A Petrov–Galerkin spectral element method for fractional elliptic problems

Abstract

We develop a new C0-continuous Petrov–Galerkin spectral element method for one-dimensional fractional elliptic problems of the form 0Dxαu(x)−λu(x)=f(x), α∈(1,2], subject to homogeneous boundary conditions. We employ the standard (modal) spectral element basis and the Jacobi poly-fractonomials as the test functions (Zayernouri and Karniadakis (2013)). We formulate a new procedure for assembling the global linear system from elemental (local) mass and stiffness matrices. The Petrov–Galerkin formulation requires performing elemental (local) construction of mass and stiffness matrices in the standard domain only once. Moreover, we efficiently obtain the non-local (history) stiffness matrices, in which the non-locality is presented analytically for uniform grids. We also investigate two distinct choices of basis/test functions: (i) local basis/test functions, and (ii) local basis with global test functions. We show that the former choice leads to a better-conditioned system and accuracy, while the latter results in ill-conditioned linear systems, and therefore, that is not an efficient and a proper choice of test function. We consider smooth and singular solutions, where the singularity can occur at boundary points as well as in the interior domain. We also construct two non-uniform grids over the whole computational domain in order to capture singular solutions. Finally, we perform a systematic numerical study of non-local effects via full and partial history fading in order to further enhance the efficiency of the scheme.