| In the early twenty-first century, with the new synthetic principle and the syn-thetic technology rapid development, we constructed the composite artificial materi-als which did not exist in nature. They have some special physical properties that allmaterials in nature don’t have. Moreover, these properties can be adjusted accordingto the needs of the people. We called this kind of materials for Metamaterials. Sincethe metamaterials are constructed by people, more and more academic researchershave generated much interest, and think that they have a very high research valueand application value. In recent years, the metamaterials, also gradually becomethe research hotspot in many academic fields (such as: Optical, Materials Science,Physics and so on).In this paper, in view of the time-dependent Maxwell’s equations modelingwave propagation in Metamaterials, we use interior penalty discontinuous Galerkinmethod for processing and solving the basic equation. As compared to the origi-nal finite element method, discontinuous Galerkin method can be very flexible indealing with the boundary of the discontinuity. Moreover, it allows for generalnon-conforming meshes with variable degrees of approximation and of higher accu-racy. The discontinuous Galerkin method was originally introduced by Reed andHill for solving a neutron transport equation in the1970s. In the following thirty orforty years, this method has improved by many researchers, and has showed goodapplication value in many academic fields.So far, there are many academic researchers published a lot of papers about us-ing the discontinuous Galerkin method to solving the Maxwell’s equations in simplemedia, there are also quite a few papers discussed the modeling Maxwell’s equationsin metamaterials. But for the error estimation, there is little research. In this paper,we consider the time-dependent Maxwell’s equations modeling wave propagation inMetamaterials. Firstly we denote them into an equation via a vector (Η Magneticfield). Secondly we consider a semi-discrete interior penalty discontinuous Galerkin(DG)method for solving the governing equations, and we develop a residual-basedposteriori error estimator. Thirdly we consider a fully-discrete DG method, and weachieve optimal error estimates in both L~2norm and energy norm. |
