A model problem

The model problem I coded on the CM was a heat flow problem. The first boundary condition is known as the essential boundary condition and has to be explicitly applied in the algorithm. The second boundary conditions is know as the natural boundary condition and it is the boundary condition implicit in the FE method. This problem was assigned as part of a computer science course ``Finite Element Methods for Elliptic PDEs Using Parallel Computers.'' The problem was solved for a rectangular domain ${(x,y)\in \Omega;0\leq x \leq 0.5,0\leq y \leq 1}$ discretized into $N \times 2N$ square shaped elements. The domain and the boundary segments are shown in figure .

 

domain
Figure 1 The model domain. Essential boundary conditions are applied on the left and bottom edges; natural boundary conditions apply on the top and right edges.

A suitable form for the elements of the Galerkin operator is

$$a(\phi_j,\phi_i) = \int_\Omega {\partial\phi_j \over\partia... ...tial\phi_j \over\partial y}{\partial\phi_i \over\partial y } dV$$