Analysis and discretization of the volume penalized Laplace operator with Neumann boundary conditions

Dmitry Kolomenskiy (1, 2) , Romain Nguyen van Yen (3) , Kai Schneider (1)

1 M2P2 - Laboratoire de Mécanique, Modélisation et Procédés Propres
2 CERFACS - Centre Européen de Recherche et de Formation Avancée en Calcul Scientifique
3 LMD - Laboratoire de Météorologie Dynamique (UMR 8539)

We study the properties of an approximation of the Laplace operator with Neumann boundary conditions using volume penalization. For the one-dimensional Poisson equation we compute explicitly the exact solution of the penalized equation and quantify the penalization error. Numerical simulations using finite differences allow then to assess the discretization and penalization errors. The eigenvalue problem of the penalized Laplace operator with Neumann boundary conditions is also studied. As examples in two space dimensions, we consider a Poisson equation with Neumann boundary conditions in rectangular and circular domains.

Keywords: Volume penalization; Neumann boundary conditions; Laplace operator; Poisson equation

Domains: Fluids mechanics [physics.class-ph]

Dmitry Kolomenskiy, Romain Nguyen van Yen, Kai Schneider. Analysis and discretization of the volume penalized Laplace operator with Neumann boundary conditions. 2024. ⟨hal-01299247⟩