Finite element analysis of a spectral problem on curved meshes occurring in diffusion with high order boundary conditions
<|reference_start|>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 \textit{geometric error}. The \textit{lift operator}, which is aimed to transform a function defined on the mesh domain into a function defined on the physical one, is recalled. This \textit{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 \textit{finite element approximation error} and of \textit{geometric 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.<|reference_end|>
@article{caubet2024finite,
    title={Finite element analysis of a spectral problem on curved meshes occurring
  in diffusion with high order boundary conditions},
    author={Fabien Caubet (LMAP), Joyce Ghantous (LMAP), Charles Pierre (LMAP)},
    journal={arXiv preprint arXiv:2404.13994},
    year={2024},
    archivePrefix={arXiv},
    eprint={2404.13994},
    primaryClass={math.NA cs.NA}
}