Spectral Elliptic Solvers in a Finite Cylinder (Communications in Computational Physics, 2018)
New direct spectral solvers for the 3D Helmholtz equation in a finite cylindrical region are presented. A purely variational (no collocation) formulation of theproblem is adopted, based on Fourier series expansion of the angular dependence andLegendre polynomials for the axial dependence. A new Jacobi basis is proposed for theradial direction overcoming the main disadvantages of previously developed bases forthe Dirichlet problem. Nonhomogeneous Dirichlet boundary conditions are enforcedby a discrete lifting and the vector problem is solved by means of a classical uncoupling technique. In the considered formulation, boundary conditions on the axis ofthe cylindrical domain are never mentioned, by construction. The solution algorithmsfor the scalar equations are based on double diagonalization along the radial and axialdirections. The spectral accuracy of the proposed algorithms is verified by numericaltests.