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A priori error estimates of a diffusion equation with Ventcel boundary conditions on curved meshes

Fabien Caubet, Joyce Ghantous, Charles Pierre

TL;DR

The focus is on the construction of high order curved meshes for the discretization of the physical domain and on the denition of the lift operator, which is aimed to transform a function dened on the mesh domain into a function denied on the physical one.

Abstract

In this work is considered an elliptic problem, referred to as the Ventcel problem, involvinga second order term on the domain boundary (the Laplace-Beltrami operator). A variationalformulation of the Ventcel problem is studied, leading to a finite element discretization. Thefocus is on the construction of high order curved meshes for the discretization of the physicaldomain and on the definition of the lift operator, which is aimed to transform a functiondefined on the mesh domain into a function defined on the physical one. This lift is definedin a way as to satisfy adapted properties on the boundary, relatively to the trace operator.The Ventcel problem approximation is investigated both in terms of geometrical error and offinite element approximation error. Error estimates are obtained both in terms of the meshorder r $\ge$ 1 and to the finite element degree k $\ge$ 1, whereas such estimates usually have beenconsidered in the isoparametric case so far, involving a single parameter k = r. The numericalexperiments we led, both in dimension 2 and 3, allow us to validate the results obtained andproved on the a priori error estimates depending on the two parameters k and r. A numericalcomparison is made between the errors using the former lift definition and the lift defined inthis work establishing an improvement in the convergence rate of the error in the latter case.

A priori error estimates of a diffusion equation with Ventcel boundary conditions on curved meshes

TL;DR

The focus is on the construction of high order curved meshes for the discretization of the physical domain and on the denition of the lift operator, which is aimed to transform a function dened on the mesh domain into a function denied on the physical one.

Abstract

In this work is considered an elliptic problem, referred to as the Ventcel problem, involvinga second order term on the domain boundary (the Laplace-Beltrami operator). A variationalformulation of the Ventcel problem is studied, leading to a finite element discretization. Thefocus is on the construction of high order curved meshes for the discretization of the physicaldomain and on the definition of the lift operator, which is aimed to transform a functiondefined on the mesh domain into a function defined on the physical one. This lift is definedin a way as to satisfy adapted properties on the boundary, relatively to the trace operator.The Ventcel problem approximation is investigated both in terms of geometrical error and offinite element approximation error. Error estimates are obtained both in terms of the meshorder r 1 and to the finite element degree k 1, whereas such estimates usually have beenconsidered in the isoparametric case so far, involving a single parameter k = r. The numericalexperiments we led, both in dimension 2 and 3, allow us to validate the results obtained andproved on the a priori error estimates depending on the two parameters k and r. A numericalcomparison is made between the errors using the former lift definition and the lift defined inthis work establishing an improvement in the convergence rate of the error in the latter case.
Paper Structure (33 sections, 12 theorems, 106 equations, 10 figures, 3 tables)

This paper contains 33 sections, 12 theorems, 106 equations, 10 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Motivations.
  3. The Ventcel problem and its approximation.
  4. Main novelties.
  5. Paper organization.
  6. Notations and needed mathematical tools
  7. Curved mesh definition
  8. Affine mesh $\mathcal{T}_h^{(1)}$
  9. Exact mesh $\mathcal{T}_h^{(e)}$
  10. Curved mesh $\mathcal{T}_h^{(r)}$ of order $r$.
  11. Functional lift
  12. Surface and volume lift definitions
  13. Lift of the variational formulation
  14. Surface integrals.
  15. Volume integrals.
  16. ...and 18 more sections

Key Result

Proposition 2.2

Let $\Omega$ be a nonempty bounded connected open subset of $\mathbb{R}^{d}$ 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 $\Gamma$, of suffic

Figures (10)

  • Figure 1: Visualisation of the two functions $\hat{y}: \, \hat{T} \mapsto \hat{T}$ and $y: \, T \mapsto \partial T \cap \Gamma$ in definition \ref{['def:fte-y']} in a 2D case
  • Figure 2: Visualisation of $G_h^{(2)}: {T}^{(2)} \to {T}^{(e)}$ in a 2D case, for a quadratic case $r=2$.
  • Figure 3: Numerical solution of the Ventcel problem on affine and quadratic meshes.
  • Figure 4: Plots of the error in volume norms with respect to the mesh step h corresponding to the convergence order in Table \ref{['tab:conv-1-Omega']}: ${\rm H}^1_0(\Omega)$ norm (above) and ${\rm L}^2(\Omega)$ norm (below) for quadratic meshes (left) and cubic meshes (right).
  • Figure 5: Plots of the error in interior norms with respect to the mesh step h corresponding to the convergence order in Table \ref{['tab:conv-1-Gamma']}: ${\rm H}^1_0(\Gamma)$ norm (above) and ${\rm L}^2(\Gamma)$ norm (below) for quadratic meshes (left) and cubic meshes (right)..
  • ...and 5 more figures

Theorems & Definitions (35)

  • Definition 2.1
  • Proposition 2.2
  • Theorem 2.3
  • Remark 3.1
  • Definition 3.2
  • Definition 3.3
  • Remark 3.4
  • Definition 4.1: Surface lift
  • Definition 4.2: Volume lift
  • Proposition 4.3
  • ...and 25 more