Extrapolation cascadic multigrid method on piecewise uniform grid

The triangular linear fnite elements on piecewise uniform grid for an elliptic problem in convex polygonal domain are discussed.Global superconvergence in discrete H1-norm and global extrapolation in discrete L2-norm are proved.Based on these global estimates the conjugate gradient method(CG)is efective,which is applied to extrapolation cascadic multigrid method(EXCMG).The numerical experiments show that EXCMG is of the global higher accuracy for both function and gradient.     

Mapped Finite Elements on Hexahedra. Necessary and Sufficient Conditions for Optimal Interpolation Errors

This paper considers finite elements which are defined on hexahedral cells via a reference transformation which is in general trilinear. For affine reference mappings, the necessary and sufficient condition for an interpolation order O(h ^k+1) in the L ²-norm and O(h ^k ) in the H ¹-norm is that the finite dimensional function space on the reference cell contains all polynomials of degree less than or equal to k. The situation changes in the case of a general trilinear reference transformation. We will show that on general meshes the necessary and sufficient condition for an optimal order for the interpolation error is that the space of polynomials of degree less than or equal to k in each variable separately is contained in the function space on the reference cell. Furthermore, we will show that this condition can be weakened on special families of meshes. These families which are obtained by applying usual refinement techniques can be characterized by the asymptotic behaviour of the semi-norms of the reference mapping.     

The Crouzeix-Raviart type nonconforming finite element method for the nonstationary Navier-Stokes equations on anisotropic meshes

This paper is devoted to study the Crouzeix-Raviart （C-R） type nonconforming linear triangular finite element method （FEM） for the nonstationary Navier-Stokes equations on anisotropic meshes. By intro- ducing auxiliary finite element spaces, the error estimates for the velocity in the L2-norm and energy norm, as well as for the pressure in the L2-norm are derived.     

Undecidability in matrices over Laurent polynomials

We show that it is undecidable for finite sets S of upper triangular (4×4)-matrices over Z[x,x⁻¹] whether or not all elements in the semigroup generated by S have a nonzero constant term in some of the Laurent polynomials of the first row. This result follows from a representations of the integer weighted finite automata by matrices over Laurent polynomials.     

Superconvergence phenomenon in the finite element method arising from averaging gradients

Summary We study a superconvergence phenomenon which can be obtained when solving a 2^nd order elliptic problem by the usual linear elements. The averaged gradient is a piecewise linear continuous vector field, the value of which at any nodal point is an average of gradients of linear elements on triangles incident with this nodal point. The convergence rate of the averaged gradient to an exact gradient in theL ²-norm can locally be higher even by one than that of the original piecewise constant discrete gradient.     

Mixed covolume method for parabolic problems on triangular grids

In this paper, we present a mixed covolume method for parabolic equations on triangular grids. This method use the lowest order Raviart–Thomas (R–T) mixed finite element space as the trial space. We prove the optimal order of convergence for the approximate pressure and velocity in L²-norm. Furthermore, we obtain the quasi-optimal error estimates for the approximate pressure in L^∞-norm.     

L p -norms of Hermite polynomials and an extremal problem on Wiener chaos

We establish sharp asymptotics for theL ^p -norm of Hermite polynomials and prove convergence in distribution of suitably normalized Wick powers. The results are combined with numerical integration to study an extremal problem on Wiener chaos.     

Superconvergence analysis of approximate boundary-flux calculations

Summary Certain projection post-processing techniques have been proposed for computing the boundary flux for two-dimensional problems (e.g., see Carey, et al. [5]). In a series of numerical experiments on elliptic problems they observed that these post-processing formulas for approximate fluxes were almost (O(h ²)-accurate for linear triangular elements. In this paper we prove that the computed boundary flux isO(h ² ln 1/h)-accurate in the maximum norm for the partial method of [5]. If the solutionuH ³() then the boundary flux error isO(h ^3/2) in theL ²-norm.     

Error estimates for a finite element-finite volume discretization of convection-diffusion equations

We consider a time-dependent linear convection-diffusion equation. This equation is approximated by a combined finite element-finite volume method: the diffusion term is discretized by Crouzeix-Raviart piecewise linear finite elements, and the convection term by upwind barycentric finite volumes on a triangular grid. An implicit Euler approach is used for time discretization. It is shown that the error associated with this scheme, measured by a discrete L^∞-L²- and L²-H¹-norm, respectively, decays linearly with the mesh size and the time step. This result holds without any link between mesh size and time step. The dependence of the corresponding error bound on the diffusion coefficient is completely explicit.     

Linear and Discontinuous Approximations for Optimal Control Problems

Abstract

An optimal control problem for 2D and 3D elliptic equations is investigated with pointwise control constraints. This paper is concerned with the discretization of the control by piecewise linear but discontinuous functions. The state and the adjoint state are discretized by linear finite elements. The paper is focused on similarities and differences to piecewise constant and piecewise linear (continuous) approximation of the controls. Approximation of order h in the L ^∞-norm is proved in the main result.     

