Numerical Analysis of an -Galerkin Mixed Finite Element Method for Time Fractional Telegraph Equation
We discuss and analyze an -Galerkin mixed finite element (-GMFE) method to look for the numerical solution of time fractional telegraph equation. We introduce an auxiliary variable to reduce the original equation into lower-order coupled equations and then formulate an -GMFE scheme with two important variables. We discretize the Caputo time fractional derivatives using the finite difference methods and approximate the spatial direction by applying the -GMFE method. Based on the discussion on the theoretical error analysis in -norm for the scalar unknown and its gradient in one dimensional case, we obtain the optimal order of convergence in space-time direction. Further, we also derive the optimal error results for the scalar unknown in -norm. Moreover, we derive and analyze the stability of -GMFE scheme and give the results of a priori error estimates in two- or three-dimensional cases. In order to verify our theoretical analysis, we give some results of numerical calculation by using the Matlab procedure.
In this paper, our purpose is to present and discuss a mixed finite element method for the time fractional telegraph equation with boundary condition and initial conditions where is a bounded domain with boundary and is the time interval with . The coefficients and are two constants, is a given source function, and are two given initial functions, and the time Caputo fractional-order derivatives and are defined, respectively, by where .
In the current literatures, we can see that some numerical methods for solving fractional partial differential equations (PDEs), which include finite element methods [1–8], mixed finite element methods , finite difference methods [10–24], finite volume methods [25, 26], spectral methods , and discontinuous Galerkin methods [28–31], have been considered and analyzed. In 2014, Liu et al.  gave some theoretical error analysis for a class of fractional PDE based on a nonstandard mixed method in spatial direction and a finite difference scheme in time direction. In , Zhao and Li discussed finite element method for the fractional telegraph equation. In 2014, Wei et al.  studied the numerical solution for time fractional telegraph equation based on the LDG method. But, we have not seen any related studies on mixed finite element methods for solving the fractional telegraph equation.
Recently, some people have made use of the method to obtain the numerical solution for some partial differential equations since Pani (in 1998)  proposed an -GMFE method. This method includes some advantages, such as avoiding the LBB consistency condition, allowing different polynomial degrees of the finite element spaces, and obtaining the optimal a priori estimates in both and -norms. In , Pani and Fairweather discussed some detailed a priori error results of two numerical schemes based on the -GMFE method for linear parabolic integrodifferential equations. In , Pani et al. gave some error analysis based on -GMFE scheme for the partial differential equation of hyperbolic type. At the same time, a modified -GMFE procedure was also proposed and analyzed for the case of two or three dimensions. Guo and Chen [36, 37] obtained some theoretical error analysis and numerical results of -GMFE method for RLW equation and Sobolev equation. In 2013, Liu et al.  introduced another auxiliary variable, which is different from the one in , and then proposed and studied an explicit multistep -GMFE scheme for a RLW equation. Liu and Li  and Zhou  gave some different discussions on -GMFE method for pseudohyperbolic equations (heat transport equation), respectively. In , Che et al. studied the -GMFE method for a nonlinear integrodifferential equation. Recently, Shi et al. [42, 43] proposed some nonconforming mixed scheme based on the -GMFE method.
Based on the above review on -GMFE method, we easily see that the method was studied based on the integer-order partial differential equations. However, the theoretical results of -GMFE method for fractional telegraph equation have not been presented and analyzed.
In this paper, our aim is to give some detailed a priori error analysis and numerical results on the -GMFE method for time fractional telegraph equation. We apply the difference schemes to approximate the time fractional derivatives and use the -GMFE method to discretize spatial direction. We obtain some optimal a priori error results of one dimension for the scalar unknown in and -norms. Moreover, we also get an a priori error result of the optimal -norm for the auxiliary variable. At the same time, we use the -GMFE method to deal with the cases in several dimensions and analyze the stable results for the -GMFE scheme. We calculate some numerical results to verify the theoretical analysis of -GMFE method for fractional telegraph equation.
The layout of the paper is as follows. In Section 2, we formulate an -GMFE scheme for time fractional telegraph equation (5). In Section 3, we introduce some lemmas of two important projections and two difference approximations for time fractional derivatives and then analyze some a priori error results. In Section 4, some theoretical results are given in the cases of two and three dimensions. In Section 5, we choose a numerical example to verify the theoretical analysis of our method. In Section 6, we give some remarks and extensions about the -GMFE method for fractional PDEs.
For the need of study, we denote the natural inner product as in , . Further, we write the classical Sobolev spaces as with norm . When , we simply write the norm as .
2. An -GMFE Scheme in One Space Dimension
In this section, we first consider the -GMFE method for the following time fractional telegraph equation in 1D case: with boundary condition and initial conditions where .
