Anisotropic Hybrid Weakly Over-Penalized Symmetric Interior Penalty Method for Poisson Equation

핵심 개념

The author proposes a new hybrid-type anisotropic weakly over-penalized symmetric interior penalty (HWOPSIP) method for solving the Poisson equation on convex domains. The key idea is to apply the relation between the Raviart-Thomas finite element space and a discontinuous space, which allows for optimal error estimates of the consistency error on anisotropic meshes.

초록

The paper investigates a hybrid-type anisotropic weakly over-penalized symmetric interior penalty (HWOPSIP) method for solving the Poisson equation on convex domains. The main contributions are:

Proposal of a new HWOPSIP scheme that is simple and easy to implement, compared to the well-known hybrid discontinuous Galerkin methods.
Demonstration of a proof for the consistency term, which enables estimation of the anisotropic consistency error. The key idea is to apply the relation between the Raviart-Thomas finite element space and a discontinuous space.

The HWOPSIP method is similar to the classical Crouzeix-Raviart non-conforming finite element method and has advantages such as stability for any penalty parameter and error analysis on more general meshes than conformal meshes.
The error analysis is performed under a semi-regular condition, which is equivalent to the maximum-angle condition. The error between the exact and HWOPSIP finite element approximation solutions with the energy norm is divided into two parts: an optimal approximation error in discontinuous finite element spaces, and a consistency error term. The consistency error term is estimated using the relation between the Raviart-Thomas interpolation and the discontinuous space, which yields an optimal error estimate.
Numerical experiments are conducted to compare the calculation results for standard and anisotropic mesh partitions.

통계

None.

인용구

None.

핵심 통찰 요약

Hybrid weakly over-penalised symmetric interior penalty method on anisotropic meshes

by Hiroki Ishiz... 게시일 arxiv.org 04-25-2024

https://arxiv.org/pdf/2404.15288.pdf

더 깊은 질문

How can the stability estimates of the HWOPSIP scheme be obtained when the domain Ω is not convex

When the domain Ω is not convex, the stability estimates of the HWOPSIP scheme can still be obtained by adapting the analysis to accommodate non-convex domains. One approach is to utilize discrete Poincaré inequalities for piecewise H1 functions, as proposed in literature. By incorporating inverse, trace inequalities, and local quasi-uniformity for meshes that do not satisfy the shape-regularity condition, the stability estimates can be derived. It is essential to carefully consider and apply the results from these studies to remove the convexity assumption and ensure the stability of the HWOPSIP scheme in non-convex domains.

What are the potential limitations or drawbacks of the HWOPSIP method compared to other hybrid discontinuous Galerkin methods

One potential limitation of the HWOPSIP method compared to other hybrid discontinuous Galerkin methods is the complexity of the stability and error analysis when dealing with anisotropic meshes. While the HWOPSIP method offers simplicity and ease of implementation, the proof of stability and error estimates on anisotropic meshes can be challenging. Additionally, the HWOPSIP method may have limitations in handling complex geometries or domains with irregular boundaries, where other methods might offer more robust solutions. Furthermore, the HWOPSIP method may require additional computational resources or refinements to achieve optimal performance in certain scenarios.

Can the proposed approach be extended to solve other types of partial differential equations beyond the Poisson equation

The proposed approach of the HWOPSIP method can be extended to solve a variety of partial differential equations beyond the Poisson equation. By adapting the formulation and analysis to suit different types of PDEs, the HWOPSIP method can be applied to problems such as the heat equation, wave equation, or diffusion-reaction equations. The key lies in modifying the weak formulation, finite element settings, and error estimates to align with the specific characteristics and requirements of the new PDEs. With appropriate adjustments and considerations, the HWOPSIP method can be a versatile and effective tool for solving a wide range of partial differential equations.