Finite Element Approximation for Maxwell’s Eigenvalue Problem

Assignment Brief

Finite Element Approximation for the eigenvalue problem associated to Maxwell`s equations

Sample Answer

Finite Element Approximation for Maxwell’s Eigenvalue Problem

Maxwell’s equations are fundamental in describing the behaviour of electromagnetic fields. In the context of eigenvalue problems, they are often written in terms of the electric field E and magnetic permeability μ, as well as the permittivity ε. A simplified expression of the eigenvalue problem can be written as:

∇ × (μ⁻¹ ∇ × E) = ω² ε E

Here:

• ∇ × represents the curl operator

• μ is the magnetic permeability

• ε is the permittivity

• E is the electric field

• ω is the angular frequency (eigenvalue related)

The goal of the finite element method (FEM) is to approximate solutions to this eigenvalue problem by discretising the domain into smaller finite elements. This allows for complex geometries and boundary conditions to be handled numerically.

Weak Formulation

To apply FEM, we begin by multiplying the equation by a test function v and integrating over the domain:

∫ (μ⁻¹ (∇ × E) · (∇ × v)) dΩ = ω² ∫ (ε E · v) dΩ

This converts the differential equation into a weak form suitable for FEM.

Discretisation

Next, the domain is divided into finite elements (e.g., tetrahedral or hexahedral meshes). The electric field E is approximated as a linear combination of basis functions defined over these elements:

E ≈ Σ (Ei φi)

where φi are basis functions and Ei are the unknown coefficients.

Continued...