Spectral Approximation of Partial Differential Equations in Highly Distorted Domains

Abstract

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.

Spectral Element Methods on Simplicial Meshes

“…This test case allows to point out the requirement of using isoparametric elements when a curved boundary is involved, that means rely on a polynomial parametrization of the sides of triangles touching the curved boundary of the same order as the TSEM basis functions. For some interesting remarks on standard mapping techniques based on conformal transformations to go from OE 1; 1 d into a domain with curved boundaries, see [2]. In Fig.…”

Free-Stream Preservation for Curved Geometrically Non-conforming Discontinuous Galerkin Spectral Elements

“…Typically, the boundaries are approximated by high order polynomial interpolants [34]. From those boundary interpolants, the mapping X is often a linear transfinite map [8] that matches the element faces, although other types of maps might be useful [2], [27].…”

“…The physical domain is subdivided into elements and on those elements the solution and fluxes are approximated by high order polynomials. The high order of the methods enables them to use curved elements to accurately approximate curved physical boundaries [23], [27], [2]. If curved elements are used not just at the physical boundaries but also in the neighboring volume, then thinner and longer (anisotropic) elements can be used without physical boundaries crossing interior element boundaries [11].…”