Computing the principal eigenvalue of the Laplace operator by a stochastic method

Antoine Lejay (1, 2) , Sylvain Maire (3)

1 OMEGA - Probabilistic numerical methods
2 IECN - Institut Élie Cartan de Nancy
3 MNC - Modélisation Numérique et Couplages

We describe a Monte Carlo method for the numerical computation of the principal eigenvalue of the Laplace operator in a bounded domain with Dirichlet conditions. It is based on the estimation of the speed of absorption of the Brownian motion by the boundary of the domain. Various tools of statistical estimation and different simulation schemes are developed to optimize the method. Numerical examples are studied to check the accuracy and the robustness of our approach.

Mots clés: First eigenvalue of the Dirichlet problem; Euler scheme for Brownian motion; random walk on spheres; random walk on rectangles

Domaines: Equations aux dérivées partielles [math.AP]; Analyse numérique [math.NA]; Probabilités [math.PR]

Antoine Lejay, Sylvain Maire. Computing the principal eigenvalue of the Laplace operator by a stochastic method. Mathematics and Computers in Simulation, 2007, 73 (3), pp.351-363. ⟨10.1016/j.matcom.2006.06.011⟩. ⟨inria-00092408⟩