Discontinuous Finite Element Spaces for Velocity Approximation in a Discrete de-Rham Complex on Periodic Triangular and Cartesian Meshes

Core Concepts

Discontinuous finite element spaces can be constructed to form a discrete de-Rham complex that satisfies the harmonic gap property on periodic triangular and Cartesian meshes.

Abstract

The article focuses on deriving discontinuous finite element spaces that can be used to form a discrete de-Rham complex, where the harmonic gap property is satisfied. This is an important property that ensures the discrete and continuous cohomology spaces are isomorphic.
For triangular meshes:

The author proves that by relaxing the normal or tangential constraint of classical conforming finite element spaces, discontinuous spaces can be built that satisfy the harmonic gap property.
The dimension of the finite element spaces involved in the discrete de-Rham complex is computed.
The properties of the discrete differential operators, such as the kernel and range of the discrete gradient and divergence, are analyzed.
A decomposition of the divergence-free elements in the discontinuous vector space is provided.
For Cartesian meshes:

The author shows that the classical discontinuous finite element space for vectors does not satisfy the harmonic gap property.
An enriched discontinuous finite element space is then introduced, which, when used in the discrete de-Rham complex, recovers the harmonic gap property.
The article provides a thorough analysis of the discrete de-Rham complex involving discontinuous finite element spaces for velocities on periodic triangular and Cartesian meshes, ensuring the harmonic gap property.

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Key Insights Distilled From

Discrete de-Rham complex involving a discontinuous finite element space for velocities: the case of periodic straight triangular and Cartesian meshes

by Vincent Perr... at arxiv.org 05-01-2024

https://arxiv.org/pdf/2404.19545.pdf

Deeper Inquiries

How can the results be extended to more general polygonal or polyhedral meshes beyond triangles and Cartesians

To extend the results to more general polygonal or polyhedral meshes beyond triangles and Cartesians, we can leverage the concept of discrete de-Rham complexes and the properties of the finite element spaces involved. By considering the geometric and topological characteristics of the new mesh elements, we can adapt the construction of the finite element spaces to accommodate the specific features of these elements. This may involve defining new basis functions, modifying the discrete differential operators, and ensuring compatibility between the spaces to maintain the harmonic gap property. Additionally, the decomposition of divergence-free elements and the orthogonal projections can be generalized to handle the different shapes and connectivity patterns present in polygonal or polyhedral meshes. By carefully extending the methodology while preserving the key properties established for triangles and Cartesians, we can effectively apply the results to a broader range of mesh types.

What are the implications of the harmonic gap property for the numerical approximation of partial differential equations using the proposed discontinuous finite element spaces

The harmonic gap property plays a crucial role in the numerical approximation of partial differential equations using discontinuous finite element spaces. This property ensures that the discrete and continuous cohomology spaces are isomorphic, indicating that the discrete complex accurately captures the topological characteristics of the domain. By proving the harmonic gap property for the proposed discontinuous finite element spaces, we can guarantee that the discrete solutions obtained using these spaces closely align with the continuous solutions. This alignment is essential for maintaining the accuracy and stability of the numerical approximation, especially in problems where the topology of the domain significantly influences the solution behavior. Therefore, the harmonic gap property provides a solid foundation for reliable and efficient numerical simulations using discontinuous finite element discretization.

Can the ideas presented in this work be applied to develop efficient solvers or preconditioners for the discrete problems arising from the discontinuous finite element discretization

The ideas presented in this work can indeed be applied to develop efficient solvers or preconditioners for the discrete problems arising from the discontinuous finite element discretization. By leveraging the properties of the discrete de-Rham complexes, such as the decomposition of divergence-free elements and the orthogonal projections, we can design specialized solvers tailored to handle the specific structures of the finite element spaces. These solvers can exploit the harmonic gap property to ensure accurate and stable solutions while efficiently solving the discrete systems of equations. Additionally, the insights gained from the study of the discrete complexes can guide the development of preconditioning techniques that effectively precondition the discrete operators associated with the discontinuous finite element spaces. By incorporating these ideas into solver and preconditioner design, we can enhance the computational efficiency and robustness of numerical simulations based on discontinuous finite element discretization.