arXiv subjects

Joyce Ghantous

Publications and source records attributed to Joyce Ghantous.

4 recordsLinked to original sources

arXiv2025

Using curved meshes to derive a priori error estimates for a linear elasticity problem with Robin boundary conditions

This work concerns the numerical analysis of the linear elasticity problem with a Robin boundary condition on a smooth domain. A finite element discretization is presented using high-order curved meshes in order to accurately discretize the physical domain. The primary objective is to conduct a detailed error analysis for the elasticity problem using the vector lift operator, which maps vector-valued functions from the mesh domain to the physical domain. Error estimates are established, both in terms of the finite element approximation error and the geometric error, respectively associated to the finite element degree and to the mesh order. These theoretical a priori error estimates are validated by numerical experiments in 2D and 3D.

math.NA↗

arXiv2024

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.

math.NA↗

arXiv2023

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

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.

math.NA↗

arXiv2023

Numerical study of a diffusion equation with Ventcel boundary condition using curved meshes

In this work is provided a numerical study of a diffusion problem involving a second order term on the domain boundary (the Laplace-Beltrami operator) referred to as the \textit{Ventcel problem}.A variational formulation of the Ventcel problem is studied, leading to a finite element discretization.The focus is on the resort to high order curved meshes for the discretization of the physical domain.The computational errors are investigated both in terms of geometrical error and of finite element approximation error, respectively associated to the mesh degree $r\ge 1$ and to the finite element degree $k\ge 1$. The numerical experiments we led allow us to formulate a conjecture on the \textit{a priori} error estimates depending on the two parameters $r$ and $k$. In addition, these error estimates rely on the definition of a functional \textit{lift} with adapted properties on the boundary to move numerical solutions defined on the computational domain to the physical one.

math.NA↗