Mixed finite element approximation of the vector potential

Summary We consider a mixed finite element approximation of the three dimensional vector potential, which plays an important rôle in the simulation of perfect fluids and in the calculation of rotational corrections to transonic potential flows. The central point of our approach is a saddlepoint formulation of the essential boundary conditions. In particular, this avoids the wellknown Babuka paradox when approximating smooth domains by polyhedrons. Using piecewise linear/piecewise constant elements for the vector potential/the boundary terms, we obtain optimal error estimates under minimal regularity assumptions for the solution of the continuous problem.     

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.     

Finite element approximation on incompressible Navier-Stokes equations with slip boundary condition

Summary We consider a mixed finite element approximation of the stationary, incompressible Navier-Stokes equations with slip boundary condition, which plays an important rôle in the simulation of flows with free surfaces and incompressible viscous flows at high angles of attack and high Reynold's numbers. The central point is a saddle-point formulation of the boundary conditions which avoids the well-known Babuka paradox when approximating smooth domains by polyhedrons. We prove that for the new formulation one can use any stable mixed finite element for the Navier-Stokes equations with no-slip boundary condition provided suitable bubble functions on the boundary are added to the velocity space. We obtain optimal error estimates under minimal regularity assumptions for the solution of the continous problem. The techniques apply as well to the more general Navier boundary condition.     

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].

     

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.     

喹喔啉—2,3—二甲酰胺与3d过渡金属配合物的合成及性质研究

首次合成了喹喔啉－２，３－二甲酰胺（Ｑｘｄａ）与３ｄ过渡金属的三种固体配合物：Ｍ（Ｑｘｄａ）２ｃｌ２（Ｍ＝Ｃｏ（Ｉ），Ｎｉ（Ｉ），Ｃｕ（Ⅱ）。通过元素分析，摩尔电导，红外光谱，电子光谱，磁化率，热分析等手段对配合物的组成和性质进行了研究。     

IR multiphoton dissociation of CF3I. Pressure effect with inert and reactive gases

The IR multiphoton dissociation of CF₃I has been studied in the presence of isobutane and with isobutane and Ar and CO₂ as inert gases. The dependence of the reaction probability P(Φ) with fluence (ϕ) is confirmed. Modeling of the experimental results shows that for the energy transfer processes the average energy transferred per collision 〈†E〉_d varies with (ϕ).     

Cu(OTf)2-catalyzed Et3SiH-reductive etherification of various carbonyl compounds with trimethylsilyl ethers

A triethylsilane-reductive etherification of the trimethylsilyl ethers with a variety of carbonyl compounds in good yields at room temperature employing 0.5 mol% Cu(OTf)₂ as an extremely efficient catalyst is described here.