Adding user interactivity to FreeFem++ with EJS: From teaching to research

Conferencia plenaria por invitaciónIn this talk we propose the use of an authoring tool, Easy Java Simulations, to build interactive simulations for pdes where the computing kernel is FreeFem++. This tool allows to play with all the parameters of the pde problem at will, mesh size, pde terms, boundary conditions, etc... and creates applets for the student to play and learn in real time on the influence of these parameters on the solution of the pde.Universidad de Málaga. Campus de Excelencia Internacional Andalucia Tech. Conferencias del plan propio de investigación

Stabilization of a non standard FETI-DP mortar method for the Stokes problem

In a recent paper [E. Chacón Vera and D. Franco Coronil, J. Numer. Math. 20 (2012) 161–182.] a non standard mortar method for incompressible Stokes problem was introduced where the use of the trace spaces H1/2 and H1/2 00 and a direct computation of the pairing of the trace spaces with their duals are the main ingredients. The importance of the reduction of the number of degrees of freedom leads naturally to consider the stabilized version and this is the results we present in this work. We prove that the standard Brezzi–Pitkaranta stabilization technique is available and that it works well with this mortar method. Finally, we present some numerical tests to illustrate this behaviour.Ministerio de Ciencia e InnovaciónJunta de Andaluci

A domain decomposition method derived from the Primal Hybrid Formulations for 2nd order elliptic problems

We consider the primal hybrid formulation for second order elliptic problems introduced by Raviart-Thomas and apply the classical iterative method of Uzawa to obtain a non overlapping domain decomposition method that converges geometrically with a mesh independent ratio. The proposed method connects with the Finite Element Tearing and Interconnecting (FETI) method proposed by Farhat-Roux and collaborators. In this research work we use the detailed work on domains with corners developed by Grisvard [6], which clarifies the situation of cross-points, and the direct computation of the duality H−1/2 − H1/2 using the H1/2 scalar product; therefore no consistency error appears

A posteriori error analysis for two non-overlapping domain decomposition techniques

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 [T. Chacón Rebollo, E. Chacón Vera, A non-overlapping domain decomposition method for the Stokes equations via a penalty term on the interface, C. R. Acad. Sci. Paris, Ser. I 334 (2002) 1–16]. In the new method a penalty term replaces the L2(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.Cet 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 [Chacón Rebollo, T., Chacón Vera, E., A non-overlapping domain decomposition method for the Stokes equations via a penalty term on the interface. C.R. Acad. Sci. Paris, t. 334, Série I; pp. 1–16, 2002.6], où une pénalisation de type H 1/2 00 (Γ) remplace celle de type L2 (Γ) dans la première méthode. Le nombre d’itérations pour une erreur du mème 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.Dirección General de Investigació

Aumento de la eficiencia de un método de descomposición de dominio mediante estimaciones a posteriori

En este trabajo introducimos un método de descomposición de dominio sin solapamiento con penalización, que viene motivado a partir de un análisis del error a posteriori del método estudiado por T. Chacón y E. Chacón en [5] y [6]. Con el objetivo de mejorar la tasa de convergencia del método de [6], en este trabajo, introducimos una nueva versi´on de este método en la cual un término de penalización H 1/2 00 (Γ) reemplaza el término L 2 (Γ) del original de [6]. Usando este nuevo término, el nímero de iteraciones necesarias para alcanzar una solución con un error del mismo orden que el error de discretización, se reduce significativamente. Realizamos además un análisis de error a posteriori, que nos permite desarrollar un estrategia para determinar simultáneamente un parámetro de penalización óptimo y una malla optimal, para reducir el error por debajo de un valor prefijado. Varios test numéricos muestran los buenos resultados de nuestras aproximaciones

A non-standard FETI-DP mortar method for Stokes problem

The trace spaces H 1/2 and H 1/2 00 play a key role in the FETI and mortar families of domain decomposition methods. However, a direct numerical evaluation of these norms is usually avoided. On the other hand, and for stability issues, the subspace of functions for which their jumps across the interfaces of neighbouring subdomains belong to these trace spaces yields a more suitable framework than the standard broken Sobolev space. Finally, the nullity of these jumps is usually imposed via Lagrange multipliers and using the pairing of the trace spaces with their duals. A direct computation of these pairings can be performed using the Riesz-canonical isometry. In this work we consider all these ingredients and introduce a domain decomposition method that falls into the FETI-DP mortar family. The application is to the incompressible Stokes problem and we see that continuous bounds are replicated at the discrete level. As a consequence, no stabilization is required. Some numerical tests are finally presented.Ministerio de Educación y Cienci