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Pré-Publication, Document De Travail Année : 2024

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

Résumé

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.
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Dates et versions

hal-04552691 , version 1 (19-04-2024)

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  • HAL Id : hal-04552691 , version 1

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Fabien Caubet, Joyce Ghantous, Charles Pierre. Finite element analysis of a spectral problem on curved meshes occurring in diffusion with high order boundary conditions. 2024. ⟨hal-04552691⟩
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