Finite element analysis of a spectral problem on curved meshes occurring in diffusion with high order boundary conditions

In this work is considered a spectral problem, involving a second order term on the domain boundary: the Laplace-Beltrami operator. A variational formulation is presented, leading to a finite element discretization. For the Laplace-Beltrami operator to make sense on the boundary, the domain is smooth: consequently the computational domain (classically a polygonal domain) will not match the physical one. Thus, the physical domain is discretized using high order curved meshes so as to reduce the $\mathit{\text{geometric}}\mathit{\text{error}}$. The $\mathit{\text{lift}}\mathit{\text{operator}}$, which is aimed to transform a function defined on the mesh domain into a function defined on the physical one, is recalled. This $\mathit{\text{lift}}$ is a key ingredient in estimating errors on eigenvalues and eigenfunctions. A bootstrap method is used to prove the error estimates, which are expressed both in terms of $\mathit{\text{finite}}\mathit{\text{element}}\mathit{\text{approximation}}\mathit{\text{error}}$ and of $\mathit{\text{geometric}}\mathit{\text{error}}$, respectively associated to the finite element degree $k\ge 1$ and to the mesh order~$r\ge 1$. Numerical experiments are led on various smooth domains in 2D and 3D, which allow us to validate the presented theoretical results.