Spectral Elliptic Solvers in a Finite Cylinder

New direct spectral solvers for the 3D Helmholtz equation in a finite cylindrical region are presented. A purely variational (no collocation) formulation of the problem is adopted, based on Fourier series expansion of the angular dependence and Legendre polynomials for the axial dependence. A new Jacobi basis is proposed for the radial direction overcoming the main disadvantages of previously developed bases for the Dirichlet problem. Nonhomogeneous Dirichlet boundary conditions are enforced by 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 of the cylindrical domain are never mentioned, by construction. The solution algorithms for the scalar equations are based on double diagonalization along the radial and axial directions. The spectral accuracy of the proposed algorithms is verified by numerical tests. AMS subject classifications: 65N30, 65N35

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