# Fully Finite Element Adaptive Algebraic Multigrid Method for Time-Space Caputo-Riesz Fractional Diffusion Equations

## Abstract

The paper aims to establish a fully discrete finite element (FE) scheme and provide cost-effective solutions for one-dimensional time-space Caputo-Riesz fractional diffusion equations on a bounded domain . Firstly, we construct a fully discrete scheme of the linear FE method in both temporal and spatial directions, derive many characterizations on the coefficient matrix and numerically verify that the fully FE approximation possesses the saturation error order under norm. Secondly, we theoretically prove the estimation on the condition number of the coefficient matrix, in which and respectively denote time and space step sizes. Finally, on the grounds of the estimation and fast Fourier transform, we develop and analyze an adaptive algebraic multigrid (AMG) method with low algorithmic complexity, reveal a reference formula to measure the strength-of-connection tolerance which severely affect the robustness of AMG methods in handling fractional diffusion equations, and illustrate the well robustness and high efficiency of the proposed algorithm compared with the classical AMG, conjugate gradient and Jacobi iterative methods.

## 1Introduction

In recent years, there has been an explosion of research interest in numerical solutions for fractional differential equations, mainly due to the following two aspects: (i) the huge majority can’t be solved analytically, (ii) the analytical solution (if luckily derived) always involve certain infinite series which sharply drives up the costs of its evaluation. Various numerical methods have been proposed to approximate more accurately and faster, such as finite difference (FD) method [?], finite element (FE) method [?], finite volume [?] method and spectral (element) method [?]. An essential challenge against standard differential equations lies in the presence of the fractional differential operator, which gives rise to nonlocality (space fractional, nearly dense or full coefficient matrix) or memory-requirement (time fractional, the entire time history of evaluations) issue, resulting in a vast computational cost.

Preconditioned Krylov subspace methods are regarded as one of the potential solutions to the aforementioned challenge. Numerous preconditioners with various Krylov-subspace methods have been constructed respectively for one- and two-dimensional, linear and nonlinear space-fractional diffusion equations (SFDE) [?]. Multigrid method has been proven to be a superior solver and preconditioner for ill-conditioned Toeplitz systems as well as SFDE. Pang and Sun propose an efficient and robust geometric multigrid (GMG) with fast Fourier transform (FFT) for one-dimensional SFDE by an implicit FD scheme [?]. Bu et al. employ the GMG to one-dimensional multi-term time-fractional advection-diffusion equations via a fully discrete scheme by FD method in temporal and FE method in spatial directions [?]. Jiang and Xu construct optimal GMG for two-dimensional SFDE to get FE approximations [?]. Chen et al. make the first attempt to present an algebraic multigrid (AMG) method with line smoothers to the fractional Laplacian through localizing it into a nonuniform elliptic equation [?]. Zhao et al. invoke GMG for one-dimensional Riesz SFDE by an adaptive FE scheme using hierarchical matrices [?]. From the survey of references, in spite of quite a number of contributions to numerical methods and preconditioners, there are no calculations taking into account of fully discrete FE schemes and AMG methods for time-space Caputo-Riesz fractional diffusion equations.

In this paper, we are concerned with the following time-space Caputo-Riesz fractional diffusion equation (CR-FDE)

with orders and , the Caputo and Riesz fractional derivatives are respectively defined by

where

The remainder of this paper proceeds as follows. A fully discrete FE method of - is developed in Section 2. Section 3 comes up with the theoretical estimation and verification experiments on the condition number of the coefficient matrix. The classical AMG method is introduced in Section 4 followed by its uniform convergence analysis and the construction of an adaptive AMG method. Section 5 reports and analyzes numerical results to show the benefits. We close in Section 6 with some concluding remarks.

## 2Fully discrete finite element scheme for the CR-FDE

For simplicity, following [?], we will use the symbols , and throughout the paper. means , means while means , where , , and are generic positive constants independent of variables, time and space step sizes.

### 2.1Reminder about fractional calculus

In this subsection, we briefly introduce some fractional derivative spaces and several auxiliary results. Here the inner product and norm are denoted by

### 2.2Derivation of the fully discrete scheme

By Lemma ?, we get the variational (weak) formulation of -: given , and , to find subject to and

where , , and

In order to acquire numerical solutions of , we firstly make a (possibly nonuniform) temporal discretization by points , and a uniform spatial discretization by points (), where represents the space step size. Let

We observe that it is convenient to form the FE spaces in tensor products

where

and denotes the set of all polynomials of degree .

We obtain a fully discrete FE scheme in temporal and spatial directions of problem : given , to find such that and

where satisfying (), and

Let

and

Note that

where is the shape function at . Using

we have

Substituting - into , yields

where the coefficient matrix

the right-hand side vector

the mass matrix

the stiffness matrix with its entries

the vector

and the fully FE approximations

Next, a number of characterizations are established regarding just defined by .