Using Curved Meshes for a Diffusion Problem with a Surface Laplacian

This presentation focuses on the numerical analysis of a diffusion problem with a boundary condition involving a surface Laplacian using a high-order finite element method. In order to define this surface operator on the boundary, the domain is assumed to be smooth: hence, the meshed domain does not coincide with the initial physical domain, resulting in a geometric error. We thus use curved meshes to reduce this error and we define an adequate lift operator that allows us to compare the exact solution defined on the physical domain and the approximate solution defined on the discretized domain. We then obtain a priori error estimates, expressed in terms of finite element approximation error and geo- metric error. Numerical experiments in 2D and 3D validate and complement these theoretical results, highlighting in particular the optimality of the obtained error convergence rates. These simulations also identify a super-convergence of the errors on quadratic meshes of order 2. We conclude by presenting other related applications, perspectives, and ongoing work.