Spectral Approximation of Partial Differential Equations in Highly Distorted Domains

In this paper we discuss spectral approximations of the Poisson equation in deformed quadrilateral domains. High order polynomial approximations are used for both the solution and the representation of the geometry. Following an isoparametric approach, the four edges of the computational domain are first parametrized using high order polynomial interpolation. Transfinite interpolation is then used to construct the mapping from the square reference domain to the physical domain. Through a series of numerical examples we show the importance of representing the boundary of the domain in a careful way; the choice of interpolation points along the edges of the physical domain may significantly effect the overall discretization error. One way to ensure good interpolation points along an edge is based on the following criteria: (i) the points should be on the exact curve; (ii) the derivative of the exact curve and the interpolant should coincide at the internal points along the edge. Following this approach, we demonstrate that the discretization error for the Poisson problem may decay exponentially fast even when the boundary has low regularity.