A hp-discontinuous Galerkin method for the time-dependent Maxwell’s equation: a priori error estimate

Abstract

A discontinuous Galerkin method for the numerical approximation for the time-dependent Maxwell's equations in "stable medium" with supraconductive boundary, is introduced and analysed. its hp-analysis is carried out and error estimates that are optimal in the meshsize h and slightly suboptimal in the approximation degree p are obtained.

A new mixed discontinuous Galerkin method for the electrostatic field

“…Proof The consistency of problem (19) can be deduced from the derivation of the DG formulation. Indeed, after some integration by parts and using the fact that the exact solution (u, p) ∈ H 0 (∇×, Ω) ∩ H(∇ ε •, Ω) × H 1 0 (Ω) (we refer to [21] for the definition of this space), we can prove that the exact solution of (2) satisfies formulation (19) and the consistency follows. Since problem (19) is linear finite-dimensional space, to prove the existence and uniqueness of a solution, we only have to prove that if J = 0, then (u h , p h ) = (0, 0).…”

“…Since problem (19) is linear finite-dimensional space, to prove the existence and uniqueness of a solution, we only have to prove that if J = 0, then (u h , p h ) = (0, 0). Letting J = 0, setting (v, ψ) = (u, p) in (19), and subtracting the second equation from the first one, we get a(u, u) -2J(u, u) + C(p, p) = 0.…”

“…During recent years, the discontinuous Galerkin method was developed and has been applied for approximating solutions of many partial differential equations, for example, Maxwell's equations [14][15][16][17][18][19][20][21][22], Navier-Stokes equations [23], and Poisson's equation [24].…”

“…For the time-dependent Maxwell equations, there are works of Daveau and Zaghdani [19][20][21]. In [19] a discontinuous Galerkin scheme for the wave equation derived from Maxwell's equations with constant coefficients was discussed and analyzed, and in [22], we studied a new discontinuous Galerkin formulation for the resolution of the wave equation with variable coefficients.…”

A New Symmetric Discontinuous Galerkin Formulation for the Time-Dependent Maxwell’s Equation

“…In the paper, [1], we have proposed an nonsymmetric discontinous formulation Galerkin method to approximate in space an initial boundary value problem derived from Maxwell's equations in vacuum with perfect electric conductor boundary…”