Global Convergence in Skew Elements

  

Figure 4.3: Meshes (A-H) are consisted of three quadrilaterals and two triangles which are progressively skewed by shifting the interior vertex.

  

Figure 4.4: Convergence in the L₂ norm for modal projection of the function $u=sin(\pi x)sin(\pi y)$ on meshes A-H.

  

Figure 4.5: Convergence in the L₂ norm for mixed projection of the function $u=sin(\pi x)sin(\pi y)$ on meshes A-H.

We now consider the effect of the skewness of the physical elements on the accuracy of the projection operator. In figure 4.3 we examine 8 different meshes consisted of triangles and quadrilaterals. We start by projecting $sin(\pi x) sin(\pi y)$ onto a square domain covered with standard elements. Figures 4.4 and 4.5 show results for the modal and mixed bases that demonstrate exponential convergence is achieved. Subsequently, we make the elements covering the domain progressively more skew in the meshes B-H. In each case we see that exponential convergence is achieved, even when one of the triangular elements has a minimum angle of about 10^-3 degrees. Hence, the accuracy of the method is extremely robust to badly shaped elements. Also, we note that the similarity of the convergence curves demonstrates that the rate of exponential convergence is unaffected by the skewing.

The independence of the rate of skewing might be expected based on our previous observations about the accuracy of projection using the orthogonal basis. We saw that the coefficients in the expansion of an infinitely smooth function are bounded by exponentially decaying functions of the polynomial order which are scaled by the Jacobian of the geometrical mapping. So we infer that as an element becomes more skew the energy in each mode should reduce in proportion to the area of the element.