A Nonconforming Primal Hybrid Finite Element Method for Solving the Two-Dimensional Vector Laplacian Problem

Keskeiset käsitteet

The authors introduce a nonconforming hybrid finite element method for solving the two-dimensional vector Laplacian problem, which ensures consistency using penalty terms similar to those used in hybridizable discontinuous Galerkin (HDG) methods.

Tiivistelmä

The key highlights and insights of the content are:

The authors present a three-field primal hybridization of the nonconforming primal finite element method for the two-dimensional vector Laplacian problem. This method accommodates elements of arbitrarily high order and can be implemented efficiently using static condensation.

The lowest-order case of the proposed method recovers the P1-nonconforming method of Brenner et al., and the authors show that higher-order convergence is achieved under appropriate regularity assumptions.

The analysis makes novel use of a family of weighted Sobolev spaces, due to Kondrat'ev, to handle domains admitting corner singularities. This allows the authors to obtain error estimates without imposing mesh-grading conditions required by previous methods.

The authors establish the well-posedness of the hybrid method and demonstrate the equivalence of the three-field, two-field, and one-field formulations. They also discuss the static condensation of the hybrid method, which enables efficient global solvers.

Numerical experiments are presented to confirm the analytically obtained convergence results.

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Tärkeimmät oivallukset

A nonconforming primal hybrid finite element method for the two-dimensional vector Laplacian

by Mary Barker,... klo arxiv.org 04-30-2024

https://arxiv.org/pdf/2206.10567.pdf

Syvällisempiä Kysymyksiä

How can the proposed hybrid method be extended to solve the three-dimensional vector Laplacian problem, and what are the challenges involved

To extend the proposed hybrid method for solving the three-dimensional vector Laplacian problem, several adjustments and considerations need to be made. In the two-dimensional case, the method accommodates elements of arbitrarily high order, allowing for higher-order convergence under appropriate regularity assumptions.
In the three-dimensional scenario, the challenges primarily arise from the increased complexity of the geometry and the additional dimension. The extension would involve defining appropriate finite element spaces for three-dimensional domains, considering the connectivity of elements, and adapting the penalty parameters to account for the increased dimensionality. The formulation of the method would need to be adjusted to handle the vector Laplacian in three dimensions, ensuring consistency and stability in the numerical solution.

What are the potential applications of the nonconforming primal hybrid finite element method beyond the vector Laplacian problem, and how could it be adapted to address those applications

The nonconforming primal hybrid finite element method has potential applications beyond the vector Laplacian problem. One such application could be in the field of computational electromagnetics, particularly in solving Maxwell's equations in complex geometries. By adapting the method to handle the electromagnetic field variables, such as electric and magnetic fields, the approach could efficiently tackle problems involving wave propagation, scattering, and antenna design.
Furthermore, the method could be applied to problems in fluid dynamics, structural mechanics, and other areas where partial differential equations with complex geometries and corner singularities are prevalent. By incorporating appropriate boundary conditions and adapting the formulation to suit the specific physics of the problem, the method could provide accurate and efficient solutions for a wide range of applications.

Can the use of weighted Sobolev spaces and the associated regularity theory be further leveraged to develop efficient numerical methods for other types of partial differential equations with corner singularities

The use of weighted Sobolev spaces and the associated regularity theory can indeed be leveraged to develop efficient numerical methods for various types of partial differential equations with corner singularities. By utilizing the regularity results obtained from weighted Sobolev spaces, one can design numerical schemes that take advantage of the specific regularity properties of the solutions.
For equations with corner singularities, such as problems in domains with complex geometries, the weighted Sobolev spaces can guide the development of adaptive mesh refinement strategies, error estimators, and solution techniques that target the singular behavior near corners. This can lead to more accurate and efficient numerical methods that capture the intricate features of the solutions in challenging geometries. Additionally, the regularity results can inform the design of preconditioners and solvers tailored to the specific regularity properties of the solutions, enhancing the overall efficiency of the numerical computations.