Multigrid methods for computing eigenvalues of symmetric and positive definite cases, e.g.,, are not applicable directly for the Maxwell eigenvalue problems due to the large kernel of the differential operator curl. However, the penalty method also introduces spurious eigenvalues.įor all the methods referring to Maxwell eigenvalue problems above, a large-scale discrete eigenvalue problem has to be solved, which is very challenging and time-consuming. Compared with the discretized linear system that arises from the mixed method, the system from the penalty method is considerably smaller. Another approach is the penalty method which relies on explicitly imposing the divergence-free condition by introducing a penalty term (see, e.g., ). However, the resulting linear algebraic system is larger and of the saddle point type such that it is difficult to solve (see, e.g., ). Using the mixed discretization, no spurious eigenvalue will be introduced. In a mixed formulation, a Lagrange multiplier is introduced to impose the divergence-free constraint ( 1.2) in a weak sense. An alternative is the so-called mixed formation. Though dropping the constraint may introduce spurious eigenvalues, doing so will not affect the nonzero eigenvalues (see, e.g., ). The divergence-free constraint ( 1.2) can be dropped from the weak formulation and imposed implicitly.
Imposing the divergence-free constraint in the discretization is a challenging task. In the following sections, we will use the conventional notation λ to replace ω 2.Įdge finite element methods for solving the Maxwell eigenvalue problem are widely used and their convergences have been studied extensively (see and references therein). The coefficients μ r and ε r are the real relative magnetic permeability and electric permittivity, respectively, that satisfy the Lipschitz continuous condition, whereas ω is the resonant angular frequency of the electromagnetic wave for cavity Ω. Ω ⊂ ℝ n( n = 2, 3) is a bounded Lipschitz polyhedron domain and γ t u is the tangential trace of u.
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