Boundary layer analysis of a d-dimensional penalization method for Neumann or Robin boundary conditions
Bouchra Bensiali
École Centrale Casablanca (Maroc)
Date(s) : 18/06/2024 iCal
11h00 - 12h00
A d-dimensional extension of a fictitious domain penalization technique that we previously proposed for Neumann or Robin boundary conditions will be presented. We apply Droniou’s approach for non-coercive linear elliptic problems to obtain the existence and uniqueness of the solution of the penalized problem, and we derive a boundary layer approach to establish the convergence of the penalization method. The developed boundary layer approach is adapted from the one used for Dirichlet boundary conditions, but in contrast to the latter where coercivity enables a straightforward estimate of the remainders, we reduce the convergence of the penalization method to the existence of suitable supersolutions of a dual problem. These supersolutions are then constructed as approximate solutions of the dual problem using an additional formal boundary layer approach.
The proposed approach results in an advection-dominated problem, requiring the use of appropriate numerical methods suitable for singular perturbation problems, such as upwind finite differences or stabilized finite elements. Numerical experiments validate both the convergence rate and the boundary layer thickness, illuminating the theoretical results.
Finally, we investigate the applicability of the suggested method to the simulation of population dynamics under climate change. The fact that climate change causes the climate envelope to shift over time motivates the idea of using a penalization method to prevent the need for costly remeshing of the spatial domain at each time step.
This talk comprises joint work with Jacques Liandrat, Centrale Méditerranée, I2M.
Emplacement
I2M Saint-Charles - Salle de séminaire
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