Numerical Analysis of a Spectral Problem with High-Order Boundary Conditions on Curved Meshes

Core Concepts

This work presents a numerical analysis of a spectral problem involving a second-order tangential operator (the Laplace-Beltrami operator) on the domain boundary, known as the Ventcel boundary condition. The analysis considers the numerical errors when solving the spectral problem using a Pk-Lagrangian finite element method on curved meshes, accounting for both the error induced by the numerical method and the geometric error caused by discretizing the smooth physical domain.

Abstract

The paper focuses on the numerical analysis of a spectral elliptic problem with Ventcel boundary conditions, which involve a high-order tangential operator (the Laplace-Beltrami operator) on the domain boundary.
Key highlights:

The physical domain Ω is required to be smooth due to the presence of second-order boundary conditions, so it cannot match the mesh domain Ωh, leading to an intrinsic geometric error.
To improve the error rate, curved meshes of polynomial degree r ≥ 1 are used to better approximate the smooth physical domain.
A lift operator is defined to transform functions from the mesh domain to the physical domain, enabling the estimation of errors on eigenvalues and eigenfunctions.
A bootstrap method is used to prove a priori error estimates, which are expressed in terms of both the finite element approximation error (degree k ≥ 1) and the geometric error (mesh order r ≥ 1).
Numerical experiments on various smooth domains in 2D and 3D validate the theoretical results, including a super-convergence of the error rate on quadratic meshes.

Stats

The paper does not contain any explicit numerical data or statistics. The key results are the a priori error estimates derived for the eigenvalues and eigenfunctions.

Quotes

"To overcome this issue, the thin layer is modeled by adapted boundary conditions involving second order terms such as the Laplace-Beltrami operator. These conditions derive from the pioneering works of Ventcel in [33, 34]."
"For the second order boundary terms to make sense, the domain is assumed to be smooth. Thus, we have to deal with problems where the physical domain and the mesh domain differ, putting forward an intrinsic geometric error."
"The main novelties is the use of the new lift operator defined in [9] to estimate the eigenvalue and eigenfunction error both in terms of finite element approximation error and of geometric error, respectively, associated to the finite element degree k ≥ 1 and to the mesh order r ≥ 1."

Key Insights Distilled From

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

by Fabien Caube... at arxiv.org 04-23-2024

https://arxiv.org/pdf/2404.13994.pdf

Deeper Inquiries

How could the proposed numerical framework be extended to vector-valued problems, such as linear elasticity, that involve Ventcel-type boundary conditions

To extend the proposed numerical framework to vector-valued problems like linear elasticity with Ventcel-type boundary conditions, we would need to adapt the finite element discretization and error estimation techniques to handle vector fields. This would involve working with vector-valued function spaces, defining appropriate bilinear forms for the vector problem, and extending the lift operator to handle vector functions. The Ventcel conditions would need to be reformulated for vector fields, taking into account the tangential components and normal derivatives of the vector functions on the boundary. The Riesz projection and orthogonal projections would also need to be generalized to vector spaces to ensure accurate approximation of the eigenfunctions. Overall, the extension to vector-valued problems would require a comprehensive understanding of vector calculus, functional analysis, and numerical methods for solving systems of partial differential equations.

What are the potential implications of the observed super-convergence of the error rate on quadratic meshes, and how could this phenomenon be further investigated and explained

The observed super-convergence of the error rate on quadratic meshes has significant implications for the accuracy and efficiency of the numerical method. Super-convergence indicates that the error decreases at a faster rate than expected based on the mesh size and the order of the finite elements. This phenomenon can lead to more accurate solutions with fewer degrees of freedom, reducing computational costs and time. To further investigate and explain this super-convergence, one could conduct a detailed analysis of the interpolation properties of the high-order curved meshes, study the behavior of the error in different regions of the domain, and explore the impact of the Ventcel boundary conditions on the error convergence. Additionally, conducting sensitivity analyses by varying the mesh parameters and the order of the finite elements could provide insights into the factors influencing the super-convergence effect.

What other types of high-order boundary conditions, beyond the Ventcel conditions considered here, could be analyzed using the presented approach of combining finite element discretization and geometric error estimation

The presented approach of combining finite element discretization and geometric error estimation can be applied to analyze other types of high-order boundary conditions beyond the Ventcel conditions. Some potential high-order boundary conditions that could be investigated using this approach include:

Robin-type boundary conditions with high-order derivatives: These boundary conditions involve derivatives of the solution up to a certain order and are commonly used in various physical problems. Analyzing the error behavior and convergence rates for problems with Robin-type boundary conditions would be valuable for understanding the impact of high-order terms on the numerical solution.
Mixed-type boundary conditions with high-order coupling terms: Boundary conditions that couple different variables or involve high-order coupling terms can pose challenges in numerical simulations. By extending the presented approach to analyze problems with mixed-type boundary conditions, researchers can gain insights into the accuracy and stability of the numerical method in such scenarios.
Nonlocal boundary conditions with high-order integral terms: Nonlocal boundary conditions, which involve integral operators on the boundary, are encountered in many applications, including fractional differential equations. Investigating the error estimates and convergence properties for problems with nonlocal boundary conditions and high-order integral terms would contribute to the understanding of the numerical behavior in these settings.