Space-time spectral methods for partial differential equations

Abstract

Spectral methods for solving partial differential equations (PDEs) depict a high order of convergence, which is exponential when the solution is analytic. However, their applications to time-dependent PDEs typically enforce a finite difference scheme in time. The slower decay of error in time overwhelms the super-algebraic convergence of error in space. A relatively new class of techniques is space-time spectral methods converging spectrally in both space and time. We devise and analyze a space-time spectral method for the Stokes problem. The main objectives of the research are estimating the condition number of the global spectral operators and proving the spectral convergence of this scheme in space and time. Numerical experiments of this scheme verify the theoretical results. Furthermore, we discuss two space-time spectral methods for the Navier-Stokes problem.

The discrete systems resulting from classical space-time spectral methods are dense, ill-conditioned, and coupled in all time steps. A new class of spectral methods, called the ultraspherical spectral (US) methods, are applied to time-dependent PDEs, which along with spectral convergence, lead to the resultant discrete systems constituting sparse and well-conditioned matrices. %We present spectral condition number estimates for the heat, Schr\"{o}dinger, and wave equations.

Additionally, we join the long tradition of estimating the eigenvalues of a sum of two symmetric matrices, say $P+Q$, in terms of the eigenvalues of $P$ and $Q$. We derive two new lower bounds on $\lambda_{\min}(P +Q)$ in terms of the minimum positive eigenvalues of $P$ and $Q$. The bounds incorporate geometric information by utilizing the Friedrichs angles between certain subspaces. Such estimates lead to new lower bounds on the minimum singular value of some full-rank block matrices in terms of the minimum positive singular value of their subblocks.