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## A Poisson Equation.

In this example, we solve the Poisson equation

in , where is given in Figure . The boundary conditions are u=0 on the inner circle and on the outer circle. Since this is a linear problem with no parameter dependence, it does not make use of the continuation options in PLTMG, and it does not require the solution of nonlinear algebraic equations.

The description of was provided as a skeleton consisting of 24 subregions. Thus we must begin by calling TRIGEN to generate a triangulation. The use of so many subregions in the skeleton was unnecessary to define such a simple domain, but we created this skeleton to illustrate some features of TRIGEN. In particular, the 24 subregions can be grouped into three sets of eight similar subregions. In defining the skeleton input, we specified that similar regions should be triangulated in a similar fashion. The resulting triangulation produced by TRIGEN is shown in Figure . The triangulation has 224 vertices and 408 triangles. Options in INPLT were set such that in the skeleton, each subregion is given its own color, while in the triangulation, regions that share the same (user specified) label are given the same color.

Figure: GPHPLT output.

Using this triangulation, we next call PLTMG to solve the problem on the initial mesh. We then adaptively refine the mesh through a call to TRIGEN with IADAPT=1 and NVTRGT=540. Next we solve the problem on the refined mesh using PLTMG. We continue the adaptive refinement process with a call to TRIGEN with IADAPT=1, NVTRGT=1100. We then solve this problem using PLTMG, and then call TRIGEN once more, this time only to compute an a posteriori error estimate for the current solution. In Figure , we illustrate the error and the solution on the mesh with NVF=1100 using two calls to TRIPLT.

Finally, we call GPHPLT to provide some data about the state of the calculation. The result is shown in Figure . On the left are log-log plots of the relative error in and norms, as a function of the number of vertices. In the top middle is a plot of the logarithms of the Newton residual and search direction as a function of iteration number. The hierarchical basis multigrid method was used to solve the large sparse linear systems of equations arising in the calculation. Its convergence history for the last Newton step is shown in the lower middle of Figure . Finally, a bar graph showing the execution time of PLTMG as a function of the major subroutines it calls is shown in the lower right corner. A complete description of GPHPLT and its options is given in Section .

Next: A Nonlinear Eigenvalue Problem. Up: Examples. Previous: Examples.

Randolph E. Bank
Fri Apr 4 12:02:05 PST 1997