Abstract. We propose a three-field formulation for efficiently solving a twodimensional Stokes problem in the case of nonstandard boundary conditions. More specifically, we consider the case where the pressure and either normal or tangential components of the velocity are prescribed at some given parts of the boundary. The proposed computational methodology consists in reformulating the considered boundary value problem via a mixed-type formulation where the pressure and the vorticity are the principal unknowns while the velocity is the Lagrange multiplier. The obtained formulation is then discretized and a convergence analysis is performed. A priori error estimates are established, and some numerical results are presented to highlight the perfomance of the proposed computational methodology.
This paper presents new correlations to calculate natural convection heat transfer coefficients (CHTC) in enclosures for building performance simulation. Current work related to the development of correlations is not oriented to building enclosures, and the influence of high numbers of Rayleigh (Ra) and aspect ratio on the CHTC has not been studied in detail. In this work, two new correlations have been developed for a vertical wall, one in laminar regime (Ra < 10 7) and another one in turbulent regime (10 7 < Ra < 10 11). Moreover, two new correlations have been developed for floor and ceiling, one in laminar regime (Ra < 10 6) and another one in turbulent regime (Ra > 10 6). All correlations have been developed as function of the aspect ratio (H/L) and Ra numbers calculated from the enclosure average air temperature and wall surface temperature. By contrast, in previous works the Ra numbers have been calculated from the temperature difference of opposite walls. Computer Fluid Dynamics (CFD) simulations for different aspect ratios (H/L = 0.5-1-2) and Ra numbers (10 3-10 11) have been carried out in order to obtain these correlations. The SIMPLE algorithm has been used for the solution of the Navier-Stokes equations and the realizable turbulence k-ε model with an enhanced wall-function treatment has been used. The correlations developed follow the expected trend for the low number of Ra in comparison with the expressions developed by other authors. For a high number of Ra, our correlations improve the previous correlations, because it is a function of the aspect ratio of the enclosure and the average air temperature of the enclosure. This approach is simple to implement in the construction of thermal simulation programs with low computational cost.
This paper is devoted to the construction of fast solvers for penalty domain decomposition techniques, based upon a posteriori error analysis. We introduce a penalty non-overlapping domain decomposition method (ddm) motivated by the a posteriori error analysis of the method proposed by Chacón and Chacón in [6]. In the new method a H 1/2 00 (Γ) penalty term replaces the L 2 (Γ) one in the original method. The number of iterations needed by the new ddm to yield a solution with an error of the same order as the discretization error is remarkably reduced. We develop an a posteriori error analysis that we use to determine an optimal value of the penalty parameter for a given grid, and also to jointly determine an optimal grid and a penalty parameter to reduce the error below a targeted value. Several numerical tests for model problems exhibit the good performances of our approach and provide to a numerical comparison of the two penalty methods. RésuméCet article a pour but la construction de solveurs rapides pour les techniques de décomposition de domaine avec pénalisation et repose sur une analyse a posteriori. Nous introduisons une méthode de décomposition de domaine sans recouvrement, issue de l'analyse a posteriori de la méthode proposée par Chacón et Chacón [6], où une pénalisation de type H 1/2 00 (Γ) remplace celle de type L 2 (Γ) dans la première méthode. Le nombre d'itérations pour une erreur du même ordre que l'erreur de discrétisation est considérablement réduit. Nous prouvons des estimations d'erreur a posteriori qui permettent d'optimiser le choix du paramètre de pénalisation pour une grille donnée, et aussi lors de l'adaptation de maillage. Plusieurs expériences numériques sur des problèmes académiques montrent les bonnes performances de notre approche et permettent une comparaison numérique des deux méthodes.
Abstract. We introduce a framework for FETI methods using ideas from the decomposition via Lagrange multipliers of H 1 0 (Ω) derived by and complemented with the detailed work on polygonal domains developed by Grisvard [17]. We compute the action of the Lagrange multipliers using the natural H 1/2 00 scalar product, therefore no consistency error appears. As a byproduct, we obtain that the condition number for the iteration matrix is independent of the mesh size and there is no need for preconditioning. This result improves the standard asymptotic bound for this condition number shown by Mandel-Tezaur in [19]. Numerical results that confirm our theoretical analysis are presented.Résumé. Nous proposons une nouvelle approche des méthodes FETI: la décomposition de domaine fait appel aux multiplicateurs de Lagrange tels qu'introduits par Raviart- Thomas [22] et au traitement des domaines polygonaux dûà Grisvard [17]. Ces multiplicateurs utilisent le produit scalaire de H 1/2 00 , de sorte qu'aucune erreur de consistance n'apparaît. En outre, nous prouvons que le nombre de condition de la matrice liéeà chaque itération est indépendant de la taille du maillage, ce qui améliore le résultat de MandelTezaur [19]; par suite, aucun préconditionnement n'est nécessaire. Nous présentons des expériences numériques qui confirment notre analyse.
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