Treatment of Elliptic Equations

We are interested in the approximate solution of an elliptic boundary value problem of the type 3.1$$Lu = f{\text{ in }}\Omega\subset {\mathbb{R}^{d}},$$$$\begin{gathered} , \hfill \\ Bu = g{\text{ on }}\partial \Omega {\text{,}} \hfill \\ \end{gathered}$$ where L is a symmetric partial differential operator of second order and B expresses suitable boundary conditions. Here, we are faced with two major computational tasks: We need to discretize the partial differential operator efficiently and we need to deal with boundary conditions properly.