On the construction of stable curvilinear pp version elements for mixed formulations of elasticity and Stokes flow

Summary. The use of mixed finite element methods is well-established in the numerical approximation of the problem of nearly incompressible elasticity, and its limit, Stokes flow. The question of stability over curved elements for such methods is of particular significance in the p version, where, since the element size remains fixed, exact representation of the curved boundary by (large) elements is often used. We identify a mixed element which we show to be optimally stable in both p and h refinement over curvilinear meshes. We prove optimal p version (up to ) and h version (p = 2, 3) convergence for our element, and illustrate its optimality through numerical experiments. Received August 25, 1998 / Revised version received February 16, 1999 / Published online April 20, 2000 –? Springer-Verlag 2000     

A non-conforming finite element method with anisotropic mesh grading for the Stokes problem in domains with edges

The solution of the Stokes problem in three-dimensional domainswith edges has anisotropic singular behaviour which is treatednumerically by using anisotropic finite element meshes. Thevelocity is approximated by Crouzeix–Raviart (nonconformingP₁ ) elements and the pressure by piecewise constants. Thismethod is stable for general meshes (without minimal or maximalangle condition). Denoting by N_e the number of elements in themesh, the interpolation and consistency errors are of the optimalorder h N_e^–1/3 which is proved for tensor product meshes.As a by-product, we analyse also nonconforming prismatic elementswith P₁ [oplus ] span {x₃²} as the local space for the velocitywhere x₃ is the direction of the edge.     

A fully discrete stabilized finite-element method for the time-dependent Navier-Stokes problem

A fully discrete stabilized finite-element method is presentedfor the two-dimensional time-dependent Navier–Stokes problem.The spatial discretization is based on a finite-element spacepair (X_h, M_h) for the approximation of the velocity and thepressure, constructed by using the Q₁ – P₀ quadrilateralelement or the P₁ – P₀ triangular element; the time discretizationis based on the Euler semi-implicit scheme. It is shown thatthe proposed fully discrete stabilized finite-element methodresults in the optimal order error bounds for the velocity andthe pressure.     

On approximating the solution of the non-stationary Stokes equations using the cell discretization algorithm

The cell discretization algorithm, a nonconforming extension of the finite element method, is used to obtain approximations to the velocity and pressure satisfying the nonstationary Stokes equations. Error estimates show convergence of the approximations. An implementation using polynomial bases is described that permits the use of the continuous approximations of the h–p finite element method and exactly satisfies the solenoidal requirement. We express the error estimates in terms of the diameter h of a cell and the degree p of the approximation on each cell. Results of an experiment with p10 are presented that confirm the theoretical estimates.     

The approximation theory for the p-version finite element method and application to non-linear elliptic PDEs

Approximation theoretic results are obtained for approximation using continuous piecewise polynomials of degree p on meshes of triangular and quadrilateral elements. Estimates for the rate of convergence in Sobolev spaces , are given. The results are applied to estimate the rate of convergence when the p-version finite element method is used to approximate the -Laplacian. It is shown that the rate of convergence of the p-version is always at least that of the h-version (measured in terms of number of degrees of freedom used). If the solution is very smooth then the p-version attains an exponential rate of convergence. If the solution has certain types of singularity, the rate of convergence of the p-version is twice that of the h-version. The analysis generalises the work of Babuska and others to the case . In addition, the approximation theoretic results find immediate application for some types of spectral and spectral element methods. Received August 2, 1995 / Revised version received January 26, 1998     

Existence of unique solutions for perturbed autonomous differential equations

We consider the existence of unique absolutely continuous solutionsfor x' = p(t)f(x) + p(t)h(t), t 0, x(0) = 0, where p, f, andh are positive almost everywhere, but none of them needs becontinuous or monotone. Moreover, p and f can be unbounded aroundzero. Our uniqueness results are not based on assumptions onthe differences f(x) – f(y), as it is usual in most uniquenessresults, and they are new even when p, f, and h are continuous.     

