Error Analysis for Mapped Jacobi Spectral Methods

Standard spectral methods are capable of providing very accurate approximations to well-behaved smooth functions with significantly less degrees of freedom when compared with finite difference or finite element methods (cf. [6,7,11]). However, if a function exhibits localized behaviors such as sharp interfaces, very thin internal or boundary layers, using a standard Gauss-type grid usually fails to produce an accurate approximation with a reasonable number of degrees of freedom. Thus, it is advisable to use a grid which is adapted to the local behaviors of the underlying function. Since spectral methods can not gracefully handle an arbitrarily locally refined grid, a popular strategy is to use a suitable mapping which transforms a function having sharp interfaces in the physical domain to a well behaved function on the computational domain. Thus, to better understand what are the impacts of the mapping on the approximation, it is necessary to study the properties of the mapped polynomials.