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Anisotropic Weakly Over-Penalized Symmetric Interior Penalty Method for the Stokes Equation


Core Concepts
The author proposes an anisotropic weakly over-penalized symmetric interior penalty (WOPSIP) method for solving the Stokes 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 an optimal error estimate of the consistency error on anisotropic meshes.
Abstract
The author investigates an anisotropic WOPSIP method for the Stokes equation on convex domains. The WOPSIP method is a simple discontinuous Galerkin method similar to the Crouzeix-Raviart finite element method. The main contributions are: The author shows a new proof for the consistency term, which allows for an estimate 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 author shows that the Stokes element satisfies the inf-sup condition on anisotropic meshes, even though inf-sup stable schemes of the discontinuous Galerkin method on shape-regular mesh partitions have been widely discussed. The author provides an error estimate in an energy norm on anisotropic meshes. In the numerical experiments, the author compares calculation results for standard and anisotropic mesh partitions.
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Deeper Inquiries

What are the potential applications of the proposed anisotropic WOPSIP method beyond the Stokes equation

The proposed anisotropic WOPSIP method has potential applications beyond the Stokes equation in various fields of science and engineering. One potential application is in fluid dynamics, where anisotropic behavior is common in flows around obstacles or in complex geometries. The method could be used to accurately simulate fluid flow with varying levels of anisotropy, providing insights into turbulence, boundary layer separation, and other flow phenomena. Additionally, in structural mechanics, the method could be applied to analyze materials with anisotropic properties, such as composites or layered structures. This could help in optimizing designs for strength, stiffness, and other mechanical properties. Furthermore, in geophysics, the method could be used to model seismic wave propagation in anisotropic media, aiding in the exploration of subsurface structures and geological formations.

How does the performance of the anisotropic WOPSIP method compare to other anisotropic finite element methods for the Stokes equation in terms of accuracy and computational efficiency

In terms of accuracy and computational efficiency, the performance of the anisotropic WOPSIP method can be compared to other anisotropic finite element methods for the Stokes equation. The method's accuracy can be evaluated by analyzing the error estimates in energy norms on anisotropic meshes. By conducting numerical experiments and benchmarking against standard and other anisotropic methods, the method's accuracy can be assessed. Computational efficiency can be evaluated by analyzing the method's implementation complexity, memory usage, and convergence rates. Comparing the method's performance to other anisotropic finite element methods on a variety of test cases and mesh configurations can provide insights into its strengths and weaknesses.

Can the proposed approach be extended to other types of partial differential equations with anisotropic solutions

The proposed approach can be extended to other types of partial differential equations with anisotropic solutions by adapting the formulation and analysis to suit the specific characteristics of the new equations. For example, in the case of anisotropic diffusion equations, the method can be modified to handle the anisotropic diffusion tensor appropriately. Similarly, for anisotropic elasticity equations, the method can be adjusted to account for the directional dependence of material properties. By considering the unique features of each equation and incorporating anisotropic elements into the formulation, the proposed approach can be generalized to a broader class of partial differential equations with anisotropic solutions.
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