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A stabilized mixed finite element method is proposed for solving the Maxwell eigenproblem in terms of the electric field and the multiplier. Using the Bochev-Dohrmann-Gunzburger stabilization, we introduce some ad hoc stabilizing parameters for stabilizing the kernel-coercivity of the electric field and for stabilizing the inf-sup condition of the multiplier. We show that the stabilized mixed method is stable and convergent, with applications to some lowest-order edge elements on affine rectangular and cuboid mesh and on nonaffine quadrilateral mesh which fail in the classical methods. In particular, we prove the uniform convergence for guaranteeing spectral-correct and spurious-free discrete eigenmodes from the Babuska-Osborn spectral theory for compact operators. Numerical results have illustrated the performance of the stabilized method and confirmed the theoretical results obtained.

The purpose of this paper is to develop a general theory on how the inf‐sup stable and convergent elements of the velocity Dirichlet boundary (VDB)‐Stokes problem with no‐slip VDB are still inf‐sup stable and convergent for the pressure Dirichlet boundary (PDB)‐Stokes problem with PDB in Lipschitz domain. The PDB‐Stokes problem in a Lipschitz domain usually only has a singular velocity solution which does not belong to (H1(Ω))2, sharply in contrast to the VDB‐Stokes problem whose velocity solution still belongs to (H1(Ω))2, and unexpectedly, some well‐known inf‐sup stable and convergent VDB‐Stokes elements may or may no longer correctly converge. It turns out that the inf‐sup condition of the PDB‐Stokes problem in Lipschitz domain relies on an unusual variational problem and requires adequate degrees of freedom on the domain boundary. In this paper we propose two families of staggered elements: staggered Taylor–Hood elements ()scriptCPℓ+22−scriptPnormalℓ with ℓ ≥ 1 (continuous in both velocity and pressure) and staggered Fortin elements ()scriptCPm+22−scriptPmdisc with m ≥ 1 (continuous in velocity and discontinuous in pressure) on triangles, for solving the PDB‐Stokes problem in Lipschitz domain. We show that the two families are inf‐sup stable and are correctly convergent for the non‐H1 singular velocity. Numerical results illustrate the proposed elements and the theoretical results.

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