Numerical Algorithm For Solving Three-dimensional Inhomogeneous Anisotropic Maxwell’s Equations

Maxwell’s equations are widely used in many engineering fields such as electromag-netic scattering,optical fiber communication,oil and gas exploration,etc.It is of great significance to accurately solve Maxwell’s equations for practical engineering problems.In many cases,multiple media coexist in practical problems,so inhomogeneous Maxwell interface problem has always been a very important topic in the field of science and engi-neering.Compared with the solution of homogeneous Maxwell’s equations,how to deal with complex interface conditions is one of the difficulties.With the wide application of anisotropic media in microwave materials and military fields,the problem of inhomoge-neous anisotropic Maxwell interface has attracted more attention.However,when the per-mittivity and permeability are anisotropic media,the scale of solving Maxwell equations are greatly increased which is difficult to solve.This problem poses an important challenge to both engineering and mathematics.In this paper,a three-dimensional Petrov-Galerkin finite element interface（3D PGFEI）method with non-body-fitted grids is proposed for solving the inhomogeneous anisotrop-ic time-harmonic Maxwell’s equations in irregular regions.In the first part,the model of the inhomogeneous anisotropic Maxwell interface problem is derived.The boundary value problem of the three-dimensional inhomogeneous anisotropic Maxwell’s equations is con-structed by using the divergence-free condition,the interface jump condition generated by inhomogeneous anisotropic media,the boundary condition and the classical Maxwell’s e-quations.In order to facilitate the subsequent section,we further derive the corresponding weak variation form.In the second part,a three-dimensional Petrov-Galerkin finite element interface method is proposed to solve the inhomogeneous anisotropic Maxwell’s equations.Firstly,the anisot-ropic Maxwell interface model is deduced as elliptic equations.The irregular region is meshed by uniform tetrahedral grids while the media interface is not aligned with the grids.This method allows the use of non-body-fitted meshes that are not aligned with the media interface,which can effectively reduce the cost of mesh generation compared with body-fitted meshes.The construction of level-set function can effectively identify the relation-ship between tetrahedral element and anisotropic media interface,and the construction of boundary function is used to judge the relationship between element and boundary.Apply-ing the above two functions,all cases of tetrahedral element truncated by inhomogeneous media interface and boundary can be obtained.Meanwhile,we construct the special ba-sis functions according to the jump conditions across the inhomogeneous interface,where the permittivity and permeability are discontinuous three-dimensional tensors.The result-ing linear system is a banded sparse matrix and demands less memory resources.Thereby substantially increasing the computing accuracy and efficiency.In the third part,numerical examples are given to verify the superior performance of 3D PGFEI method.Maxwell’s equations with homogeneous and inhomogeneous anisotropic media are solved respectively.Maxwell’s equations for magnetized plasma and ferrite with practical engineering significance are also solved.We introduce the L_∞norm and L₂norm of error.Numerical experiments illustrate the efficiency and accuracy of the proposed 3D PGFEI method and the solutions can achieve second-order accuracy in terms of the L_∞and L₂error norm.