B-spline discontinuous Galerkin method for Maxwell equations

In the field of computational electromagnetics, the Discontinuous Galerkin Time Domain (DGTD) method typically relies on irregular grid partitioning in model space and high-order polynomial interpolation calculations on elements. When comparing two-dimensional spatial quadrilateral mesh partitioning to triangular mesh partitioning at the same interpolation order, quadrilateral meshing offers fewer degrees of freedom and higher computational efficiency. However, traditional basis function spaces, relying on isoparametric transformations and polynomial tensor product interpolation, only possess low-order completeness on quadrilateral elements. Consequently, their stability and accuracy are significantly influenced by grid distortion. Addressing this challenge, this thesis proposes a high-order B-spline interpolation DGTD method based on irregular quadrilateral meshes for solving Maxwell's equations. The advantage of B-spline interpolation lies in its high-order completeness on irregular elements, effectively eliminating internal degrees of freedom within the elements. Furthermore, the coefficient matrices of the discrete system for Maxwell's equations also possess exact analytical forms.. Utilizing this method to analyze the eigenmodes of cavities and the electromagnetic scattering of wedge structures, the results indicate that increasing the maximum allowable time step by 2.5 times, and reducing the required unknowns by 25% compared to COMSOL software, the proposed algorithm exhibits notable advantages in terms of higher stability and precision.