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Efficient Discretization of the Laplacian Operator on Complex Geometries Using Continuous Summation-by-Parts Methods


Core Concepts
The author presents a new efficient discretization method for the Laplacian operator on complex geometries by extending the continuous summation-by-parts (SBP) framework to second derivatives and combining it with spectral-type SBP operators on Gauss-Lobatto quadrature points.
Abstract
The author addresses the challenge of efficiently discretizing the Laplacian operator on complex geometries, which is of primary interest in academia and industry for problems involving second derivatives, such as the Navier-Stokes equations or wave propagation problems. The key highlights and insights are: The author extends the continuous summation-by-parts (SBP) framework to second derivatives and combines it with spectral-type SBP operators on Gauss-Lobatto quadrature points to obtain a highly efficient discretization of the Laplacian on complex domains. The resulting Laplace operator is defined on a grid without duplicated points on the interfaces, removing unnecessary degrees of freedom in the scheme, and is proven to satisfy a discrete equivalent to Green's first identity. The author proves semi-discrete stability using the new Laplace operator for the acoustic wave equation in 2D. The method can easily be coupled together with traditional finite difference operators using glue-grid interpolation operators, resulting in a method with great practical potential. Two numerical experiments are conducted on the acoustic wave equation in 2D, demonstrating the accuracy and efficiency properties of the method, as well as its potential use in a realistic problem with a complex region of the domain discretized using the new method and coupled to the rest of the domain discretized using a traditional finite difference method.
Stats
The author presents the following key figures and metrics: "The error results with the SBP GL operators are also presented in Table 1, including convergence rates estimated as q = log(e1/e2)/log(N2/N1)^(1/2), where e1 and e2 are the L2-errors with N1 and N2 DOFs."
Quotes
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Key Insights Distilled From

by Gustav Eriks... at arxiv.org 04-16-2024

https://arxiv.org/pdf/2404.09050.pdf
Efficient discretization of the Laplacian on complex geometries

Deeper Inquiries

How can the continuous SBP method be extended to handle more complex element shapes beyond quadrilaterals, such as triangles or hexagons, to further improve the flexibility in discretizing complex geometries

The continuous SBP method can be extended to handle more complex element shapes beyond quadrilaterals by utilizing the concept of simplex elements. Simplex elements, such as triangles in 2D and tetrahedra in 3D, offer a flexible approach to discretizing irregular geometries. By defining SBP operators on simplex elements, the continuous SBP method can be adapted to handle these shapes. The key idea is to map the complex geometry to a reference domain where the SBP operators are well-defined, similar to the approach used for quadrilateral blocks. The interpolation and projection operators needed to impose interface conditions can be constructed based on the geometry of the simplex elements. This extension would enhance the method's versatility and applicability to a wider range of geometries.

What are the potential limitations or drawbacks of the glue-grid framework used to couple the new Laplace operator with traditional finite difference operators, and how could these be addressed

The glue-grid framework used to couple the new Laplace operator with traditional finite difference operators may have some limitations and drawbacks. One potential limitation is the complexity of constructing accurate interpolation operators between different grid structures, especially when dealing with non-conforming interfaces. Inaccurate interpolation can introduce errors and affect the overall stability and accuracy of the numerical scheme. Additionally, the computational cost of implementing and applying multiple interpolation operators can be significant, especially for large-scale simulations. To address these limitations, one approach could be to optimize the interpolation algorithms to ensure accuracy and efficiency. This may involve using higher-order interpolation techniques or adaptive methods to minimize interpolation errors. Furthermore, exploring alternative coupling strategies, such as domain decomposition methods or overlapping grids, could provide more robust and efficient ways to handle complex geometries with different grid structures. By carefully designing the interpolation and coupling procedures, the drawbacks of the glue-grid framework can be mitigated, leading to improved performance and reliability in numerical simulations.

Could the efficient discretization of the Laplacian presented in this work be applied to other types of partial differential equations beyond the acoustic wave equation, such as the Navier-Stokes equations or electromagnetic wave propagation problems

The efficient discretization of the Laplacian presented in this work can be applied to a wide range of partial differential equations beyond the acoustic wave equation. The method's flexibility and accuracy make it suitable for various problems involving second derivatives on complex geometries. For example, the approach can be extended to solve the Navier-Stokes equations, which describe fluid flow behavior, by incorporating appropriate boundary conditions and adapting the discretization to handle the specific requirements of fluid dynamics. Similarly, electromagnetic wave propagation problems, such as Maxwell's equations, can benefit from the efficient Laplacian discretization on complex domains. By integrating the Laplace operator into numerical schemes for these equations, researchers can achieve highly accurate simulations with reduced computational costs, making the method applicable to a diverse set of PDEs in different fields of study.
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