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

Fabien Caubet, Joyce Ghantous, Charles Pierre

Abstract

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.

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

Abstract

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 and to the mesh order~. Numerical experiments are led on various smooth domains in 2D and 3D, which allow us to validate the presented theoretical results.
Paper Structure (32 sections, 14 theorems, 148 equations, 6 figures, 3 tables)

This paper contains 32 sections, 14 theorems, 148 equations, 6 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Motivation ad objective.
  3. The Ventcel eigenvalue problem.
  4. State of the art and main results.
  5. Paper organization.
  6. The continuous problem
  7. Needed mathematical tools.
  8. Well posedness of the spectral problem.
  9. Curved mesh and lift definition
  10. Curved mesh $\mathcal{T}_h^{(r)}$ of order $r$.
  11. Functional lift.
  12. Lift properties.
  13. Finite element approximation
  14. Discrete formulation.
  15. Lifted discrete formulation.
  16. ...and 17 more sections

Key Result

Proposition 2.1

Let $\Omega$ be a nonempty bounded connected open subset of $\mathbb{R}^{d}$$(d~=~2,3)$ with a $\mathcal{C}^2$ boundary $\Gamma= \partial \Omega$. Let $\mathrm{d} : \mathbb{R}^d \to \mathbb{R}$ be the signed distance function with respect to $\Gamma$ defined by, Then there exists a tubular neighborhood $\mathcal{U}_{\Gamma}:= \{ x \in \mathbb{R}^d ; |\mathrm{d}(x)| < \delta_\Gamma \}$ of $\Gamma$

Figures (6)

  • Figure 1: Representation of the $6^{\rm th}$ eigenfunction computed using $\mathbb{P}^ 3$ finite element on a (coarse) mesh of $\Omega$: affine mesh (left) and quadratic mesh (right).
  • Figure 2: Display of the eigenfunction $U_6$ associated to the computed eigenvalue $\Lambda_6$ using $\mathbb{P}^ 3$ method on an affine mesh (left) and a quadratic mesh (right).
  • Figure 3: Display of the eigenfunction associated with the eigenvalue $\Lambda_{10}$ using $\mathbb{P}^ 2$ finite element on an affine mesh (left) and a quadratic mesh (right).
  • Figure 4: Display of the convergence rate of $e_{\lambda_{10}}=|\lambda_{10}-\Lambda_{10}|$ using $\mathbb{P}^ 2$ and $\mathbb{P}^ 3$ finite element on a quadratic mesh (left) and a cubic mesh (right).
  • Figure 5: Display of the convergence rate of $e_{\mathrm{L}^2}$ using $\mathbb{P}^ 2$ and $\mathbb{P}^ 3$ finite element on a quadratic mesh (left) and a cubic mesh (right).
  • ...and 1 more figures

Theorems & Definitions (37)

  • Proposition 2.1
  • Definition 3.1
  • Remark 3.1
  • Remark 4.1
  • Remark 4.2
  • Remark 5.1
  • Corollary 5.1
  • Proposition 5.1
  • proof
  • Corollary 5.2
  • ...and 27 more