On a least-squares formulation for hyperelastic,transversely isotropic problems

In the present work a mixed finite element based on a least-squares formulation is proposed. In detail, the provided constitutive relation is based on a hyperelastic free energy including terms describing a transversely isotropic material behavior. Basis for the element formulation is a weak form resulting from a least-squares method, see e.g. [1]. The L₂-norm minimization of the residuals of the given first-order system of differential equations leads to a functional depending on displacements and stresses. The interpolation of the unknowns is executed using different approximation spaces for the stresses (W^q (div, Ω)) and the displacements (W^1,p(Ω)), under consideration of suitable p and q. For the approximation of the stresses vector-valued shape functions of Raviart-Thomas type, related to the edges of the respective triangular element, are applied. Standard interpolation polynomials are used for the continuous approximation of the displacements. The performance of the proposed formulation will be investigated considering a numerical example. (© 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A Priori Error Estimates for Finite Element Methods for H (2,1)-Elliptic Equations

The convergence of finite element methods for linear elliptic boundary value problems of second and forth order is well understood. In this article, we introduce finite element approximations of some linear semi-elliptic boundary value problem of mixed order on a two-dimensional rectangular domain Q. The equation is of second order in one direction and forth order in the other and appears in the optimal control of parabolic partial differential equations if one eliminates the control and the state (or the adjoint state) in the first order optimality conditions. We establish a regularity result and estimate for the finite element error of conforming approximations of this equation. The finite elements in use have a tensor product structure, in one dimension we use linear, quadratic or cubic Lagrange elements in the other dimension cubic Hermite elements. For these elements, we prove the error bound O(h ² + τ ^k ) in the energy norm and O((h ² + τ ^k )(h ² + τ)) in the L ²(Q)-norm.     

A Class of Extremal Functions and Trigonometric Polynomials

In this paper we present two classes of extremal approximating functions. These functions have the property that they are entire, have finite exponential type, and provide excellent approximations along the real line for a specific set of functions. One class of functions provides majorants and minorants, while the other class minimizes theL¹-norm on the real line. As applications we construct extremal trigonometric polynomials and obtain an inequality involving almost periodic trigonometric polynomials.     

On the L 1-condition number of the univariate Bernstein basis

We show that the size of the 1-norm condition number of the univariate Bernstein basis for polynomials of degree n is O (2ⁿ / √n). This is consistent with known estimates [3], [5] for p = 2 and p = ∞ and leads to asymptotically correct results for the p-norm condition number of the Bernstein basis for any p with 1 ≤ p ≤ ∞.     

Variational Problems in Domains with Cusp-Points and the Finite Element Method

A variational problem in a two-dimensional domain with cusp-points corresponding to a linear elliptic boundary value problem is formulated and the unique existence of its solution is proved. The corresponding finite element method using triangular finite C ⁰-elements with polynomials of the first degree is analyzed and both the convergence (under the assumptions sufficient for the existence of the exact solution) and the maximal rate of convergence 𝒪(h) are proved.     

Estimates of the norm of the Mercer kernel matrices with discrete orthogonal transforms

The paper is related to the lower and upper estimates of the norm for Mercer kernel matrices. We first give a presentation of the Lagrange interpolating operators from the view of reproducing kernel space. Then, we modify the Lagrange interpolating operators to make them bounded in the space of continuous function and be of the de la Vallée Poussin type. The order of approximation by the reproducing kernel spaces for the continuous functions is thus obtained, from which the lower and upper bounds of the Rayleigh entropy and the l ²-norm for some general Mercer kernel matrices are provided. As an example, we give the l ²-norm estimate for the Mercer kernel matrix presented by the Jacobi algebraic polynomials. The discussions indicate that the l ²-norm of the Mercer kernel matrices may be estimated with discrete orthogonal transforms. Supported by the national NSF (No: 10871226) of P.R. China.     

A new stabilized finite volume method for the stationary Stokes equations

In this paper we develop and study a new stabilized finite volume method for the two-dimensional Stokes equations. This method is based on a local Gauss integration technique and the conforming elements of the lowest-equal order pair (i.e., the P ₁–P ₁ pair). After a relationship between this method and a stabilized finite element method is established, an error estimate of optimal order in the H ¹-norm for velocity and an estimate in the L ²-norm for pressure are obtained. An optimal error estimate in the L ²-norm for the velocity is derived under an additional assumption on the body force. This work is supported in part by the NSF of China 10701001 and by the US National Science Foundation grant DMS-0609995 and CMG Chair Funds in Reservoir Simulation.     

A simple technique for computing theH ∞-norm of polynomials

In this note, computation of theH -norm of polynomials is considered. It is shown that direct computation of theH -norm of polynomials, based on the definition of the norm, results in a simple and inexpensive technique for computing the norm.     

The heterogeneous multiscale finite element method FE-HMM for the homogenization of linear elastic solids

The objective of the present paper is a formulation of the Heterogeneous Multiscale Finite Element Method (FE-HMM) for the homogenization of linear elastic solids in a geometrical linear frame, and doing so, for the first time, of a vector-valued field problem. The macro stiffness is estimated by stiffness sampling on heterogeneous microdomains in terms of a modified quadrature formula, which implies an equivalence of energy densities of the microscale with the macroscale. Existing a-priori estimates are assessed and used for optimal micro-macro refinement strategies in the H¹-norm and the L²-norm. (© 2016 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Direct Evaluation of Nonlinear Gradient Elasticity in 2D with C1 Continuous Discretization Methods

In this contribution, we present a study on the performance of four different subparametric triangular finite element formulations for linear and nonlinear strain gradient elastic continua. Fully C¹ and nonconforming C^* finite elements are considered for a numerical analysis. The results are demonstrated by means of two computational examples together with the L², H¹ and H² error norms. An interesting aspect concerning the linear geometry approximation is revealed. (© 2010 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)