Least-squares spectral collocation for discontinuous and singular perturbation problems

A least-squares spectral collocation scheme for discontinuous problems is proposed. For the first derivative operator the domain is decomposed in subintervals where the jumps are imposed at the discontinuities. Equal order polynomials are used on all subdomains. For the discretization spectral collocation with Chebyshev polynomials is employed. Fast Fourier transforms are now available. The collocation conditions and the interface conditions lead to an overdetermined system which can be efficiently solved by least-squares. The solution technique will only involve symmetric positive definite linear systems. This approach is further extended to singular perturbation problems where least-squares are used for stabilization. By a suitable decomposition of the domain the boundary layer is well resolved. Numerical simulations confirm the high accuracy of our spectral least-squares scheme.