Asymptotic expansions of finite element solutions to Robin problems in H 3 and their application in extrapolation cascadic multigrid method

For the Poisson equation with Robin boundary conditions,by using a few techniques such as orthogonal expansion(M-type),separation of the main part and the finite element projection,we prove for the first time that the asymptotic error expansions of bilinear finite element have the accuracy of O(h3)for u∈H3.Based on the obtained asymptotic error expansions for linear finite elements,extrapolation cascadic multigrid method(EXCMG)can be used to solve Robin problems effectively.Furthermore,by virtue of Richardson not only the accuracy of the approximation is improved,but also a posteriori error estimation is obtained.Finally,some numerical experiments that confirm the theoretical analysis are presented.     

A posteriori error estimates for nonconforming finite element methods for fourth-order problems on rectangles

The a posteriori error analysis of conforming finite element discretisations of the biharmonic problem for plates is well established, but nonconforming discretisations are more easy to implement in practice. The a posteriori error analysis for the Morley plate element appears very particular because two edge contributions from an integration by parts vanish simultaneously. This crucial property is lacking for popular rectangular nonconforming finite element schemes like the nonconforming rectangular Morley finite element, the incomplete biquadratic finite element, and the Adini finite element. This paper introduces a novel methodology and utilises some conforming discrete space on macro elements to prove reliability and efficiency of an explicit residual-based a posteriori error estimator. An application to the Morley triangular finite element shows the surprising result that all averaging techniques yield reliable error bounds. Numerical experiments confirm the reliability and efficiency for the established a posteriori error control on uniform and graded tensor-product meshes.     

Superconvergence analysis and extrapolation of quasi-Wilson nonconforming finite element method for nonlinear Sobolev equations

Quasi-Wilson nonconforming finite element approximation for a class of nonlinear Sobolev equa-tions is discussed on rectangular meshes. We first prove that this element has two special characters by novel approaches. One is that (▽h ( u-Ihu )1, ▽hvh) h may be estimated as order O ( h2 ) when u ∈ H3 (Ω), where Ihu denotes the bilinear interpolation of u , vh is a polynomial belongs to quasi-Wilson finite element space and ▽h denotes the piecewise defined gradient operator, h is the mesh size tending to zero. The other is that the consistency error of this element is of order O ( h2 ) /O ( h3 ) in broken H 1-norm, which is one/two order higher than its interpolation error when u ∈ H3 (Ω) /H4 (Ω). Then we derive the optimal order error estimate and su- perclose property via mean-value method and the known high accuracy result of bilinear element. Furthermore, we deduce the global superconvergence through interpolation post processing technique. At last, an extrapola- tion result of order O ( h3 ), two order higher than traditional error estimate, is obtained by constructing a new suitable extrapolation scheme.     

A posteriori error estimates for hp finite element solutions of convex optimal control problems

In this paper, we present a posteriori error analysis for hp finite element approximation of convex optimal control problems. We derive a new quasi-interpolation operator of Clément type and a new quasi-interpolation operator of Scott-Zhang type that preserves homogeneous boundary condition. The Scott-Zhang type quasi-interpolation is suitable for an application in bounding the errors in L²-norm. Then hp a posteriori error estimators are obtained for the coupled state and control approximations. Such estimators can be used to construct reliable adaptive finite elements for the control problems.     

The finite element method for a degenerate hyperbolic partial differential equation

Cahlon has recently considered finite difference methods for the degenerate Cauchy problem (tu _t)_t=Δu,u(x, 0)=u ₀(x) withu ₀ analytic. In this paper existence of a unique solution is shown for a more general class of initial data. A smoothing property of the solution operator is then exhibited. The usual semidiscrete finite element method is considered. The approximation is shown to be stable and superconvergent with orderO(h ^2μ?2) inl ₂ andl _∞, whereμ?1 is the degree of the polynomials used. OptimalL ² andL ^∞ estimates are also derived.     

A posteriori error estimators for the first-order least-squares finite element method

In this paper, we propose a posteriori error estimators for certain quantities of interest for a first-order least-squares finite element method. In particular, we propose an a posteriori error estimator for when one is interested in where . Our a posteriori error estimators are obtained by assigning proper weight (in terms of local mesh size h_T) to the terms of the least-squares functional. An a posteriori error analysis yields reliable and efficient estimates based on residuals. Numerical examples are presented to show the effectivity of our error estimators.     

Superconvergence of tricubic block finite elements

In this paper, we first introduce interpolation operator of projection type in three dimen- sions, from which we derive weak estimates for tricubic block finite elements. Then using the estimate for the W 2, 1-seminorm of the discrete derivative Green’s function and the weak estimates, we show that the tricubic block finite element solution uh and the tricubic interpolant of projection type Πh3u have superclose gradient in the pointwise sense of the L∞-norm. Finally, this supercloseness is applied to superc...     

