Spectral element method with geometric mesh for two-sided fractional differential equations

Solutions of two-sided fractional differential equations (FDEs) usually exhibit singularities at the both endpoints, so it can not be well approximated by a usual polynomial based method. Furthermore, the singular behaviors are usually not known a priori, making it difficult to construct special spectral methods tailored for given singularities. We construct a spectral element approximation with geometric mesh, describe its efficient implementation, and derive corresponding error estimates. We also present ample numerical examples to validate our error analysis.