A Petrov–Galerkin spectral element method for fractional elliptic problems

• Published in: Computer methods in applied mechanics and engineering Vol. 324; pp. 512 - 536
• Main Authors: Kharazmi, Ehsan, Zayernouri, Mohsen, Karniadakis, George Em
• Format: Journal Article
• Language: English
• Published: Amsterdam Elsevier B.V 01.09.2017; Elsevier BV
• Subjects: [formula omitted]-continuous element; Accuracy; Boundary conditions; Boundary/interior singularities; Conditioning; Construction standards; Efficiency; Galerkin method; History fading analysis; Linear systems; Matrix methods; Modal basis/test functions; Non-local assembling/scattering; Spectra; Spectral convergence; Spectral element method; Stiffness matrix; Studies; History fading analysis; Modal basis/test functions; Non-local assembling/scattering; Spectral convergence; Boundary/interior singularities; C0-continuous element
• ISSN: 0045-7825; 1879-2138
• DOI: 10.1016/j.cma.2017.06.006

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. •Development of a new fast and accurate C0-continuous Petrov–Galerkin spectral element method, employing local basis/test functions, where the test functions are Jacobi poly-fractonomials.•Reducing the number of history matrix calculation from Nel(Nel−1)2 to (Nel−1) for a uniformly partitioned domain.•Analytical expressions of non-local effects in uniform grids leading to fast computation of the history matrices.•A new procedure for the assembly of the global linear system.•Performing off-line computation of history matrices and on-line retrieval of the stored matrices.•Boundary and interior singularity capturing using adaptive hp-refinement.•Non-uniform “kernel-based” grid generation for resolving steep gradients and singularities.