Spectral Methods in Non-Square Geometries

Abstract

With the ability to map the reference square to non-square domains and to modify the equations to reflect those mappings, we can use spectral methods to compute solutions to PDEs in geometries more complex than the square. In this chapter, we retrace our steps in Chap. 5 to develop spectral methods for non-square geometries. 7.1 Steady Potentials in a Quadrilateral Domain The most generally applicable approximations for the solution of potential problems on a quadrilateral domain are the collocation or nodal Galerkin methods. The potential equation for a quadrilateral domain will generally have non-constant coefficients. The variable coefficients will limit the exact Galerkin approximation to special cases, such as cylindrical coordinates, where the integrals can be evaluated analytically. The two nodal approximations have the advantage of being generally applicable, at the expense of some spectrally small aliasing and quadrature errors. 7.1.1 The Collocation Approximation To derive a collocation approximation of the potential equation ∇ 2 ϕ = s (7.1) we follow the derivation of the approximation of the variable coefficient equation (5.25). This is necessary because we showed in Sect. 6.2.1 that under the mapping from the reference square to the physical, quadrilateral domain, the potential equation is transformed to the variable coefficient problem ∇ 2 ϕ = ∇ · F = 1 J ∂ ∂ξ F 1 + ∂ ∂η F 2 (7.2) where the contravariant fluxes are F 1 = Y η J Y η ∂ϕ ∂ξ − Y ξ ∂ϕ ∂η − X η J −X η ∂ϕ ∂ξ + X ξ ∂ϕ ∂η , F 2 = −Y ξ J Y η ∂ϕ ∂ξ − Y ξ ∂ϕ ∂η + X ξ J −X η ∂ϕ ∂ξ + X ξ ∂ϕ ∂η .