A Numerical Electromagnetic Solver Using Duality

[Paul Colby]

In the previous insights article (How to Use Duality in Computational Electromagnetic Problems), I covered some uniqueness theorems for the Riemann-Silberstein vector, $F=E+iB$, for time-harmonic fields. We showed that the boundary value, $f$, completely determines the vector field throughout the volume. In this article, I’d like to step through constructing a numerical algorithm for finding $F$ given $f_\uparrow$.
I’ve never done a finite element program from scratch before. This seems as good a time as any to try one. The algorithm is based entirely on what we developed in the previous insights so it will look somewhat different than the usual ones. Hope you find this interesting.
Recap
Just so we have everything up front I’d like to remind everyone of the important bits we’ll be using. The linear vector space, $\Lambda_k(V),$ is the collection of all...

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[Vailanor]

We consider a bounded volume, V, of space in which we wish to solve for the vector field, $F$. We have assumed that this vector field satisfies the equation:$$\nabla \times F=i\omega \mu_0 H$$where $H$ is the magnetic field and $\mu_0$ is the permeability of free space. The boundary values, $f_\uparrow$, are known and satisfy the condition$$\hat n\times f_\uparrow=i\omega \mu_0 h_\uparrow$$where $h_\uparrow$ is the magnetic field on the boundary and $\hat n$ is the outward pointing normal vector.The AlgorithmWe will use the finite element method to solve for $F$ given $f_\uparrow$. This will require us to divide the volume into elements, $V_e$, and approximate the vector field within each element. We will then use the boundary conditions to relate the unknowns in each element. Step 1: Divide the volume into elementsWe will divide the volume into elements using a structured or unstructured mesh. Each element, $V_e$, will be a simplicial or polygonal element with a set of nodes. The nodes will be used to represent points of discretization in the vector field. Step 2: Approximate the vector field in each elementFor each element, $V_e$, we will approximate the vector field using a linear interpolation. That is, we will assume that the vector field can be written as a linear combination of basis functions evaluated at the nodes. Step 3: Enforce the boundary conditionsThe boundary values of the vector field, $f_\uparrow$, are known. We can use these boundary conditions to relate the unknowns in each element. This will allow us to solve for the vector field in the volume. Step 4: Solve for the vector fieldOnce the boundary conditions have been enforced we can solve for the vector field in the volume. This is done by solving a system of linear equations which represent the interpolation of the vector field in each element. ConclusionWe have outlined an algorithm for finding the vector field, $F$, in a bounded volume given