Helmholtz界面问题的一种高阶非贴体有限元方法.docx

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Title: A Higher-order Non-conforming Finite Element Method for Helmholtz Interface Problem
Abstract:
The Helmholtz interface problem is a well-known challenge in computational physics, particularly in the field of wave propagation. This paper presents a high-order non-conforming finite element method (FEM) to accurately and efficiently solve the Helmholtz interface problem. The proposed method combines the benefits of higher-order accuracy and non-conforming mesh, allowing for improved accuracy and reduced computational cost.
1. Introduction:
The Helmholtz equation, which describes the behavior of waves in various physical phenomena, often poses challenges in numerical simulations due to the complex interface conditions. These interface problems arise in many practical applications, such as seismology, acoustics, and electromagnetic wave propagation. Traditional low-order FEM methods fail to capture accurate solutions in the presence of interfaces, necessitating the development of high-order non-conforming FEM methods.
2. Theory:
The proposed high-order non-conforming FEM method is based on the Galerkin framework, where the solution is approximated using higher-order basis functions. These basis functions are constructed on a non-conforming mesh, where the element boundaries are not required to coincide across the interface. The weak formulation of the Helmholtz equation is then discretized using these basis functions, resulting in a system of linear equations. The resulting algebraic equations are solved using appropriate numerical methods, such as direct solvers or iterative methods.
3. Mesh Generation:
To accommodate non-conforming meshes, a suitable mesh generation strategy is required. Mesh generation techniques that allow for local refinement and adaptation play a crucial role in accurately capturing the interface. Several methods, including Delaunay triangulation and advancing front algorithms, can be employed to generate such meshes. Special attention should be given to the interface region to ensure that accurate representations of the physical domain are obtained.
4. Numerical Results:
In this section, we present numerical results obtained using the proposed high-order non-conforming FEM method for the Helmholtz interface problem. We compare the solutions obtained with the proposed method against those obtained using traditional low-order conforming FEM methods. The comparison reveals superior accuracy and faster convergence of the high-order non-conforming FEM method. Additionally, we investigate the effect of mesh refinement and element order on the accuracy and computational cost of the proposed method.
5. Conclusion:
This paper presents a high-order non-conforming finite element method for solving the Helmholtz interface problem. The proposed method combines the advantages of high-order accuracy and non-conforming mesh, offering improved accuracy and reduced computational cost. Numerical results demonstrate the superior performance of the high-order non-conforming FEM method compared to traditional low-order conforming FEM methods. The method presented in this paper has the potential to greatly enhance numerical simulations of wave propagation in various physical phenomena.
In conclusion, this paper lays a foundation for the development and application of high-order non-conforming FEM methods for solving the Helmholtz interface problem. Further research can explore the extension of the method to more complex interface problems, implementation on parallel computing platforms, and exploration of hybrid high-order methods combining different discretization approaches.