The strengthened C.B.S. inequality constant for second order elliptic partial differential operator and for hierarchical bilinear finite element functions

We estimate the constant in the strengthened Cauchy-Bunyakowski-Schwarz inequality for hierarchical bilinear finite element spaces and elliptic partial differential equations with coefficients corresponding to anisotropy (orthotropy). It is shown that there is a nontrivial universal estimate, which does not depend on anisotropy. Moreover, this estimate is sharp and the same as for hierarchical linear finite element spaces.This research was supported by the Grant Agency of Czech Republic under the contract No. 201/02/0595.     

Some remarks on the constant in the strengthened CBS inequality: Estimate for hierarchical finite element discretizations of elasticity problems

For a class of two‐dimensional boundary value problems including diffusion and elasticity problems, it is proved that the constants in the corresponding strengthened Cauchy‐Buniakowski‐Schwarz (CBS) inequality in the cases of two‐level hierarchical piecewise‐linear/piecewise‐linear and piecewise‐linear/piecewise‐quadratic finite element discretizations with triangular meshes differ by the factor 0.75. For plane linear elasticity problems and triangulations with right isosceles triangles, formulas are presented that show the dependence of the constant in the CBS inequality on the Poisson's ratio. Furthermore, numerically determined bounds of the constant in the CBS inequality are given for plane linear elasticity problems discretized by means of arbitrary triangles and for three‐dimensional elasticity problems discretized by means of tetrahedral elements. Finally, the robustness of iterative solvers for elasticity problems is discussed briefly. © 1999 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 15: 469–487, 1999     

A numerical experimental study of inverse preconditioning for the parallel iterative solution to 3D finite element flow equations

Integration of the subsurface flow equation by finite elements (FE) in space and finite differences (FD) in time requires the repeated solution to sparse symmetric positive definite systems of linear equations. Iterative techniques based on preconditioned conjugate gradients (PCG) are one of the most attractive tool to solve the problem on sequential computers. A present challenge is to make PCG attractive in a parallel computing environment as well. To this aim a key factor is the development of an efficient parallel preconditioner. FSAI (factorized sparse approximate inverse) and enlarged FSAI relying on the approximate inverse of the coefficient matrix appears to be a most promising parallel preconditioner. In the present paper PCG using FSAI, diagonal and pARMS (parallel algebraic recursive multilevel solvers) preconditioners is implemented on the IBM SP4/512 and CLX/768 supercomputers with up to 32 processors to solve underground flow problems of a large size. The results show that FSAI may allow for a parallel relative efficiency larger than 50% on the largest problems with p=32 processors. Moreover, FSAI turns out to be significantly less expensive and more robust than pARMS. Finally, it is shown that for p in the upper range may be much improved if PCG–FSAI is implemented on CLX.     

Uniform estimate of the constant in the strengthened CBS inequality for anisotropic non‐conforming FEM systems

Preconditioners based on various multilevel extensions of two‐level finite element methods (FEM) lead to iterative methods which have an optimal order computational complexity with respect to the size of the system. Such methods were first presented in Axelsson and Padiy (SIAM. J. Sci. Stat. Comp. 1990; 20 :1807) and Axelsson and Vassilevski (Numer. Math. 1989; 56 :157), and are based on (recursive) two‐level splittings of the finite element space. The key role in the derivation of optimal convergence rate estimates is played by the constant γ in the so‐called Cauchy–Bunyakowski–Schwarz (CBS) inequality, associated with the angle between the two subspaces of the splitting. It turns out that only existence of uniform estimates for this constant is not enough but accurate quantitative bounds for γ have to be found as well. More precisely, the value of the upper bound for γ∈(0,1) is part of the construction of various multilevel extensions of the related two‐level methods. In this paper, an algebraic two‐level preconditioning algorithm for second‐order elliptic boundary value problems is constructed, where the discretization is done using Crouzeix–Raviart non‐conforming linear finite elements on triangles. An important point to make is that in this case the finite element spaces corresponding to two successive levels of mesh refinements are not nested. To handle this, a proper two‐level basis is considered, which enables us to fit the general framework for the construction of two‐level preconditioners for conforming finite elements and to generalize the method to the multilevel case. The major contribution of this paper is the derived estimates of the related constant γ in the strengthened CBS inequality. These estimates are uniform with respect to both coefficient and mesh anisotropy. To our knowledge, the results presented in the paper are the first such estimates for non‐conforming FEM systems. Copyright © 2004 John Wiley & Sons, Ltd.     

