Using high-$\kappa$ dielectrics or materials with high dielectric constants is increasingly popular in emerging antenna and microwave components. The electromagnetic analysis of them often encounters dense mesh and massive unknowns, which results in cumbersome simulation and optimization. To alleviate this difficulty, the discontinuous Galerkin surface integral equation (DG-SIE) and an independent mesh procedure are presented so that the metallic sheets attached to the dielectric surface are mesh independently for two sets of unknown currents. Two numerical examples, including a high dielectric constant array antenna and a frequency-selective structure, are carried out to verify that the proposed method can improve computational efficiency, reduce the number of unknowns and save computational resources while guaranteeing accuracy.