7. Test Problems
Society for Industrial and Applied Mathematics eBooks · 1998
7.1. Overview. In this chapter, we briefly document the test problem data sets included with the PLTMG source code. These problems encompass a variety of applications and exercise most features of the package. Each data set minimally consists of functions A1XY, A2XY, FXY, GXY, P1XY, P2XY, and QXY and subroutines USRCMD and GDATA. Problem specific routines are also included. 7.2. Test Problem CIRCLE. In this problem, we solve the Laplace equation −Δu=0 in the unit circle, with a crack along the positive x axis. Homogeneous Dirichlet boundary conditions are imposed on the top of the crack, and homogeneous Neumann boundary conditions are imposed below the crack. On the boundary of the circle, nonhomogeneous boundary conditions are imposed such that the true solution is given by u= rα sin αθ, 7.1 where . This is the leading term of the singularity arising from the interior angle of 2π coupled with the change of boundary conditions at the origin. The domain is defined by a triangulation consisting of eight similar triangles, shown in Figure 2.1. The USRCMD for this test problem has three parameters that can be set. IBC determines the boundary conditions. if , the boundary conditions on the outer boundary of the circle are nonhomogeneous Dirichlet as described above; if , nonhomogeneous Neumann boundary conditions are imposed on the circular part of the boundary such that (7.1) remains the exact solution. One can also alter the domain, and the severity of the singularity, using the parameter NTRI, where . if the entire circle is used as the domain; if , only the first NTRI triangles are used. Some examples are shown in Figure 7.1.