Diffusion Operator

 

We now consider the two-dimensional elliptic Helmholtz equation:

$$\left( \nabla^2 - \lambda \right) u = f, \quad \lambda \ge 0.$$

Again using the Galerkin formulation and integrating by parts we obtain:

$$\left[\left(\nabla \phi_m,\nabla \phi_n\right) + \lambda \l... ...\int_{\partial \Omega} \phi_m \frac{\partial u}{\partial n} d s$$

We will define a new set of K operators:

$$L_k = \left[(\nabla \phi_m^k, \nabla \phi_n^k) + \lambda (\phi_m,\phi_n) \right]$$

and a new set of K vectors:

$$F_k = (\phi_m,f) + \int_{\partial \Omega^k} \phi_m \frac{\partial u}{\partial n} \partial s$$

Then repeating the process to assemble the weak convective operator we can assemble the subdomain operators into a global operation to obtain:

$$\vect{\hat{u}}^{g} = (Z^t L Z)^{-1} Z^t {\bf F},$$

where the bold letters denote vectors.