In order to get the -GMFE formulation, we first introduce an auxiliary variable and split (5) into the following first-order system by
Multiplying (8) by , , integrating with respect to space from to , and using an integration by parts with and , we easily get Multiply (9) by , , and integrate with respect to space from to to obtain For formulating finite element scheme, we now choose the finite element spaces and , which satisfy the following approximation properties: for and , positive integers , Based on the chosen finite element spaces, the semidiscrete -GMFE scheme is described by
3. Full Discrete Scheme and A Priori Error Estimates
3.1. Two Projection Lemmas
Lemma 1. One defines a Ritz projection for the variable by Then the following estimates hold, for :
Lemma 2. Further, one also defines an elliptic projection of as the solution of where . Here is chosen to satisfy Then the following estimates are found: for ,
3.2. Approximation of Time-Fractional Derivative
For formulating fully discrete scheme, let be a given partition of the time interval with step length and nodes , for some positive integer . For a smooth function on , define . In the following analysis, for deriving the convenience of theoretical process, we now denote Now, we will introduce two lemmas on the approximations of time fractional derivatives.
Lemma 3 (see ). The time fractional derivative at is approximated by, for , and then holds
Lemma 4 (see ). The time fractional order derivative at is estimated by, for , and then holds
In the next analysis, we will derive and prove some a priori error results for and .
3.3. Error Estimates for Fully Discrete Scheme
Now, we formulate a fully discrete procedure: find such that
In the following discussion, we will derive the proof for the fully discrete a priori error estimates.
Theorem 5. With and , suppose that , , , and , where is the projection defined by , . Then there exists a positive constant free of space-time discrete parameters and such that, for , and for
Proof. For the need of error analysis, we first consider the -norm and the -norm . Taking in (26) and using Poincaré inequality based on the -space and Cauchy-Schwarz inequality, we easily get
In the next analysis, we will give the estimates of and in -norm. Noting that
then (27) may be rewritten as
We take in (35) and multiply by
to arrive at
By the simple calculation, we get the following equality:
By applying the similar process of calculation to (38), we use Cauchy-Schwarz inequality to get
Noting (33), we use Cauchy-Schwarz inequality and Young inequality to have
Substitute (38), (39), and (40) into (37) and use Cauchy-Schwarz inequality to arrive at
By an application of Young inequality, we get
Noting that , we arrive at At the same time, noting that , we use the similar method to have By a combination of (43) and (44) with (18) and noting that , we arrive at Substitute (45) into (42) and note that to get In the following discussion, we apply mathematical induction to obtain the error result Take in (46) and note that ; it is easy to find that the following inequality holds: Assuming that the inequalities hold for , we now prove that the inequality (47) holds. Noting that , , and combining (43) and (44) with (46), we have Based on the process of mathematical induction, we claim that (47) holds.
For the need of the next proof, we have to estimate the term . We now use Taylor formula to arrive at where and . Noting that and (50), we have Substitute (51) into (33) and use (18) to get By combining (51) and (18) with triangle inequality, the -norm estimate is got. Similarly, the estimate in the -norm and the estimate in the -norm also are obtained by a combination of (52) and (15) with triangle inequality.
Now we mainly analyze the case . We can find that the error inequality (29) in this case has no meaning since the coefficient as . So, we have to look for another error estimate’s process. Noting the fact that (), we can obtain the following error inequality:
Now we can use induction to prove the inequality (53). The detailed proof is similar to the above process of analysis, so we do not give the detailed proof again. Making a combination of (53) and (18) with triangle inequality, we can get the estimate (31). A similar discussion for (32) can also be made.
4. An -GMFE Scheme for Several Spaces Variables
In this section, we consider (1) with two and three spaces variables.
Let , ( or 3) with inner product and norm Further, let with norm Taking , we use a similar process to the system (10) and (11) to get the -Galerkin mixed weak formulation by The corresponding time semidiscrete system is defined by where , which can be estimated by a similar process to .
In order to get fully discrete mixed finite element scheme, we now choose the finite element spaces and , which satisfy the following approximation properties: for and , positive integers , The fully discrete -GMFE scheme is to find such that
4.1. The Analysis of Stability
Lemma 6. With and , the following inequality holds:
Proof. In (60), we take and use Cauchy-Schwarz inequality and Poincaré inequality to arrive at In (61), we choose and use Cauchy-Schwarz inequality, Young inequality, and (66) to get By the simple simplification for (67), we easily get For the case in (68), we multiply (36) to get easily Noting that and Lemma 6, we use Cauchy-Schwarz inequality to get By the simple calculation for (70), we arrive at For the case , using a similar process to the proof of Theorem 5 based on the mathematical induction, we can get By a combination of (72) and (66), we get the conclusion of Theorem 7.
4.2. A Priori Error Results
For deriving the a priori error analysis, we define the Ritz projection by Further, let be the standard finite element interpolant of .
Theorem 8. With , , , and , there exists a positive constant free of space-time meshes and such that, for , and for
5. Some Numerical Results
Here, in order to show the numerical performances on the rate of convergence and a priori error estimates, we now choose and , the exact solution , for all , and the determined source term by the exact solution in (5) and then calculate some numerical results by using Matlab procedure. In the numerical calculation, the time direction is approximated by finite difference schemes and the spatial direction is discretized by the -GMFE method. Now, we divide interval into equal-length intervals , . Now we choose the piecewise linear spaces and with index . Let Based on the above expressions (77), we can get the following equivalent algebraic equation of (25): where