finest space‐time mesh. © 2005 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2005     

Wavelets‐Galerkin scheme for a Stokes problem

This note presents a wavelets‐Galerkin scheme for the numerical solution of a Stokes problem by using the scaling function of a symmetric biorthogonal spline wavelets that can be modified to generate the divergence‐free wavelets. © 2003 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 20: 193–198, 2004     

Least‐squares mixed finite element methods for the incompressible Navier‐Stokes equations

Least‐squares mixed finite element schemes are formulated to solve the evolutionary Navier‐Stokes equations and the convergence is analyzed. We recast the Navier‐Stokes equations as a first‐order system by introducing a vorticity flux variable, and show that a least‐squares principle based on L² norms applied to this system yields optimal discretization error estimates. © 2002 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 18: 441–453, 2002; Published online in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/num.10015     

Semi‐analytical integration of the elastic stiffness matrix of an axisymmetric eight‐noded finite element

The finite element method (FEM) is a numerical method for approximate solution of partial differential equations with appropriate boundary conditions. This work describes a methodology for generating the elastic stiffness matrix of an axisymmetric eight‐noded finite element with the help of Computer Algebra Systems. The approach is described as “semi analytical” because the formulation mimics the steps taken using Gaussian numerical integration techniques. The semianalytical subroutines developed herein run 50[percnt] faster than the conventional Gaussian integration approach. The routines, which are made publically available for download,¹ should help FEM researchers and engineers by providing significant reductions of CPU times when dealing with large finite element models. © 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2010     

Superconvergence of discontinuous Galerkin finite element method for the stationary Navier‐Stokes equations

This article focuses on discontinuous Galerkin method for the two‐ or three‐dimensional stationary incompressible Navier‐Stokes equations. The velocity field is approximated by discontinuous locally solenoidal finite element, and the pressure is approximated by the standard conforming finite element. Then, superconvergence of nonconforming finite element approximations is applied by using least‐squares surface fitting for the stationary Navier‐Stokes equations. The method ameliorates the two noticeable disadvantages about the given finite element pair. Finally, the superconvergence result is provided under some regular assumptions. © 2006 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 23: 421–436, 2007     

Analysis and finite element simulations of a second‐order fluid model in a bounded domain

This article is concerned with the equations governing the steady motion of a viscoelastic incompressible second‐order fluid in a bounded domain. A new proof of existence and uniqueness of strong solutions is given. In addition, using appropriate finite element methods to approximate a coupled equivalent problem, sharp error estimates are obtained using a fixed point argument. The method is applied to the two‐dimensional lid‐driven cavity problem, at low Reynolds number and in a certain range of values of the viscoelastic parameters, to analyze the combined effects of inertia and viscoelasticity on the flow. © 2007 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2007     

Second‐order Galerkin‐Lagrange method for the Navier‐Stokes equations (retracted article)

It has come to the attention of the editors and publisher that an article published in Numerical Methods and Partial Differential Equations, “Second‐order Galerkin‐Lagrange method for the Navier‐Stokes equations,” by Mohamed Bensaada, Driss Esselaoui, and Pierre Saramito, Numer Methods Partial Differential Eq 21(6) (2005), 1099–1121 included large portions that were copied from the following paper without proper citation: “Convergence and nonlinear stability of the Lagrange‐Galerkin method for the Navier‐Stokes equations,” Endre Suli, Numerische Mathematik, Vol. 53, No. 4, pp. 459–486 (July, 1988). We have retracted the paper and apologize to Dr. Suli Numer Methods Partial Differential Eq (2007)23(1)211 .     

On a mixed finite element formulation of a second‐order quasilinear problem in the plane

We analyze a mixed finite element discretization of a second‐order quasilinear problem based on the Raviart‐Thomas space. We prove that the discrete problem is solvable and provide a local uniqueness result for the solution. We also obtain optimal order L²‐error estimates for both the scalar variable and the associated flux. The main feature of our method is that it is free from the boundness conditions required in previous works on the coefficients of the quasilinear operator. © 2003 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 20: 90–103, 2004.     

Stability of a combined finite element ‐ finite volume discretization of convection‐diffusion equations

We consider a time‐dependent and a stationary convection‐diffusion equation. These equations are approximated by a combined finite element – finite volume method: the diffusion term is discretized by Crouzeix‐Raviart piecewise linear finite elements on a triangular grid, and the convection term by upwind barycentric finite volumes. In the nonstationary case, we use an implicit Euler approach for time discretization. This scheme is shown to be L²‐stable uniformly with respect to the diffusion coefficient. In addition, it turns out that stability is unconditional in the time‐dependent case. These results hold if the underlying grid satisfies a condition that is fulfilled, for example, by some structured meshes. © 2012 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 28: 402–424, 2012