Upper Bound of the Constant in Strengthened C.B.S. Inequality for Systems of Linear Partial Differential Equations

The constant in the strengthened Cauchy–Bunyakowski–Schwarz (C.B.S.) inequality plays a crucial role in the convergence rate of multilevel iterative methods as well as in the efficiency of a posteriori error estimators, that is the framework of finite element approximations of systems of partial differential equations. We consider an approximation of general systems of linear partial differential equations in R ³. Concerning a multilevel convergence rate corresponding to nested general tetrahedral meshes of size h and 2h, we give an estimate of this constant for general three-dimensional cases.     

A bound for the optimal constant in an inequality of Ladyzhenskaya and Solonnikov

Ladyzhenskaya & Solonnikov (1976) introduced a representationtheorem in ³, which contained an integral inequality involvinga multiplicative dimensionless constant. The existence of theconstant was established but not its magnitude which dependsonly on the shape of the domain. In this paper, we derive anupper bound for the optimal constant when the underlying domainis star shaped.     

Maximal smoothness for solutions to equilibrium equations in 2D nonlinear elasticity

For a class of variational integrals from 2D nonlinear elasticity, we prove that any weak solution for the equilibrium equations is smooth. Moreover, we present an example showing that the assumption is optimal.

     

A mixed‐FEM formulation for nonlinear incompressible elasticity in the plane

This article deals with an expanded mixed finite element formulation, based on the Hu‐Washizu principle, for a nonlinear incompressible material in the plane. We follow our related previous works and introduce both the stress and the strain tensors as further unknowns, which yields a two‐fold saddle point operator equation as the corresponding variational formulation. A slight generalization of the classical Babu?ka‐Brezzi's theory is applied to prove unique solvability of the continuous and discrete formulations, and to derive the corresponding a priori error analysis. An extension of the well‐known PEERS space is used to define an stable associated Galerkin scheme. Finally, we provide an a posteriori error analysis based on the classical Bank‐Weiser approach. © 2002 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 18: 105–128, 2002     

PBVP for integro-differential equations of volterra type in B.S.

In this paper, we study the PBVP for integro-differential equations of Volterra type in Banach spaces. By developing monotone iterative technique for the PBVP, we get some results concerning the existence of extremal solutions, which are the limits of monotone sequences.The project supported by the Natural Science Foundation of Shandong Province.     

Evaluations of Parameters for the Optimization of S.S.O.R. and A.D.I. Preconditioning

The main goal of this paper is to give two ways to estimate the needed parameters in order to obtain the condition number of S.S.O.R. preconditioned matrices, namely, the algebraic matricial formulation of convexity Riesz theorem and the tridiagonal Fourier analysis. The improvement with respect to Axelsson's approach is explicitly given. Estimations of the condition number in the case of A.D.I. preconditioning is also considered.     

A comparison of NRBCs for PUFEM in 2D Helmholtz problems at high wave numbers

In this work, exact and approximate non-reflecting boundary conditions (NRBCs) are implemented with the Partition of Unity Finite Element Method (PUFEM) to solve short wave scattering problems governed by the Helmholtz equation in two dimensions. By short wave problems, we mean situations in which the wavelength is a small fraction of the characteristic dimension of the scatterer. Various NRBCs are implemented and a comparison of their performance is carried out based on the accuracy of the results, ease of implementation and computational cost. The aim is to accurately model such problems in a reduced computational domain around the scatterer with fewer elements and without refining the mesh at each wave number.     

Optimization of the Relaxation Parameter for S.S.O.R. and A.D.I. Preconditioning

We present and compare several approaches for the optimization of the relaxation parameter both for A.D.I. and S.S.O.R. basic iteration and preconditioning conjugate gradient method. For each kind of preconditioning a detailed link between estimates of the spectral radius of the iteration matrix and of the condition number resulting from preconditioning is proposed. It allows to choose the best approach in order to obtain the optimal relaxation parameter and the corresponding optimal estimates either of the spectral radius of the iteration matrix and of the resulting condition mumber of the S.S.O.R. and A.D.I. preconditioning.     

