Superconvergence of Differential Structure on Perturbed Surface Meshes

Core Concepts

The paper studies general geometric conditions for developing numerical analysis on discretized manifolds, especially when the solutions involve the differential structure of the manifolds. It introduces the concept of geometric supercloseness and shows that it enables superconvergence of differential structure, such as gradient recovery, on deviated discretizations of manifolds.

Abstract

The paper focuses on the numerical analysis of solutions involving the differential structure of manifolds, such as tangential vector fields and gradients of scalar functions. It addresses the gap between the standard requirement of exact geometric information for a priori or a posteriori error analysis, and the practical violation of this requirement.
The key contributions are:

Introduction of the concept of geometric supercloseness, which provides the necessary condition for superconvergence of differential structure on deviated discretizations of manifolds.

Development of a family of isoparametric gradient recovery schemes that can achieve superconvergence on deviated discretizations without requiring exact geometric information.

Demonstration of how the superconvergence of differential structure enables the development of optimal convergent numerical methods for solving vector Laplace problems on discretized manifolds without the need for superparametric elements or exact normal fields.

The paper first provides counterexamples showing that the standard O(h^2) vertex condition is not sufficient to guarantee superconvergence of gradient recovery on deviated manifolds. It then introduces the geometric supercloseness condition and shows that it implies superconvergence of differential structure.
The isoparametric gradient recovery schemes are developed based on a two-level strategy: recovering the Jacobian of the local geometric mapping and then recovering the gradient isoparametrically. The superconvergence of this recovery scheme is proven under the geometric supercloseness condition.
Finally, the paper presents an application of the superconvergence of differential structure to develop an optimal convergent numerical method for solving vector Laplace problems on discretized manifolds, without requiring superparametric elements or exact normal fields.

Stats

The paper does not contain any specific numerical data or statistics to support the key claims. The analysis is primarily theoretical, focusing on establishing the necessary geometric conditions for superconvergence of differential structure on discretized manifolds.

Quotes

"Such gap is filled using the superconvergence of differential structure on manifolds."
"We use counterexamples to show that there exist cases where the superconvergence is not guaranteed given barely the O(h^2) vertex condition."
"With geometric supercloseness, we can completely solve the two open questions in [29]."

Key Insights Distilled From

Superconvergence of differential structure for finite element methods on perturbed surface meshes

by Guozhi Dong,... at arxiv.org 05-03-2024

https://arxiv.org/pdf/1910.14093.pdf

Deeper Inquiries

How can the geometric supercloseness condition be generalized or relaxed to apply to a wider range of discretization schemes for manifolds

The geometric supercloseness condition can be generalized or relaxed to apply to a wider range of discretization schemes for manifolds by considering alternative approaches to ensure the accuracy of the geometric approximation. One way to generalize this condition is to incorporate higher-order interpolation methods or adaptive mesh refinement techniques. By using higher-order interpolation, such as quadratic or cubic polynomials, the geometric approximation can be improved, leading to a higher level of supercloseness. Additionally, adaptive mesh refinement allows for the local refinement of the mesh in areas where the geometric approximation deviates the most from the actual manifold, thereby enhancing the supercloseness condition in those regions.
Furthermore, the relaxation of the geometric supercloseness condition can involve allowing for a certain degree of error or deviation in the geometric approximation, especially in areas where the curvature of the manifold is low or the solution is less sensitive to geometric accuracy. By relaxing the condition in such regions, computational efficiency can be improved without significantly compromising the overall accuracy of the numerical solution. This approach balances the trade-off between accuracy and computational cost, making the method more practical for real-world applications.

What are the practical implications and limitations of the proposed isoparametric gradient recovery schemes in terms of computational cost and implementation complexity

The proposed isoparametric gradient recovery schemes have both practical implications and limitations in terms of computational cost and implementation complexity.
Practical Implications:

Accuracy: The isoparametric gradient recovery schemes aim to improve the accuracy of the recovered gradient by considering the differential structure of the manifolds. This can lead to more precise numerical solutions, especially in problems where the solution involves the differential structure directly.

Optimal Convergence: By leveraging the superconvergence of the differential structure, the numerical methods developed using these schemes can achieve optimal convergence rates, providing more efficient and accurate solutions to partial differential equations on discretized manifolds.

Flexibility: The framework allows for the use of various recovery methods, providing flexibility in choosing the most suitable approach based on the specific characteristics of the problem or the discretization scheme.

Limitations:

Computational Cost: Implementing isoparametric gradient recovery schemes may require additional computational resources, especially when higher-order interpolation or adaptive mesh refinement techniques are used to enhance the geometric approximation. This can increase the overall computational cost of the numerical simulations.

Complexity: The implementation of these schemes may introduce additional complexity to the numerical algorithms, requiring a deeper understanding of differential geometry and interpolation techniques. This complexity can make the implementation and debugging process more challenging.

Mesh Quality: The effectiveness of the recovery schemes heavily relies on the quality of the mesh and the accuracy of the geometric approximation. In cases where the mesh quality is poor or the geometric approximation deviates significantly from the actual manifold, the recovery schemes may not perform optimally, leading to potential inaccuracies in the numerical solutions.

Can the superconvergence of differential structure be leveraged to develop optimal numerical methods for other types of partial differential equations on discretized manifolds, beyond the vector Laplace problem considered in the paper

The superconvergence of the differential structure can indeed be leveraged to develop optimal numerical methods for various types of partial differential equations on discretized manifolds beyond the vector Laplace problem considered in the paper.

Scalar PDEs: For scalar partial differential equations, such as the Poisson equation or the heat equation, the superconvergence of the differential structure can be utilized to design gradient recovery schemes that enhance the accuracy of the numerical solutions. This can lead to improved convergence rates and more precise solutions for scalar PDEs on manifolds.

Higher-Order PDEs: The superconvergence property can also be applied to higher-order partial differential equations, such as the biharmonic equation or the elasticity equations, where the solutions involve higher-order derivatives. By incorporating the superconvergence of the differential structure into the numerical methods, optimal convergence rates can be achieved for these types of PDEs on discretized manifolds.

Multi-Dimensional Problems: The concept of superconvergence can be extended to multi-dimensional problems, where the solutions are vector fields or tensor fields on manifolds. By developing specialized recovery schemes that leverage the superconvergence of the differential structure, optimal numerical methods can be designed for solving multi-dimensional PDEs efficiently and accurately.

In conclusion, the superconvergence of the differential structure opens up opportunities to enhance the accuracy and efficiency of numerical methods for a wide range of partial differential equations on discretized manifolds, making them more robust and reliable for practical applications.