A posteriori bounds for linear functional outputs of Crouzeix–Raviart finite element discretizations of the incompressible Stokes problem

A finite element technique is presented for the efficient generation of lower and upper bounds to outputs which are linear functionals of the solutions to the incompressible Stokes equations in two space dimensions. The finite element discretization is effected by Crouzeix–Raviart elements, the discontinuous pressure approximation of which is central to this approach. The bounds are based upon the construction of an augmented Lagrangian: the objective is a quadratic ‘energy’ reformulation of the desired output, the constraints are the finite element equilibrium equations (including the incompressibility constraint), and the inter‐sub‐domain continuity conditions on velocity. Appealing to the dual max–min problem for appropriately chosen candidate Lagrange multipliers then yields inexpensive bounds for the output associated with a fine‐mesh discretization. The Lagrange multipliers are generated by exploiting an associated coarse‐mesh approximation. In addition to the requisite coarse‐mesh calculations, the bound technique requires the solution of only local sub‐domain Stokes problems on the fine mesh. The method is illustrated for the Stokes equations, in which the outputs of interest are the flow rate past and the lift force on a body immersed in a channel. Copyright © 2000 John Wiley & Sons, Ltd.     

Stereochemical Course of the Reaction between Thiocarbonyl Compounds and Oxiranes: Reaction with cis‐ and trans‐2,3‐Dimethyloxirane

The reactions of thiocarbonyl compounds with cis‐2,3‐dimethyloxirane ( 1a ) in CH₂Cl₂ in the presence of BF₃⋅Et₂O or SnCl₄ led to trans‐4,5‐dimethyl‐1,3‐oxathiolanes, whereas with trans‐2,3‐dimethyloxirane ( 1b ) cis‐4,5‐dimethyl‐1,3‐oxathiolanes were formed. With the stronger Lewis acid SnCl₄, the formation of side‐products was also observed. In the case of 1,3‐thiazole‐5(4H)‐thione 2 , these side‐products are the corresponding 1,3‐thiazol‐5(4H)‐one 5 and the 1 : 2 adduct 8 (Schemes 2 – 4). Their formation can be rationalized by the decomposition of the initially formed spirocyclic 1,3‐oxathiolane and by a second addition onto the C=N bond of the 1 : 1 adduct, respectively. The secondary epimerization by inversion of the configuration of the spiro‐C‐atom (Schemes 5 – 7) can be explained by a Lewis‐acid‐catalyzed ring opening of the 1,3‐oxathiolane ring and subsequent ring closure to the thermodynamically more stable isomer (Scheme 12). In the case of 2,2,4,4‐tetramethyl‐3‐thioxocyclobutanone ( 20 ), apart from the expected spirocyclic 1,3‐oxathiolanes 21 and 23 , dispirocyclic 1 : 2 adducts were formed by a secondary addition onto the C=O group of the four‐membered ring (Schemes 9 and 10).     

Desulfurization of 4-Nitro-N,2-diphenyl-3-(phenylamino)isothiazol- 5(2H)-imine: Formation of a 3-Imino-2-nitroprop-2-enamidine

The course of the desulfurization reaction of 4-nitro-N,2-diphenyl-3-(phenylamino)isothiazol-5(2H)-imine ( 3 ) is investigated and the formation of the unstable 3-imino-2-nitroprop-2-enamidine ( A ) as intermediate is discussed. Addition of amines and thiophenol to the reaction mixture yielded the amidine derivatives 5 and the thioimidate 6 , respectively, via nucleophilic addition of the respective reagent to A (Scheme 2). Benzoic acid and thiobenzoic acid afforded the amide 7 and the thioamide 8 , respectively, as secondary products of the expected adducts 7a and 8a (Schemes 3 and 4). The presence of (benzylidene)(methyl)amine in the reaction mixture of the desulfurization of 3 led to the 1,2,4-oxadiazole derivative 10 , together with the quinoxaline N-oxide 4 as a minor product. Reaction mechanisms involving an intermediate ketene imine and participation of the NO₂ group in the reaction leading to 1,2,4-oxadiazole 10 are proposed. Ab initio calculations of model structures for the nitroketene imine were performed and the results correlated with the experimental results. The structures of 8 and 10 were established by X-ray crystal-structure analysis.