Convergence for the Galerkin Helmholtz Operator

  

Figure 4.21: Convergence test for the Helmholtz equation using quadrilaterals and triangles, with Dirichlet boundary conditions. The exact solution is $u=sin(\pi x)cos(\pi y)$ and forcing function $f=-(\lambda + 2 \pi^2)sin(\pi x)cos(\pi y)$.

In figure 4.21 we demonstrate convergence to the exact solution with p-refinement and h-refinement for the Helmholtz equation with $\lambda=1$; the exact solution is:

$$u=sin(\pi cos(\pi r^2))$$

and forcing function $f=-(\lambda + 4\pi^4 r^2 sin(\pi r^2)^2)sin(\pi( cos(\pi r^2))) -4\pi^2(\pi r^2 cos(\pi r^2) + sin(\pi r^2))cos(\pi cos(\pi r^2))$, where r² = x²+y² for Dirichlet boundary conditions.

  

Figure 4.22: Convergence test for the Helmholtz, using a triangle and a quadrilateral, with Dirichlet boundary conditions: u=sin(pi cos(pi r**2)).

In figure 4.22 we show p-type convergence for a more complicated exact solution. This example demonstrates that the method is stable to at least N=64.