Spectral element methods on triangles and quadrilaterals: comparisons and applications

In this paper, we compare a triangle based spectral element method (SEM) with the classical quadrangle based SEM and with a standard spectral method. For the sake of completeness, the triangle-SEM, making use of the Fekete points of the triangle, is first revisited. The requirement of a highly accurate quadrature rule is particularly emphasized. Then it is shown that the convergence properties of the triangle-SEM compare well with those of the classical SEM, by solving an elliptic equation with smooth (but steep) analytical solution. It is also proved numerically that the condition number grows significantly faster for triangles than for quadrilaterals. Finally, the attention is focused on a diffraction problem to show the high flexibility of the triangle-SEM.