Adaptive Finite Element Method for fractional differential equations using Hierarchical Matrices

Abstract

A robust and fast solver for the fractional differential equation (FDEs) involving the Riesz fractional derivative is developed using an adaptive finite element method on non-uniform meshes. It is based on the utilization of hierarchical matrices ($\mathcal{H}$-Matrices) for the representation of the stiffness matrix resulting from the finite element discretization of the FDEs. We employ a geometric multigrid method for the solution of the algebraic system of equations. We combine it with an adaptive algorithm based on a posteriori error estimation to deal with general-type singularities arising in the solution of the FDEs. Through various test examples we demonstrate the efficiency of the method and the high-accuracy of the numerical solution even in the presence of singularities. The proposed technique has been verified effectively through fundamental examples including Riesz, Left/Right Riemann-Liouville fractional derivative and, furthermore, it can be readily extended to more general fractional differential equations with different boundary conditions and low-order terms. To the best of our knowledge, there are currently no other methods for FDEs that resolve singularities accurately at linear complexity as the one we propose here.

Cited by in corpus (8)

• A novel Hermite RBF-based differential quadrature method for solving two-dimensional variable-order time fractional advection-diffusion equation with Neumann boundary condition
• A fast semi-discrete Kansa method to solve the two-dimensional spatiotemporal fractional diffusion
• $\mathcal{H}$-matrix approximability of inverses of discretizations of the fractional Laplacian
• An a posteriori error estimator for the spectral fractional power of the Laplacian
• Novel numerical analysis for simulating the generalized 2D multi-term time fractional Oldroyd-B fluid model
• A fast solver for spectral element approximation applied to fractional differential equations using hierarchical matrix approximation
• Accurate analytic approximation for a fractional differential equation with a modified Bessel function term
• A scale-dependent finite difference method for time fractional derivative relaxation type equations