Annals of Mathematical Sciences and Applications

Volume 7 (2022)

Number 1

Local discontinuous Galerkin methods for the carpet cloak model

Pages: 97 – 137



Xinyue Yu (Division of Applied Mathematics, Brown University, Providence, Rhode Island, U.S.A.)

Jichun Li (Department of Mathematical Sciences, University of Nevada, Las Vegas, Nv., U.S.A.)

Chi-Wang Shu (Division of Applied Mathematics, Brown University, Providence, Rhode Island, U.S.A.)


The DG methods have been shown to have good performance in numerical simulations of the carpet cloak model in [32]. However, the stability analysis and the error estimate are left to be done. In this paper, we introduce the leap-frog DG methods to solve the carpet cloak model. We prove the stability of the semi-discrete scheme, the sub-optimal error estimate for unstructured meshes, and the optimal error estimate for tensor-product meshes. Then, the fully discrete scheme is stated and the stability is proved. Finally, the numerical accuracy tests on rectangular and triangular meshes are given respectively, and the results of numerical simulations of the wave propagation in the carpet cloak model using the DG scheme are presented.


discontinuous Galerkin method, Maxwell’s equations, metamaterials, leap-frog scheme

2010 Mathematics Subject Classification

65M12, 65M60

The research of Jichun Li was supported by NSF grant DMS-2011943.

The research of Chi-Wang Shu was supported by AFOSR grant FA9550-20-1-0055 and NSF grant DMS-2010107.

Received 31 January 2022

Accepted 25 February 2022

Published 7 April 2022