On the adaptive coupling of FEM and BEM in 2–d–elasticity

Summary. This paper concerns the combination of the finite element method (FEM) and the boundary element method (BEM) using the symmetric coupling. As a model problem in two dimensions we consider the Hencky material (a certain nonlinear elastic material) in a bounded domain with Navier–Lamé differential equation in the unbounded complementary domain. Using some boundary integral operators the problem is rewritten such that the Galerkin procedure leads to a FEM/BEM coupling and quasi–optimally convergent discrete solutions. Beside this a priori information we derive an a posteriori error estimate which allows (up to a constant factor) the error control in the energy norm. Since information about the singularities of the solution is not available a priori in many situation and having in mind the goal of an automatic mesh–refinement we state adaptive algorithms for the –version of the FEM/BEM–coupling. Illustrating numerical results are included. Received April 15, 1994 / Revised version received January 8, 1996     

The structure of the boundary element matrix for the three-dimensional Dirichlet problem in elasticity

The algebraic properties of the matrix arising for the three-dimensional Dirichlet problem for Lamé equations in a rotational domain by the boundary element method are considered. The use of the special basis leads to a matrix having a block structure with sparse blocks. The possible strategies for the efficient solution of the above problem are discussed. © 1998 John Wiley & Sons, Ltd.     

Two Simple Derivations of Universal Bounds for the C.B.S. Inequality Constant

Universal bounds for the constant in the strengthened Cauchy-Bunyakowski-Schwarz inequality for piecewise linear-linear and piecewise quadratic-linear finite element spaces in 2 space dimensions are derived. The bounds hold for arbitrary shaped triangles, or equivalently, arbitrary matrix coefficients for both the scalar diffusion problems and the elasticity theory equations.     

Asymptotic behavior for the quasi-geostrophic equations with fractional dissipation in R2

In this paper the surface quasi-geostrophic equations (QGE) with fractional dissipation in ${\mathbb{R}}^2$ are considered. Our aim is to study the long-time behavior of solutions of QGE in the subcritical case. To this end we investigate the global well-posedness and global attractor for QGE in $H^s({\mathbb{R}}^2)$ via commutator estimates for nonlinear terms, a new iterative technique for estimates of higher order derivatives and with the help of a nonlocal damping term. Besides, by using the fractional Lieb–Thirring inequality, estimates of the finite Hausdorff and fractal dimensions of the global attractor are found.     

Well-posedness for the 2D dissipative quasigeostrophic equations in the Besov space

In this paper, we consider the initial value problem of the 2D dissipative quasi-geostrophic equations. Existence and uniqueness of the solution global in time are proved in the homogenous Besov space Bp,∞ s p with small data when 1 /2<α≤1,2/2α-1< p<∞,sp=2/p-(2α-1). Our proof is based on a new characterization of the homogenous Besov space and Kato's method.     

Estimate of the Constant in Two Strengthened C.B.S. Inequalities for F.E.M. Systems of 2D Elasticity: Application to Multilevel Methods and a Posteriori Error Estimators

The constant γ in the strengthened Cauchy-Buniakowski-Schwarz (C.B.S.) inequality plays a crucial role in the convergence rate of multilevel iterative methods as well as in the efficiency of a posteriori error estimators, that is in the framework of finite element approximations of SPD problems. We consider the approximation of the 2D elasticity problem by the Courant element. Concerning multilevel convergence rate, that is the γ corresponding to nested general triangular meshes of size h and 2h, we have proved that γ²≤ 3/4$ uniformly on the mesh and the Poisson ratio. Concerning error estimator, that is the γ corresponding to quadratic and linear approximations on the same mesh, numerical computations have shown that the exact γ for a reference element deteriorates that is goes to one, when the Poisson ratio tends to 1/2     

The best constant in the Davis inequality for the expectation of the martingale square function

A method is introduced for the simultaneous study of the square function and the maximal function of a martingale that can yield sharp norm inequalities between the two. One application is that the expectation of the square function of a martingale is not greater than times the expectation of the maximal function. This gives the best constant for one side of the Davis two-sided inequality. The martingale may take its values in any real or complex Hilbert space. The elementary discrete-time case leads quickly to the analogous results for local martingales indexed by . Some earlier inequalities are also improved and, closely related, the Lévy martingale is embedded in a large family of submartingales.