Adaptive Finite Element Method for fractional differential equations using Hierarchical Matrices

TL;DR

This paper introduces a robust adaptive finite element method utilizing hierarchical matrices and multigrid techniques to efficiently solve fractional differential equations with singularities, achieving high accuracy and linear complexity.

Contribution

The paper presents a novel adaptive finite element approach with hierarchical matrices and multigrid methods for solving FDEs, capable of accurately handling singularities at linear computational complexity.

Findings

High-accuracy solutions even with singularities

Efficient linear complexity computational method

Versatile extension to various fractional derivatives

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.