1 Discretization of the Simple Diffusion Model

The discretization of the simple diffusion models follows directly from the discussion presented in Section 2.3. In order to construct the global matrix of the system, we write nucleus matrix of the simple diffusion model for each $T\in T_h(\Omega)$, $rank(\mathbf{\Pi}(T))= 4$,

$$\mathbf{\Pi}(T) = \mathbf{K}(T) + D \Delta t \mathbf{M}(T).$$ (139)

$\Delta t$ is the time step of the discretisized time and $\mathbf{K}(T)$ and $\mathbf{M}(T)$ the stiffness and mass matrices defined on single tetrahedra $T$ from $T_h(\Omega)$,

$$K_{pq}(T)=\int\limits_T \nabla \varphi_p(\mathbf{x}) \nabla \varphi_q(\mathbf{x}) d\Omega,$$

$$M_{pq}(T)=\int\limits_T \varphi_p(\mathbf{x}) \varphi_q(\mathbf{x}) d\Omega,$$ (140)

for $p,q\in\{1,2,3,4\}$. This problem is linear and the calculation of the nucleus matrix $\mathbf{\Pi}(T)$ as the Jacobi matrix of the operator $\ell^e : \Bbb{R}^{4}\rightarrow \Bbb{R}^{4}$ (introduced previously in Section 2.5) is trivial.
the global matrix $\mathbf{G}$ is assembled out of the $\mathbf{\Pi}(T)$ matrices and has obviosly a dimension equal to the number of all points of discretization.