Continuous Finite Element Method For Maxwell Eigenvalue Problems With Regular Decomposition Technique

With the regular decomposition technique, we decompose the space $\mathbf{H}_0^s(\mathbf{curl}; Ω)$ into the sum of a vector potential space and the gradient of a scalar space, both possessing higher regularity. Based on this new high order regular decomposition, a novel numerical method using standard high order Lagrange finite elements is designed for solving Maxwell eigenvalue problems. Specifically, the full convergence orders of the eigenpair approximations are proved for the proposed numerical method. Finally, numerical examples are provided to validate the proposed scheme and confirm the theoretical convergence results.

💡 Research Summary

The paper addresses the long‑standing challenge of solving Maxwell eigenvalue problems with standard continuous (C⁰) Lagrange finite elements. Traditional high‑order Nédélec edge elements, while conforming to H(curl), suffer from complex basis construction and difficult implementation, especially for higher polynomial degrees. Moreover, existing remedies such as divergence‑stabilization, special composite meshes, or mixed formulations introduce additional computational overhead and often require careful tuning.
The authors revisit the regular decomposition technique, which expresses any field in H₀ˢ(curl;Ω) as the sum of a vector potential and the gradient of a scalar potential. Building on recent results by Costabel and McIntosh on de Rham complexes, they prove a new high‑order regular decomposition for s > ½ on bounded Lipschitz polyhedral domains. Crucially, they relax the boundary condition on the vector potential to a tangential‑zero condition (u × n = 0) while retaining a Dirichlet condition on the scalar potential. Under these conditions, any ξ ∈ H₀ˢ(curl;Ω) can be written as ξ = u + ∇p with u ∈ Hˢ⁺¹(Ω)∩H₀(curl;Ω) and p ∈ Hˢ⁺¹(Ω)∩H₀¹(Ω). Both potentials enjoy one extra Sobolev regularity compared with the original field.
Exploiting this decomposition, the authors design a discretization scheme that uses two continuous finite‑element spaces: a degree‑k Lagrange space V_hᵏ for the vector potential u and a degree‑(k+1) Lagrange space W_h^{k+1} for the scalar potential p. The Maxwell eigenvalue problem is reformulated as a generalized eigenvalue problem a((u_h,p_h),(v_h,q_h)) = λ_h b((u_h,p_h),(v_h,q_h)), where a and b are bilinear forms derived from the curl‑curl operator and the L² inner product, respectively. Because both spaces are subspaces of H¹, the implementation fits directly into any standard FEM package without the need for edge‑based degrees of freedom.
Theoretical analysis proceeds in two stages. First, using the compactness of the continuous solution operator combined with the regular decomposition, the authors prove that the discrete operator is also compact and free of spurious zero modes. This guarantees spectral convergence of the discrete eigenvalues to the true spectrum. Second, they derive optimal error estimates. By standard interpolation theory and the additional regularity of u and p, they obtain ‖E − E_h‖_{H(curl)} ≤ C h^{min(k,s)} and |λ − λ_h| ≤ C h^{2 min(k,s)}. Here s denotes the Sobolev regularity of the exact eigenfunctions, which is at least ½ + ε on polyhedral domains. Consequently, for smooth eigenfunctions the method achieves the full convergence order dictated by the polynomial degree.
Numerical experiments are conducted on three representative polyhedral domains: a unit cube, an L‑shaped domain, and a domain with mixed convex and concave faces. For polynomial degrees k = 1, 2, 3, uniform tetrahedral refinements are performed. The computed eigenvalues converge with rates matching the theoretical predictions, and the H(curl)‑norm errors exhibit the expected O(h^{k}) behavior. Importantly, no spurious eigenvalues appear, confirming the theoretical claim of zero‑mode freeness. The results also demonstrate that the higher‑order scheme (k = 3) attains very high accuracy with relatively modest mesh refinement, highlighting the efficiency of the approach.
In conclusion, the paper presents a conceptually simple yet mathematically rigorous framework for Maxwell eigenvalue problems that leverages continuous Lagrange elements. By introducing a novel regular decomposition with relaxed boundary conditions, the authors circumvent the need for edge elements, special meshes, or stabilization terms. The method is readily implementable in existing FEM software, achieves optimal convergence rates, and avoids spurious modes, making it a compelling alternative for electromagnetic eigenvalue simulations. Future work suggested includes extending the analysis to heterogeneous media, non‑polyhedral geometries, and time‑dependent Maxwell equations.