14.9 Example

As an example consider the diffusion problem with the following parameter setting. We consider the domain ? : 0 ? x ? 1, with prescribed essential boundary conditions at x = 0 and x = 1. These conditions are: u(0) = 0 and u(1) = 0. There are no natural boundary conditions. The material constant satisfies: c = 1 and the source term: f = 1.

The domain ? is divided into five elements of equal length. Fig. 14.9 shows the solution. The left part displays the computed solution u (solid line) as well as the exact solution (dashed line). Remarkably, in this one-dimensional case with the current choice of parameters, the nodal solutions are exact. The right part of the figure shows the computed flux, say flux p = c du/ dx. Again, the solid line denotes the computed flux p, which is clearly discontinuous from one element to the next, and the dashed line denotes the exact solution. The discontinuity of the computed flux field is obvious: the field u is piecewise linear, therefore the derivative du/ dx is piecewise constant. The flux p does not necessarily have to be piecewise constant: if the parameter c is a function of x, the flux p will be varying within an element.

Figure 14.9: Five element solution. Left: (solid line) approximate solution u _h(x),...