A posteriori error estimation in the conforming finite element method based on its local conservativity and using local minimization

We present in this Note fully computable a posteriori error estimates allowing for accurate error control in the conforming finite element discretization of pure diffusion problems. The derived estimates are based on the local conservativity of the conforming finite element method on a dual grid associated with simplex vertices rather than directly on the Galerkin orthogonality. To cite this article: M. Vohralík, C. R. Acad. Sci. Paris, Ser. I 346 (2008).     

A weak Galerkin finite element method for the stokes equations

This paper introduces a weak Galerkin (WG) finite element method for the Stokes equations in the primal velocity-pressure formulation. This WG method is equipped with stable finite elements consisting of usual polynomials of degree k≥1 for the velocity and polynomials of degree k?1 for the pressure, both are discontinuous. The velocity element is enhanced by polynomials of degree k?1 on the interface of the finite element partition. All the finite element functions are discontinuous for which the usual gradient and divergence operators are implemented as distributions in properly-defined spaces. Optimal-order error estimates are established for the corresponding numerical approximation in various norms. It must be emphasized that the WG finite element method is designed on finite element partitions consisting of arbitrary shape of polygons or polyhedra which are shape regular.     

Superconvergence analysis of splitting positive definite nonconforming mixed finite element method for pseudo-hyperbolic equations

In this paper, a new splitting positive definite nonconforming mixed finite element method is proposed for pseudo-hyperbolic equations, in which a quasi-Wilson quadrilateral element is used for the flux p, and the bilinear element is used for u. Superconvergence results in ｜｜·｜｜div,h norm for p and optimal error estimates in L2 norm for u are derived for both semi-discrete and fully discrete schemes under almost uniform meshes.     

A posteriori FE error control for p-Laplacian by gradient recovery in quasi-norm

A posteriori error estimators based on quasi-norm gradient recovery are established for the finite element approximation of the p-Laplacian on unstructured meshes. The new a posteriori error estimators provide both upper and lower bounds in the quasi-norm for the discretization error. The main tools for the proofs of reliability are approximation error estimates for a local approximation operator in the quasi-norm.

     

Numerical analysis of the primal problem of elastoplasticity with hardening

Summary. The finite element method is a reasonable and frequently utilised tool for the spatial discretization within one time-step in an elastoplastic evolution problem. In this paper, we analyse the finite element discretization and prove a priori and a posteriori error estimates for variational inequalities corresponding to the primal formulation of (Hencky) plasticity. The finite element method of lowest order consists in minimising a convex function on a subspace of continuous piecewise linear resp. piecewise constant trial functions. An a priori error estimate is established for the fully-discrete method which shows linear convergence as the mesh-size tends to zero, provided the exact displacement field u is smooth. Near the boundary of the plastic domain, which is unknown a priori, it is most likely that u is non-smooth. In this situation, automatic mesh-refinement strategies are believed to improve the quality of the finite element approximation. We suggest such an adaptive algorithm on the basis of a computable a posteriori error estimate. This estimate is reliable and efficient in the sense that the quotient of the error by the estimate and its inverse are bounded from above. The constants depend on the hardening involved and become larger for decreasing hardening. Received May 7, 1997 / Revised version received August 31, 1998     

New a posteriori L∞(L2) and L2(L2)-error estimates of mixed finite element methods for general nonlinear parabolic optimal control problems

We study new a posteriori error estimates of the mixed finite element methods for general optimal control problems governed by nonlinear parabolic equations. The state and the co-state are discretized by the high order Raviart-Thomas mixed finite element spaces and the control is approximated by piecewise constant functions. We derive a posteriori error estimates in L^∞(J; L²Ω)-norm and L²(J; L²Ω)-norm for both the state, the co-state and the control approximation. Such estimates, which seem to be new, are an important step towards developing a reliable adaptive mixed finite element approximation for optimal control problems. Finally, the performance of the posteriori error estimators is assessed by two numerical examples.     

Supraconvergence and Supercloseness of a Scheme for Elliptic Equations on Nonuniform Grids

In this paper, we study the convergence of a finite difference scheme on nonuniform grids for the solution of second-order elliptic equations with mixed derivatives and variable coefficients in polygonal domains subjected to Dirichlet boundary conditions. We show that the scheme is equivalent to a fully discrete linear finite element approximation with quadrature. It exhibits the phenomenon of supraconvergence, more precisely, for s ∈ [1,2] order O(h ^s )-convergence of the finite difference solution, and its gradient is shown if the exact solution is in the Sobolev space H ^1+s (Ω). In the case of an equation with mixed derivatives in a domain containing oblique boundary sections, the convergence order is reduced to O(h ^3/2?ε) with ε > 0 if u ∈ H ³(Ω). The second-order accuracy of the finite difference gradient is in the finite element context nothing else than the supercloseness of the gradient. For s ∈ {1,2}, the given error estimates are strictly local.     

