研究目的
To couple FEM and MMP to model Maxwell's equations for materials with spatially varying properties in an unbounded domain.
研究成果
The FEM-MMP coupling offers advantages such as simpler assembly without singular integrals and exponential convergence when the coupling boundary is far from field sources. Among the coupling approaches, PDE-constrained coupling is recommended for reliability, while multifield and DG-based methods are less expensive but more prone to ill-conditioning. Future work will involve full numerical analysis for Maxwell's equations.
研究不足
The method suffers from ill-conditioning and lacks a rigorous theory on the placement of multipoles in 3D. It is susceptible to errors with coarse meshes and high numbers of multipoles, and requires user input for parameters like penalty in DG coupling.
1：Experimental Design and Method Selection:
The study involves coupling the Finite Element Method (FEM) and the Multiple Multipole Program (MMP) for solving Maxwell's equations in unbounded domains with spatially varying material properties. Four coupling approaches are developed: least-squares-based coupling, multifield variational formulation, discontinuous Galerkin coupling, and coupling by tangential traces.
2：Sample Selection and Data Sources:
Numerical experiments are conducted using exact solutions and a magnetostatic inductor setup. Domains include bounded FEM regions and unbounded MMP regions with specific geometries (e.g., spheres, torus, cylinders, prism).
3：List of Experimental Equipment and Materials:
Computational tools include COMSOL for mesh generation, C++14 code with Eigen v
4：4 for linear algebra, HyDi for FEM components, and PARDISO v0 for matrix solving. No physical equipment is mentioned. Experimental Procedures and Operational Workflow:
Meshes are generated and refined (h-refinement for FEM, p-refinement for MMP). Coupling approaches are implemented, and convergence is studied by monitoring L2-errors in FEM and MMP domains. Iterations are performed for nonlinear cases.
5：Data Analysis Methods:
Relative L2-errors are computed for FEM and MMP parts. Convergence plots and surface plots are generated to analyze error behavior with mesh refinement and multipole numbers.
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