I have a task for an assignment.
Let $\Omega \subset \mathbb{R}^2$ be a domain with boundary $\partial \Omega$. Consider the boundary value problem, $$ -\frac{\partial }{\partial x}\left[\lambda_{11}(x,y) \frac{\partial u}{\partial x} + \lambda_{12}(x,y) \frac{\partial u}{\partial y}\right] - \frac{\partial }{\partial y}\left[\lambda_{12}(x,y) \frac{\partial u}{\partial x} + \lambda_{22}(x,y) \frac{\partial u}{\partial y}\right]= f(x,y) \text{ in } \Omega$$ $$ n_x\left[\lambda_{11}(x,y) \frac{\partial u}{\partial x} + \lambda_{12}(x,y) \frac{\partial u}{\partial y}\right] + n_y\left[\lambda_{12}(x,y) \frac{\partial u}{\partial x} + \lambda_{22}(x,y) \frac{\partial u}{\partial y}\right]= g(x,y) \text{ in } \partial \Omega$$ where $\lambda_{11}, \lambda_{12},\lambda_{22},f$ and $g$ are functions of $x$ and $y$. The outward unit normal vector is denoted by $n$ with components $n_{x}$ and $n_{y}$. (Hint: Solve using linear triangular elements. Use Newton Cotes for the numerical integration)
a) Derive the weak formulation corresponding to the above equation.
b) Compute an arbitrary element $S_{ij}$ of the element matrix of the internal triangle.
c) Compute the element vector of an internal triangle.
d) Compute the element matrix and element vector of a boundary element on $\Gamma$ (boundary).
This is a PDE with boundary conditions: $$- \nabla ( A \nabla u) = f \quad \text{in} \quad \Omega$$ $$ \bar n .(A \nabla u) + u = g \quad \text{in} \quad \partial \Omega$$ $$ A=\left [ \begin{matrix} \lambda_{11} & \lambda_{12} \\ \lambda_{12} & \lambda_{22} \\ \end{matrix} \right ]$$
So far, for b) I have obtained the equation for the Element Matrix($S_{ij}$) and Element vector($f_i$) as follows $$S_{ij} = \Sigma_{j=1}^{n}(\int_{\Omega} \langle A \nabla \phi_j , \nabla \phi_i \rangle d\Omega + \int_{\partial \Omega} \phi_j \phi_id\Gamma $$ and, $$f_i = \int_{\Omega}f\phi_i d\Omega + \int_{\partial\Omega}g\phi_i d\Gamma $$ I am not sure what to do next to compute the Element matrix and vector of the boundary element on $\Gamma$? Any guidance for c) and d) will be really helpful.
Thanks in Advance