Inhomogeneous Dirichlet conditions in a priori and a posteriori finite element error analysis

Summary. The numerical solution of elliptic boundary value problems with finite element methods requires the approximation of given Dirichlet data u_D by functions u_D,h in the trace space of a finite element space on _D. In this paper, quantitative a priori and a posteriori estimates are presented for two choices of u_D,h, namely the nodal interpolation and the orthogonal projection in L²(_D) onto the trace space. Two corresponding extension operators allow for an estimate of the boundary data approximation in global H¹ and L² a priori and a posteriori error estimates. The results imply that the orthogonal projection leads to better estimates in the sense that the influence of the approximation error on the estimates is of higher order than for the nodal interpolation.Mathematics Subject Classification (1991): 65N30, 65R20, 73C50This work was initiated while C. Carstensen was visiting the Max Planck Institute for Mathematics in the Sciences, Leipzig. S. Bartels acknowledges support by the German Research Foundation (DFG) within the Graduiertenkolleg Effiziente Algorithmen und Mehrskalenmethoden and the priority program Analysis, Modeling, and Simulation of Multiscale Problems. G. Dolzmann gratefully acknowledges partial support by the Max Planck Society and by the NSF through grant DMS0104118.     

Galerkin discontinuous approximation of the transport equation and viscoelastic fluid flow on quadrilaterals

Numerical simulation of industrial processes involving viscoelastic liquids is often based on finite element methods on quadrilateral meshes. However, numerical analysis of these methods has so far been limited to triangular meshes. In this work, we consider quadrilateral meshes. We first study the approximation of the transport equation by a Galerkin discontinuous method and prove an 𝒪(h^k+1/2) error estimates for the Q_k finite element. Then we study a differential model for viscoelastic flow with unknowns u the velocity, p the pressure, and σ the viscoelastic part of the extra-stress tensor. The approximations are ((Q₁)² transforms of) Q_k+1 continuous for u, Q_k discontinuous for σ, and P_k discontinuous for p, with k ≥ 1. Upwinding for σ is obtained by the Galerkin discontinuous method. We show that an error estimate of order 𝒪(h^k+1/2) is valid in the energy norm for the three unknowns. © 1998 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 14: 97–114, 1998     

Asymptotically exact a posteriori estimators for the pointwise gradient error on each element in irregular meshes. Part 1: A smooth problem and globally quasi-uniform meshes

A class of a posteriori estimators is studied for the error in the maximum-norm of the gradient on single elements when the finite element method is used to approximate solutions of second order elliptic problems. The meshes are unstructured and, in particular, it is not assumed that there are any known superconvergent points. The estimators are based on averaging operators which are approximate gradients, ``recovered gradients', which are then compared to the actual gradient of the approximation on each element. Conditions are given under which they are asympotically exact or equivalent estimators on each single element of the underlying meshes. Asymptotic exactness is accomplished by letting the approximate gradient operator average over domains that are large, in a controlled fashion to be detailed below, compared to the size of the elements.

     

On the error bounds of nonconforming finite elements

We prove that the error estimates of a large class of nonconforming finite elements are dominated by their approximation errors, which means that the well-known Cea's lemma is still valid for these nonconforming finite element methods. Furthermore, we derive the error estimates in both energy and L2 norms under the regularity assumption u ∈ H1+s(Ω) with any s 0. The extensions to other related problems are possible.     

Recovery type superconvergence and a posteriori error estimates for control problems governed by Stokes equations

In this paper, we derive recovery type superconvergence analysis and a posteriori error estimates for the finite element approximation of the distributed optimal control governed by Stokes equations. We obtain superconvergence results and asymptotically exact a posteriori error estimates by applying two recovery methods, which are the patch recovery technique and the least-squares surface fitting method. Our results are based on some regularity assumption for the Stokes control problems and are applicable to the first order conforming finite element method with regular but nonuniform partitions.     

Superconvergence and a posteriori error estimates of RT1 mixed methods for elliptic control problems with an integral constraint

In this paper, we investigate the superconvergence property and a posteriori error estimates of mixed finite element methods for a linear elliptic control problem with an integral constraint. The state and co-state are approximated by the order k = 1 Raviart-Thomas mixed finite element spaces and the control variable is approximated by piecewise constant functions. Approximations of the optimal control of the continuous optimal control problem will be constructed by a projection of the discrete adjoint state. It is proved that these approximations have convergence order h ². Moreover, we derive a posteriori error estimates both for the control variable and the state variables. Finally, a numerical example is given to demonstrate the theoretical results.