Spectral schemes on triangular elements

The Poisson problem with homogeneous Dirichlet boundary conditions is considered on a triangle. The mapping between square and triangle is realized by mapping an edge of the square onto a corner of the triangle. Then standard Chebyshev collocation techniques can be applied. Numerical experiments demonstrate the expected high spectral accuracy. Further it is shown that finite difference preconditioning can be successfully applied in order to construct an efficient iterative solver. Then the convection-diffusion equation is considered. Here finite difference preconditioning with central differences does not overcome instability. However, applying the first order upstream scheme, we obtain a stable system. Finally a domain decomposition technique is applied to the patching of rectangular and triangular elements.