Convergence in 3D Skew Elements

We have shown in chapter 3 that the approximation properties of the orthogonal bases should not be adversely affected by the shape of triangles or tetrahedra. We constructed a twelve element tetrahedral mesh, shown in figure 4.24, for a box. All the tetrahedra share a vertex at the center or the box. For the first test of the Helmholtz solver this vertex was at the center of the box. We then moved the vertex progressively closer to one of the corners of the box and ran the solver for each position. In figure 4.24 we show that the convergence of the $L_\infty$ error norm was only minimally effected by the element skewing. This is very good news because we can use unstructured meshes without worrying too much about mesh quality for accuracy. We should however still be careful about using elements with small angles because we have already shown that these elements have over-resolution in their shortest vertex to edge direction. This can lead to a very restrictive Courant-Friedrichs-Lax (CFL) condition in wave propogation problems.

  

Figure 4.24: Convergence is guaranteed for the Helmholtz problem even on very skewed tetrahedra. Dirichlet boundary conditions $u=sin(\frac{\pi x}{6})sin(\frac{\pi y}{6})sin(\frac{\pi z}{6})$ and $\lambda=1$.