Rate-Optimal Higher-Order Conforming FEM for Biharmonic Eigenvalue Problems on Polygonal Domains

Core Concepts

Establishing optimal convergence rates for adaptive finite element schemes in biharmonic eigenvalue problems.

Abstract

The article introduces a novel adaptive scheme for biharmonic eigenvalue problems, proving optimal convergence rates towards simple eigenvalues. It highlights the necessity of extending classical finite element spaces to hierarchical structures. The paper emphasizes the competitive nature of the suggested algorithm despite higher computational costs compared to nonconforming schemes. Numerical experiments demonstrate that higher polynomial degrees lead to increased convergence rates, emphasizing the importance of adaptive mesh-refining. The content is structured into sections focusing on introduction, state of the art, overview, and specific examples illustrating the effectiveness of the proposed method.

Stats

"It took a long time to realise the necessity of an extension of the classical finite element spaces to make them hierarchical."
"The numerical experiments provide overwhelming evidence that higher polynomial degrees pay off with higher convergence rates."
"Five computational benchmarks display accurate reference eigenvalues up to 30 digits."

Quotes

"We refer to textbooks [Bre11, Rud91, Yos95, Zei92] for functional analysis and spectral theory facts."
"Recall Λstab = √2(1 + Λsep)Λdual from key Lemma 2.1."

Key Insights Distilled From

Rate-optimal higher-order adaptive conforming FEM for biharmonic eigenvalue problems on polygonal domains

by Cars... at arxiv.org 03-20-2024

https://arxiv.org/pdf/2403.12577.pdf

Deeper Inquiries

How does extending classical finite element spaces hierarchically impact computational efficiency

Extending classical finite element spaces hierarchically can have a significant impact on computational efficiency. By introducing hierarchical structures in the finite element spaces, such as in the case of the Argyris FEM, it becomes possible to capture more complex behavior and higher-order features of the solution. This hierarchical extension allows for better approximation of functions with increased accuracy, leading to improved convergence rates. Additionally, hierarchical finite element spaces enable adaptive mesh refinement strategies that focus computational resources where they are most needed, reducing overall computational costs by concentrating efforts on areas with high error contributions.

What are potential drawbacks or limitations of using higher polynomial degrees in achieving convergence rates

While using higher polynomial degrees can lead to improved convergence rates and accuracy in numerical simulations, there are potential drawbacks and limitations to consider. One limitation is the increased complexity and computational cost associated with higher-degree polynomials. Higher polynomial degrees require more degrees of freedom, resulting in larger system matrices that need to be solved iteratively during each simulation step. This can lead to longer computation times and greater memory requirements.
Another drawback is the potential for oscillations or instability when using very high polynomial degrees in certain types of problems or geometries. High-degree polynomials may introduce spurious oscillations or artifacts in the solution if not properly controlled through stabilization techniques or appropriate mesh refinement strategies.
Furthermore, increasing the degree of polynomials may also result in increased sensitivity to geometric distortions or irregularities in the mesh. In some cases, higher-degree elements may struggle to accurately represent solutions on distorted meshes, leading to reduced robustness and reliability of the numerical method.

How can concepts from functional analysis be applied in other mathematical domains beyond finite element methods

Concepts from functional analysis play a crucial role not only in finite element methods but also across various mathematical domains due to their foundational nature and applicability.
In differential equations and calculus, functional analysis provides tools for studying function spaces where solutions reside. Understanding properties like completeness, compactness, continuity helps analyze existence and uniqueness results for differential equations.
In optimization theory, functional analysis aids in formulating optimization problems over infinite-dimensional spaces by defining functionals that need optimizing subject to constraints expressed as inequalities or equalities.
In probability theory and statistics, concepts from functional analysis underpin theories related to stochastic processes defined over function spaces enabling rigorous treatment of random variables taking values within these abstract structures.
Overall Functional Analysis serves as a unifying framework connecting diverse mathematical disciplines through its study of vector spaces equipped with suitable topologies allowing generalization beyond finitely many dimensions facilitating deeper insights into complex mathematical phenomena.