Problème de Dirichlet

Résumé

En mathématiques, le problème de Dirichlet est de trouver une fonction harmonique définie sur un ouvert \Omega, de \mathbb R^n prolongeant une fonction continue définie sur la frontière de l'ouvert \Omega,. Ce problème porte le nom du mathématicien allemand Johann Peter Gustav Lejeune Dirichlet. Exposé du problème Il n'existe pas toujours de solution au problème de Dirichlet. Solutions au problème Exemple : solution sur un disque dans ℝ² Dans cette partie, \Omega =D(0,1), où D(0,1) est le disque de centre 0 et de rayon 1. Il existe alors une solution au problème de Dirichlet, définie ci-dessous. On a toujours G : \partial \Omega \to \mathbb R continue sur \partial \Omega . On pose : g : \begin{matrix} \mathbb R & \to & \mathbb R \ \theta & \mapsto & G(\cos \theta, \sin \theta) \end{matrix}. La solution est \Phi : \Omega \to \mathbb

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https://fr.wikipedia.org/wiki/Probl%C3%A8me_de_Dirichlet

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The aim of the project is double: to understand the flexibility of the Isogeometric Analysis tools through the solution of some PDEs problems; to test the improvement in the computational time given by a partial loops vectorization at compile-time of the LifeV IGA code. Three different appli- cations have been selected: the potential flow problem around an airfoil profile, the heat equation problem in a bent cylinder and the Laplace problem in a multi-patches geometry representing a blood vessels bifurcation. The geometries used are built through the NURBS package available with the software GeoPDEs. The numerical analysis of the first application is performed with both GeoPDEs and LifeV IGA code. The comparison between different implementations shows that the degrees of freedom loop vectorization at compile-time is able to reduce the matrix assembling time of around 20%. The automatic vectorization at compile-time of the loop on the elements requires too much computational effort without having a reasonable improvement in the running time performances. Unsteady problems and multi-patches geometry have not been tested with LifeV IGA code, but GeoPDEs results show the expected solutions.

2012

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Sur quelques inclusions différentielles de premier et de deuxième ordre

Gisella Croce

The principal subject of this thesis is the study of some Dirichlet problems related to differential inclusions of first and second order. We have studied the two following problems : where E ⊂ R2×2 is a compact isotropic set and where We show some existence theorems of W1,∞ and W2,∞ solutions respectively. Moreover we have studied a minimization problem : the computation of α(p, q, r) defined by

EPFL2004

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A Least-Squares/Relaxation Method for the Numerical Solution of the Three-Dimensional Elliptic Monge-Ampere Equation

Alexandre Caboussat, Dimitrios Gourzoulidis

In this article, we address the numerical solution of the Dirichlet problem for the three-dimensional elliptic Monge-Ampere equation using a least-squares/relaxation approach. The relaxation algorithm allows the decoupling of the differential operators from the nonlinearities. Dedicated numerical solvers are derived for the efficient solution of the local optimization problems with cubicly nonlinear equality constraints. The approximation relies on mixed low order finite element methods with regularization techniques. The results of numerical experiments show the convergence of our relaxation method to a convex classical solution if such a solution exists; otherwise they show convergence to a generalized solution in a least-squares sense. These results show also the robustness of our methodology and its ability at handling curved boundaries and non-convex domains.

SPRINGER/PLENUM PUBLISHERS2018

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