In this section numerical results will be reported for the application of the method outlined in the previous section to two parabolic control problems from [1,9] and a total of ten elliptic control problems from [16][19]. The first parabolic problem is particularly interesting because for it in [1] an analytical exact solution is given and the continuous SSC conditions are verified. For the sake of completeness, the specification will be given for each problem. The discretizations used are those defined in section 2 for the parabolic and in [16][19] for the elliptic problems.
Problem (P) from section 2 is solved with the following data


In Table 1 the maximum errors over the grid points are listed for the computed control and state functions while in Table 2 the smallest eigenvalue is listed as well as the value scaled by . The errors exhibit quadratic convergence while the scaled eigenvalue stays nearly constant with a slightly increasing tendency. These facts justify the application of the technique to other control problems for which no exact solution is known and it also shows which scaling of is appropriate for problem . This scaling is also suggested by known quadratic convergence estimates for fixed control in since the differential equation as well as all boundary conditions were discretized with secondorder consistency. An argument could be made that the smallness of the eigenvalue may indicate that it is, in fact, negative and the relatively coarse discretization prevents it from exhibiting this fact. This seems highly unlikely in the light of the very uniform and even slightly growing behavior of the scaled value combined with the fact that all the eigenvalues were computed and the smallest one is not considerably smaller than the next ones but they grow only gradually.
As a second parabolic case one from [9] was chosen. The data for
(P) are


For 5.2II we also include a plot of the optimal control and state in Figure 1.
Next, we present the data for the elliptic boundary control problems from [16,19] and the eigenvalues obtained. The domain is the unit square in all cases. The problem (EB) together with (2.1) and (2.2) is considered and in all cases.
(EB1)

(EB2) as above, but , , .

(EB3)

(EB4)

(EB5)

In all cases except (EB5) which is different from the others in key elements, the scaling of by was appropriate. The following five elliptic distributed control examples were considered in [17,19].
(ED1)
(ED2)
(ED3)
(ED4)
(ED5) as (ED1) except