An Absolutely Stable Discontinuous Galerkin Method for the Indefinite Time-Harmonic Maxwell Equations with Large Wave Number

Abstract

This paper develops and analyzes an interior penalty discontinuous Galerkin (IPDG) method using piecewise linear polynomials for the indefinite time harmonic Maxwell equations with the impedance boundary condition in the three-dimensional space. The main novelties of the proposed IPDG method include the following: first, the method penalizes not only the jumps of the tangential component of the electric field across the element faces but also the jumps of the tangential component of its vorticity field; second, the penalty parameters are taken as complex numbers of negative imaginary parts. For the differential problem, we prove that the sesquilinear form associated with the Maxwell problem satisfies a generalized weak stability (i.e., inf-sup condition) for star-shaped domains. Such a generalized weak stability readily infers wave-number explicit a priori estimates for the solution of the Maxwell problem, which plays an important role in the error analysis for the IPDG method. For the proposed IPDG method, we show that the discrete sesquilinear form satisfies a coercivity for all positive mesh size $h,$ wave number $k$, and for general domains including nonstar-shaped ones. In turn, the coercivity estimate easily yields the well-posedness and stability estimates (i.e., a priori estimates) for the discrete problem without imposing any mesh constraint. Based on these discrete stability estimates, by adapting a nonstandard error estimate technique of [X. Feng and H. Wu, SIAM J. Numer. Anal., 47 (2009), pp. 2872--2896, X. Feng and H. Wu, Math. Comp., 80 (2011), pp. 1997--2024], we derive both the energy-norm and the $L^2$-norm error estimates for the IPDG method in all mesh parameter regimes including preasymptotic regime (i.e., $k^2 h\gtrsim 1$). Numerical experiments are also presented to gauge the theoretical results and to numerically examine the pollution effect (with respect to $k$) in the error bounds.