I've been asked by my advisor to solve Dirac's Equation using some finite element method. Furthermore, he's asked me to formulate everything using Geometric Calculus.
The differential form of Dirac's equations has been put in Geometric Calculus formalism by David Hestenes in Real Dirac Theory, and it goes like this:
\begin{equation}\nabla\psi\gamma_2\gamma_1 = m \psi \gamma_0\end{equation}
where $\nabla$ is the vector derivative ($\nabla = \sum_{i=0}^3 \gamma^i\partial_{x_i}$ in coordinate form), $\psi$ is a multivector field valued in the even spacetime algebra $Cl_{1,3}^{+}(\mathbb{R})$ and, even though $\gamma_2\gamma_1$ and $\gamma_0$ appears in the equation, it is independent from the reference frame, as Hestenes said,
"These constants need not be associated with vectors in a particular reference frame, though it is often convenient to do so. It is only required that $\gamma_0$ be a fixed, future-pointing, timelike unit vector while $\gamma_2\gamma_1$ is a spacelike unit bivector which commutes with $\gamma_0$".
Usually, to derive a finite element method for real valued functions, I have to start with a weak or variational form, which I obtain by multiplying the differential equation by a test function $v$ (as I am interested in Discontinuous Galerkin methods, lets not assume it vanishes on the boundary), integrating over the domain and applying some kind of divergence theorem. For example,
\begin{equation} \frac{\partial u (t,x)}{\partial t} + \frac{\partial u (t,x)}{\partial x} = 0 \end{equation}
ends up like this,
\begin{equation} \int_{I}\frac{\partial u (t,x)}{\partial t}vdx -\int_I u\frac{\partial v}{\partial x}dx +\int_{\partial I} uv = 0 \qquad \forall v \subset V \end{equation} where the integration is in space $I\subset \mathbb{R}$, and $v = v(x)$ and $V$ is some appropriate function space.
The question I want to ask is how to do something similar in Geometric Calculus to the Dirac equation I showed above, using the fundamental theorem of Geometric Calculus instead of the divergence theorem and using Clifford valued test functions insted of real valued $v$. Is this even the correct approach, or should I try to look at Clifford valued Lagrangian and action and try to minimize that ( in the sense that every multivector entry should be minimized).
I don't even know where to start, given I am absolutely new to this subject. I have been trying to read books and articles but some are really complicated, and my background is in physics so that doesn't help much. I would really appreciate book recommendations on this subject in particular.