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Efficient spectral-Galerkin algorithms for direct solution of second-order equations using ultraspherical polynomials. (English) Zbl 1020.65088
A ultra-spherical Galerkin method (UGM) for elliptic second-order differential equations in one and two space dimensions is presented. Starting point is the construction of appropriate bases for UGM. The properties of ultraspherical polynomial are reviewed and the UGM is applied to the Helmholtz equation in one and two space dimensions in Cartesian geometry with homogeneous Dirichlet boundary conditions.
Extensions to more general problems for mixed type boundary conditions and nonhomogeneous cases are presented for one-dimensional problems. Finally, numerical results obtained with the UGM scheme are presented for a one-dimensional nonhomogeneous Helmholtz equation and a two-dimensional Poisson equation.

65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
65N22 Numerical solution of discretized equations for boundary value problems involving PDEs
35J25 Boundary value problems for second-order elliptic equations
35J05 Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation
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