A Petrov-Galerkin Spectral Element Method for Fractional Elliptic Problems

Abstract

We develop a new $C^0$-continuous Petrov-Galerkin spectral element method for one-dimensional fractional elliptic problems of the form ${}_0{\mathcal{D}}_x^{\alpha }u(x)-\lambda u(x)=f(x)$, $\alpha \in (1,2]$, subject to homogeneous boundary conditions. We employ the standard (modal) spectral element bases and the Jacobi poly-fractonomials as the test functions [1]. 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. 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.

40 pages, 15 figures, 7 Tables

Cited by in corpus (9)

• Multi-domain Spectral Collocation Method for Variable-Order Nonlinear Fractional Differential Equations
• A Unified Spectral Method for FPDEs with Two-sided Derivatives; A Fast Solver
• A Unified Spectral Method for FPDEs with Two-sided Derivatives; Stability, and Error Analysis
• Petrov-Galerkin Method for Fully Distributed-Order Fractional Partial Differential Equations
• An Automated Singularity-Capturing Scheme for Fractional Differential Equations
• Fast IMEX Time Integration of Nonlinear Stiff Fractional Differential Equations
• Operator-Based Uncertainty Quantification of Stochastic Fractional PDEs
• Fractional Sensitivity Equation Method: Applications to Fractional Model Construction
• Fractional Buffer Layers: Absorbing Boundary Conditions for Wave Propagation