This program was contributed by Jake J. Harmon <jake.harmon@ieee.org>.
It comes without any warranty or support by its authors or the authors of deal.II.

This program is part of the deal.II code gallery and consists of the following files (click to inspect):

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Annotated version of Readme.md

Readme file for Maxwell-Eigenvallue-hp-Refinement

Note: The implementation of this program is in part based on [1].

Motivation for project

From the source free Maxwell equations in differential form, we may find the following eigenvalue problem involving the electric field \(\mathbf{E}\), \(\) \nabla\times(\mu_r^{-1}\nabla\times\mathbf{E})-k_0^2\varepsilon_r\mathbf{E} = 0 \textrm{ in } \Omega, \(\) where \(\mu_r\) and \(\varepsilon_r\) denote, respectively, the relative permeability and permitivity of the medium (which we assume to be homogeneous), and \(k_0\) signifies the free space wavenumber, for some \(\Omega \subset \mathbb{R}^d, \, d = 2,3.\) Finding (approximate) solutions of this eigenvalue problem poses a number of challenges computationally; for those interested, we refer to the excellent thesis of S. Zaglmayr [2].

In the remainder of this project, we assume \(d=2\), though the methodology is largely unaffected by this choice. We further assume perfect electrical conductor (PEC) boundary conditions: \(\hat{\textbf{n}}\times\textbf{E}=0 \textrm{ on }\partial\Omega\), \(\hat{\textbf{n}}\) being the outward normal vector.

In the standard way, we consider weak solutions by solving the variational form of the eigenvalue problem, which, in the 2-D case, is found to be the following after Galerkin testing: \(\) \textrm{Find } U_{hp}=\left\{ \mathbf{u}_{hp},\,\lambda_{hp}\right\}\in V_{hp}\times \mathbb{R}_{>0} \textrm{ such that} \(\) \(\) a(\textbf{u}_{hp},\,\boldsymbol{\phi}_{hp}) = \lambda_{hp} m(\textbf{u}_{hp},\,\boldsymbol{\phi}_{hp}) \quad \forall\boldsymbol{\phi}\in V_{hp}, \(\) with \(a(\textbf{u}_{hp},\,\boldsymbol{\phi}_{hp}) = \langle \nabla_t\times\textbf{u}_{hp},\,\nabla_t\times\boldsymbol{\phi}_{hp}\rangle\) (note: \(\nabla_t\) represents the transversal gradient operator and \(\langle \cdot ,\, \cdot \rangle\) represents the \(L^2\) inner-product), and \(m(\textbf{u}_{hp},\,\boldsymbol{\phi}_{hp}) = \langle \textbf{u}_{hp},\,\boldsymbol{\phi}_{hp} \rangle\). The finite dimensional subspace \(V_{hp}\) will be further specified below along with its infinite dimensional analog \(V\) associated with an exact solution \(U=\left\{ \textbf{u},\, \lambda \right\}\).

For this problem, we consider a single family of quantities of interest (QoIs), namely the approximation error of the approximate eigenvalue \(\lambda_{hp}\), i.e., \(\) e_{\lambda_{hp}} := \lambda-\lambda_{hp}. \(\)

Some comments on the discretization of Maxwell's equations

In the proper solution of variational problem, \(V_{hp}\) is not arbitrary, but should instead a subspace of \(H(\mathrm{curl};\,\Omega)\) or, with the boundary conditions indicated above, \(H_{0}(\mathrm{curl};\,\Omega)\), where \(\) H(\mathrm{curl};\,\Omega) = \left\textbf{u}\in \left[ L_2(\Omega)\right]^d \, \mathrm{s.t.} \, \nabla\times\textbf{u}\in \left[ L_2(\Omega)\right]^{2d-3} \right