A note on least-squares mixed finite elements in relation to standard and mixed finite elements

^** Email: brandts{at}science.uva.nl The least-squares mixed finite-element method for second-orderelliptic problems yields an approximation u_h V_h H₀¹() of thepotential u together with an approximation p_{h h} H(div ; )of the vector field p = – Au. Comparing u_h with the standardfinite-element approximation of u in V_h, and p_h with the mixedfinite-element approximation of p, it turns out that they arehigher-order perturbations of each other. In other words, theyare ‘superclose’. Refined a priori bounds and superconvergenceresults can now be proved. Also, the local mass conservationerror is of higher order than could be concluded from the standarda priori analysis.     

On solenoidal high‐degree polynomial approximations to solutions of the stationary Stokes equations

The cell discretization algorithm, a nonconforming extension of the finite element method, is used to obtain approximations to the velocity and pressure functions satisfying the Stokes equations. Error estimates show convergence of the method. An implementation using polynomial bases is described that permits the use of the continuous approximations of the h‐p finite element method and exactly satisfies the solenoidal requirement. We express the error estimates in terms of the diameter h of a cell and degree p of the approximation on each cell. Examples of 10^th degree polynomial approximations are described that substantiate the theoretical estimates. © 2000 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 16: 480–493, 2000     

On optimal convergence rates for higher-order Navier-Stokes approximations. I. Error estimates for the spatial discretization

^** Email: bause{at}am.uni-erlangen.de Due to the increasing use of higher-order methods in computationalfluid dynamics, the question of optimal approximability of theNavier–Stokes equations under realistic assumptions onthe data has become important. It is well known that the regularitycustomarily hypothesized in the error analysis for parabolicproblems cannot be assumed for the Navier–Stokes equations,as it depends on non-local compatibility conditions for thedata at time t = 0, which cannot be verified in practice. Takinginto account this loss of regularity at t = 0, improved convergenceof the order (min{h^(5/2)–,h³/t^(1/4)+}), for any >0, is shown locally in time for the spatial discretization ofthe velocity field by (non-)conforming finite elements of third-orderapproximability properties. The error estimate itself is provedby energy methods, but it is based on sharp a priori estimatesfor the Navier–Stokes solution in fractional-order spacesthat are derived by semigroup methods and complex interpolationtheory and reflect the optimal regularity of the solution ast 0.     

Optimal error estimates for the hp-version interior penalty discontinuous Galerkin finite element method

We consider the hp-version interior penalty discontinuous Galerkinfinite-element method (hp-DGFEM) for second-order linear reaction–diffusionequations. To the best of our knowledge, the sharpest knownerror bounds for the hp-DGFEM are due to Rivière et al.(1999,Comput. Geosci., 3, 337–360) and Houston et al.(2002,SIAM J. Numer. Anal., 99, 2133–2163). These are optimalwith respect to the meshsize h but suboptimal with respect tothe polynomial degree p by half an order of p. We present improvederror bounds in the energy norm, by introducing a new functionspace framework. More specifically, assuming that the solutionsbelong element-wise to an augmented Sobolev space, we deducefully hp-optimal error bounds.     

Convergence rates for the coupling of FEM and collocation BEM

We prove convergence of the coupling of finite and boundaryelements where Galerkin's methd is used for finite elementsand collocation for boundary elements. We consider linear ellipticboundary value problems in two dimensions, in particular problemsin elasticity. The mesh width k of the boundary elements andthe mesh width h of the finite elements are required to satisfykßh with suitable ß. Asymptotic error estimatesin the energy norm and in the L²-norm are derived. Numericalexamples are included.     

The streamline-diffusion method for nonconforming Qrot1 elements on rectangular tensor-product meshes

When the streamline–diffusion finite element method isapplied to convection–diffusion problems using nonconformingtrial spaces, it has previously been observed that stabilityand convergence problems may occur. It has consequently beenproposed that certain jump terms should be added to the bilinearform to obtain the same stability and convergence behaviouras in the conforming case. The analysis in this paper showsthat for the Q^rot₁ 1 element on rectangular shape-regular tensor-productmeshes, no jump terms are needed to stabilize the method. Inthis case moreover, for smooth solutions we derive in the streamline–diffusionnorm convergence of order h^3/2 (uniformly in the diffusion coefficientof the problem), where h is the mesh diameter. (This estimateis already known for the conforming case.) Our analysis alsoshows that similar stability and convergence results fail tohold true for analogous piecewise linear nonconforming elements.     

