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We develop a spectral method to solve the heat equation in a closed cylinder,achieving a quasi-optimal $\mathcal{O}(N\log N)$ complexity and high-order,spectral accuracy. The algorithm relies on a Chebyshev--Chebyshev--Fourier(CCF) discretization of the cylinder, which is easily implemented and decouplesthe heat equation into a collection of smaller, sparse Sylvester equations. Inturn, each of these equations is solved using the alternating directionimplicit (ADI) method in quasi-optimal time; overall, this represents animprovement in the heat equation solver from $\mathcal{O}(N^{4/3})$ (inprevious Chebyshev-based methods) to $\mathcal{O}(N\log N)$. WhileLegendre-based methods have recently been developed to achieve similarcomputation times, our Chebyshev discretization allows for far fastercoefficient transforms; we demonstrate the application of this by outlining aspectral method to solve the incompressible Navier--Stokes equations in thecylinder in quasi-optimal time. Lastly, we provide numerical simulations of theheat equation, demonstrating significant speed-ups over traditional spectralcollocation methods and finite difference methods.

A Quasi-Optimal Spectral Solver for the Heat and Poisson Equations in a Closed Cylinder