The average a posteriori error of numerical methods

Summary The definition of the average error of numerical methods (by example of a quadrature formula to approximateS(f)= f d on a function classF) is difficult, because on many important setsF there is no natural probability measure in the sense of an equidistribution. We define the average a posteriori error of an approximation by an averaging process over the set of possible information, which is used by (in the example of a quadrature formula,N(F)={(f(a ₁), ...,f/fF} is the set of posible information). This approach has the practical advantage that the averaging process is related only to finite dimensional sets and uses only the usual Lebesgue measure. As an application of the theory I consider the numerical integration of functions of the classF={f:[0,1]/f(x)–f(y)||x–y|}. For arbitrary (fixed) knotsa _i we determine the optimal coefficientsc _i for the approximation and compute the resulting average error. The latter is minimal for the knots . (It is well known that the maximal error is minimal for the knotsa _i .) Then the adaptive methods for the same problem and methods for seeking the maximum of a Lipschitz function are considered. While adaptive methods are not better when considering the maximal error (this is valid for our examples as well as for many others) this is in general not the case with the average error.     

An asymptotic finite element method for improvement of solutions of boundary layer problems

Summary A scheme that uses singular perturbation theory to improve the performance of existing finite element methods is presented. The proposed scheme improves the error bounds of the standard Galerkin finite element scheme by a factor of O(ⁿ⁺¹) (where is the small parameter andn is the order of the asymptotic approximation). Numerical results for linear second order O.D.E.'s are given and are compared with several other schemes.     

Consistency of a linear system of inequalities

We show that a consistency check of a linear system of inequalitiesAxb reduces to check whetherQb0 for a certain matrixQ. It is a direct consequence of the Farkas-Minkowski theorem. Thus, when one has to check consistency for different values ofb, one need not run a numerical algorithm for eachb.On leave at the Electronics Research Laboratory of the University of California at Berkeley in a CNRS/NSF Exchange Program.     

A priori-Schranken für diskrete Lösungen monotoner Operatorgleichungen und Anwendungen

Summary Operator equationsTu=f are approximated by Galerkin's method, whereT is a monotone operator in the sense of Browder and Minty, so that existence results are available in a reflexive Banach spaceX. In a normed spaceY error estimates are established, which require a priori bounds for the discrete solutionsu _h in the norm of a suitable space . Sufficient conditions for the uniform boundedness u _{h Z} =O(1) ash0 are proved. Well-known error estimates in [3] for the special caseX=Y=Z are generalized by this. The theory is applied to quasilinear elliptic boundary value problems of order 2m in a bounded domain . The approximating subspaces are finite element spaces. Especially the caseX=W ₀ ^{m, p} (), 1<p<,Y=W ₀ ^{m. 2} (),Z=W ₀ ^{m. max (2,p)} ()W^m, () is treated. Some examples for 1<p<2 are considered. Forp2 a refined technique is introduced in the author's paper [7].

     

流动注射分析中样品分散与反应管长度的关系

根据串联釜模型给出样品分散度与通过管长的关系式，应用此式可以满意地解释分散度随管长的变化。     

Synthesis, crystal structure and magnetic properties of a novel complex [Cu(NIT2Py)(PDA)(H2O)]·(CH3OH)(H2O)

A novel complex [Cu(NIT2Py)(PDA)(H₂O)]·(CH₃OH)(H₂O) has been synthesized and structurally characterized by X-ray diffraction methods. It crystallizes in the monoclinic space group P2(1)/c. The structure consists of [Cu(NIT2Py)(PDA)(H₂O)] moiety, one solvent methanol molecule and one water molecule. The copper(II) ion is in a distorted octahedral environment: one nitrogen atom and one oxygen atom from the NIT2Py, one nitrogen atom from the PDA (2,6-pyridine dicarboxylic acid) and one oxygen atom from the aqueous in the basal plane; two oxygen atoms from the PDA in the axial position. The units of [Cu(NIT2Py)(PDA)(H₂O)] were connected as one dimension chain by the intermolecular hydrogen bonds. The complex exhibits intramolecular antiferromagnetic interactions between the Cu(II) ion and the NIT2Py.     

Thermal decomposition of Cd(CH3COO)2·2H2O studied by a coupled TG-DTA-MS method

The thermal decomposition of cadmium acetate dihydrate in helium and in air atmosphere has been investigated by means of a coupled TG-DTA-MS method combined with X-ray diffraction analysis. Dehydration of Cd(CH₃COO)₂·2H₂O is a two-stage process with Cd(CH₃COO)₂·H₂O as intermediate. The way of Cd(CH₃COO)₂ decomposition strongly depends on the surrounding gas atmosphere and the rate of heating. CdO, acetone and CO₂ are the primary products of decomposition in air. In helium decomposition goes by two parallel and consecutive reactions in which intermediates, Cd and CdCO₃, are formed. Metallic cadmium oxidizes and cadmium carbonate decomposes giving CdO. Some of the metallic cadmium, depending on the heating rate and the concentration of oxygen, evaporates. Acetone is partially oxidized in secondary reactions with oxygen. This revised version was published online in July 2006 with corrections to the Cover Date.     

有机锗化合物中碳氢的快速测定

报道了用Co3O4 作覆盖氧化剂分析有机锗化合物中碳氢元素的方法。通过对 9个样品的十几次分析 ,碳、氢的相对误差分别小于± 1.30 % ,4 .4 0 %。方法简便 ,结果稳定可靠 ,可用于含锗有机化合物的碳氢分析。