A posteriori error estimates based on superconvergence of FEM for fractional evolution equations

Abstract: In this paper, we consider an approximation scheme for fractional evolution equation with variable coefficient. The space derivative is approximated by triangular finite element and the time fractional derivative is evaluated by the L1 approximation. The main aim of this work is to provide convergence and superconvergence analysis and derive a posteriori error estimates. Some numerical examples are presented to demonstrate our theoretical results.

It is well known that there has been extensive research on the superconvergence of FEMs for partial differential equations (PDEs). A systematic introduction can be found in [18][19][20][21][22][23]. Generally speaking, there are three types of superconvergence. The first is pointwise superconvergence, namely in certain sampling points the values in derivatives of error have a higher convergent order than elsewhere [24]. The second is global superconvergence, namely the gradient in L 2 -norm of error between the numerical solution and the projection of exact solution have a greater accuracy than the optimal order of convergence. The third is superconvergence based on post-processing technique, namely reconstruct an improved accuracy gradient. Representative post-processing techniques are developed by interpolation [25], extrapolation [26] and gradient recovery, which include superconvergent patch recovery [27], polynomial preserving recovery [28,29] and supercovergent cluster recovery [30]. In recent years, there has been some research on superconvergence analysis of FEMs for FPDEs. Superconvergence of FEMs and mixed FEMs for FPDEs is inves-tigated in [31][32][33] and [34], respectively. In [35,36], superconvergence of nonconforming FEMs for fractional PDEs is established.
Adaptive FEMs are among the most important means to improve the accuracy and efficiency of finite element discretization. The pioneering work was carried out by Babuška and Rheinboldt in [37]. One of the key concepts in adaptive FEMs is a posteriori error estimates, which are computable quantities in terms of the discrete solution and can measure the actual discrete errors without the knowledge of exact solution. A posteriori error estimates of FEMs for elliptic problems are well-developed [38][39][40]. There are substantial research on a posteriori error estimates of FEMs for integer-order PDEs based on explicit residual [41], local problems [42,43], recovery [27,29,44,45], hierarchical basis [46][47][48][49] and equilibrated error [50,51]. However, to the best of our knowledge, a posteriori error estimates of FEMs for evolution equations are less developed, and only a few results can be found in [52][53][54].
The purpose of this work is to provide a fully discrete finite element approximation for fractional evolution equations and analyze its convergence and superconvergence. Then, we derive a posteriori error estimates based on the superconvergence results and construct an adaptive FEM algorithm for fractional evolution equations with variable coefficients.
We are interested in the following fractional evolution equation: Throughout the paper, ( (see e.g. [55]). In addition, c or C is a generic positive constant.
The rest of this paper is organized as follows. In Section 2, we give a fully discrete approximation scheme of (1). Convergence analysis results are presented in Section 3. In Section 4, we derive the superconvergence between the numerical solutions and the elliptic projection of exact solutions. In Section 5, we construct a posteriori error estimates based on the superconvergence results. Some numerical experiments are presented to support our theoretical results in Section 6.

Fully discrete finite element approximation
In this section, we present a fully discrete approximation scheme of (1). To begin with, we introduce triangular FEMs for the spatial discretization. For brevity, we denote In addition, From the assumption of coefficient matrix ( ) A x , we have We recast (1) as the following weak formulation: , Let h be a family of quasi-uniform triangulations of Ω, such that = ⋃ ∈ τ Ωτ h and where h τ is the diameter of the element τ. Furthermore, we set where 1 represents the space of all polynomials whose degree at most 1, and . Then a semi-discrete approximation scheme of (2) reads as In the second, we will consider the L1 scheme for the time discretization. . The time fractional derivative can be approximated as follows: where n k 1 and r τ n is the truncation error.
It follows from [25] for Then a fully discrete approximation scheme of (2) is as follows: , where P h will be specified later.

Convergence analysis
We will derive the convergence of the numerical scheme (6). Let → P W W : h h be the elliptic projection operator, for any ∈ v W defined by It has the following approximation properties (see [8]): The following conclusions will be used in the following error analysis.

Superconvergence analysis
In this section, we will derive the global superconvergence results between the finite element solution and the elliptic projection of exact solution.
Theorem 4.1. Let y and y h be the solutions of (2) and (6) (2) and subtracting (2) from (6), we obtain the following error equation: By using the definition of P h and (17), we have According to ( ) ∈ ∞ y W H Then, by applying Hölder's inequality, Young's inequality and (7), we arrive at Combining (18)- (22) and noting that It follows from (23), Lemma 3.2 and Poincaré's inequality that Hence, we complete the proof of Theorem 4.1. □

A posteriori error estimates
We introduce recovery operators R h and G h . Similar to the Z-Z patch recovery in [27], R v h be a continuous piecewise linear function (without zero boundary constraint), the value of R v h on the nodes is defined by least-squares argument on element patches surrounding the nodes. Then the gradient recovery operator The details can be found in [56].
Theorem 5.1. Let y and y h be the solutions of (2) and (6), respectively. Suppose that

.
Then for any integer ≤ ≤ n N 1 , we have Proof. Let y I n be the piecewise linear Lagrange interpolation of y n . According to Theorem 2.

(28)
Hence, we complete the proof of (25). □ Combining the previous results, we obtain the following a posteriori error estimates of fully discrete finite element approximation for fractional evolution equations.
Proof. According to (25) and triangle inequality, it is easy to get (29). □

Numerical experiments
In this section, we present some different numerical examples to illustrate the correctness of the convergence and superconvergence results and the reliable and efficient a posteriori error estimates.
For an acceptable iteration error Tol, we present an uniformly refined FEM algorithm for the discrete problem (6) of fractional evolution equations. Step 2. Solve the following discrete equations: Step 3. Uniformly refine the meshes obtain new meshes + For an acceptable iteration error Tol, by selecting η n in (29) as mesh refinement indicators, we construct the following adaptive FEM algorithm for the discrete problem (6): Step 2. Solve the following discretized problems: Obtain numerical solution on the current meshes i h and calculate the error estimators η i n ; Step 3. Adjust the meshes by using the estimators η i n obtain new meshes + The following examples were dealt numerically with codes developed based on AFEPack, which is freely available and the details can be found in [56]. The discretization was described in Section 2. We denote || || ( ) ⋅ ∞ L H 1 and || || ( ) ⋅ ∞ L L 2 by || || ⋅ ∞

1,
and || || ⋅ ∞ 0, , respectively. The convergence order rate is computed by the following formula:    A posteriori error estimates of FEM for TFPDE  1217 Example 6.2. This is a 2D example. The data are as follows:     We take = − τ 10 2 and solve this example by using Algorithms 6.1 and 6.2. Numerical results based on a sequence of uniformly refined meshes and adaptive meshes are listed in Tables 7 and 8, respectively. It is clear that the adaptive meshes generated via the error estimators η n are able to save substantial computational work. We plot the profile of the numerical solution y h at = t 0.5, where adaptive mesh = nodes 1,241 in Figure 3.

Conclusion
Although there has been extensive research on FEMs for FPDEs, mostly focused on convergence analysis [7][8][9][10][11]. While there is little work on a posteriori error estimates of FEM for FPDEs. Hence, our results on a posteriori error estimates and adaptive FEM for fractional evolution equations are new.