Domain decomposition algorithms for indefinite weakly singular integral equations: the h and p versions

We extend the approach of Cai and Widlund (Domain decompositionalgorithms for indefinite elliptic problems, SIAM J. Sci. Stat.Comput. 13 (1992), 243-258), which was designed for finite elementdiscretizations, to boundary element discretizations of indefiniteweakly singular integral equations. Both the h and p versionsof the Galerkin approximation are considered. We prove thatthe additive Schwarz method suggested by Cai and Widlund canbe used for this equation as an efficient preconditioner forGMRES, an iterative method of conjugate gradient type. For bothversions, the rates of convergence of this iterative methodare shown to approach 1 only logarithmically as the degreesof freedom tend to infinity.     

Error estimate for convection problem with characteristics method

We consider the numerical approximation of a first order steady-state convective problem by the method of characteristics with pseudo-time step k and P _r discontinuous finite elements on a mesh T _h. We show, when the entry flow field is Ω filling and under a technical hypothesis, the existence and the uniqueness for the continuous and the discrete variational problems (P^k) and P( _h ^k ) and we give an error estimate. This revised version was published online in June 2006 with corrections to the Cover Date.     

Mixed hp-DGFEM for incompressible flows II: Geometric edge meshes

We consider the Stokes problem of incompressible fluid flowin three-dimensional polyhedral domains discretized on hexahedralmeshes with hp-discontinuous Galerkin finite elements of typeQ_k for the velocity and Q_k–1 for the pressure. We provethat these elements are inf-sup stable on geometric edge meshesthat are refined anisotropically and non-quasiuniformly towardsedges and corners. The discrete inf-sup constant is shown tobe independent of the aspect ratio of the anisotropic elementsand is of O(k^–3/2) in the polynomial degree k, as in thecase of conforming Q_k–Q_k–2 approximations on thesame meshes.     

Improving the accuracy of the mini-element approximation to Navier-Stokes equations

^** Email: blanca{at}imati.cnr.it^*** Email: frutos{at}mac.cie.uva.es^**** Corresponding author. Email: julia.novo{at}uam.es A technique to improve the accuracy of the mini-element approximationto incompressible the Navier–Stokes equations is introduced.Once the mini-element approximation has been computed at a fixedtime, the linear part of this approximation is postprocessedby solving a discrete Stokes problem. The bubble functions neededto stabilize the approximation to the Navier–Stokes equationsare not used at the postprocessing step. This postprocessingprocedure allows us to increase by one unit (up to a logarithmicterm) the H¹ norm rate of convergence of the velocity and correspondinglythe L² norm of the pressure. An error analysis of the algorithmis performed.     

On the Solution of Some Elliptic Difference Equations

Iterative methods for the solution of some nonlinear ellipticdifference systems, approximating the first boundary value problemare considered. If h > 0 is the network step in the spaceof variables x = (x₁, x₂,..., x_p) and 2m is the order of theoriginal boundary value problem, then the iterative methodsproposed give solution of accuracy with the expenditure ofO(|In | h^–(p+m–)) and O(|In | |In h| h^–p)arithmetic operations in the case of a general region and arectangular parallelepiped respectively. In the case p = 2 theestimate O(|In | h^{–[2+ (m/2)]}) is obtained if the regionis made up of rectangles with sides parallel to the co-ordinateaxes.     

Discontinuous Stable Elements for the Incompressible Flow

In this paper, we derive a discontinuous Galerkin finite element formulation for the Stokes equations and a group of stable elements associated with the formulation. We prove that these elements satisfy the new inf–sup condition and can be used to solve incompressible flow problems. Associated with these stable elements, optimal error estimates for the approximation of both velocity and pressure in L ² norm are obtained for the Stokes problems, as well as an optimal error estimate for the approximation of velocity in a mesh dependent norm.     

Compact High-Order Finite Differences for Interface Problems in One Dimension

This paper is concerned with the construction and analysis ofcompact finite difference approximations to the model linearsource problem –(pu')' + qu = f where the functions p,q, and f can have jump discontinuities at a finite number ofpoints. Explicit formulae that give O(h²) O(h³) and O(h⁴) accuracyare derived, and a procedure for computing three-point schemesof any prescribed order of accuracy is presented. A rigoroustruncation and discretization error analysis is offered. Numericalresults are also given.