An absolutely stable discontinuous Galerkin method for the indefinite time-harmonic Maxwell equations with large wave number

TL;DR

This paper introduces a novel interior penalty discontinuous Galerkin method for solving the indefinite time-harmonic Maxwell equations with large wave numbers, ensuring stability and providing error estimates across all mesh regimes.

Contribution

The paper proposes a new IPDG method with complex penalty parameters that guarantees stability and error bounds for Maxwell equations with large wave numbers, without mesh constraints.

Findings

The method is stable for all mesh sizes and wave numbers.

Error estimates are derived in energy and L2 norms.

Numerical experiments confirm theoretical stability and error bounds.

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 $h$ and wave number $k$ and for general domains including non-star-shaped ones. In turn, the coercivity 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 Fung and Wu (2009), we derive both the energy-norm and the $L^2$-norm error estimates for the IPDG method in all mesh parameter regimes including pre-asymptotic 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.