Abstract
We consider a boundary-value problem of one-side conservative elliptic equation involving Riemann-Liouville fractional integral. The appearance of the singular term in the solution leads to lower regularity of the solution of the equation, so to the lower order convergence rate for the numerical solution. In this paper, by the dividing of equation, we drop the lower regularity term in the solution successfully and get a new fractional elliptic equation which has full regularity. We present a theoretical framework of mixed finite element approximation to the new fractional elliptic equation and derive the error estimates for unknown function, its derivative, and fractional-order flux. Some numerical results are illustrated to confirm the optimal error estimates.
1. Introduction
We consider the following one-side conservative diffusion problems of order : where is the first-order derivative operator, , defined by (2), refers to the left Riemann-Liouville fractional integral of order , is a positive constant diffusive coefficient, and is the source or sink term. Equations (1a) and (1b) reduce to the classical second-order diffusion equation when .
Equations (1a) and (1b), delicately derived from random walk model [1], fractional conservation of mass [2], and the impact of boundary [3], describe subsurface fluid flow and solute transport processes taking place in a multiscale heterogeneous medium/aquifer (see [4] and the references cited therein). These processes exhibit anomalous diffusion or non-Fickian diffusion that cannot be modeled adequately and properly by classical second-order diffusion equation and thus have contracted considerable attention in practical applications.
Since the analytic solutions to general fractional partial differential equations are rarely available, one has to turn to numerical methods in general. In the last decade, a number of numerical methods, for example, finite difference method [5], finite volume method [6], spectral method [7], fast difference method [8], finite element method [9–13], and Petrov-Galerkin formulation [14–18], have been developed consecutively for space-fractional partial differential equations.
Considering the conservation property of the diffusion processes and the application of the fractional-order flux in engineering, an ideal numerical procedure should recognize both the unknown function and its flux and obey the conservation of mass to reflect the physical character of the diffusion model ((1a) and (1b)). Hence, one should involve the state variable and the fractional-order flux to form a saddle-point formulation then design a locally conservative numerical procedure. For this purpose, by introducing two intermediate variables and the flux , Chen and Wang [19] proposed a saddle-point framework and developed a locally conservative expanded mixed finite element method recently. Although a rigorous numerical analysis theory was established in [19], the convergence rates for the unknown and the flux are not optimal with respect to regularity requirement for sufficiently smooth solution . In particular, for the solution only in with , the convergence rates are heavily destroyed to , due to the appearance of the singular term .
The aim of this paper is to express the solution of (1a) and (1b) by a full-regular solution satisfying a general fractional diffusion equation with -dependent right-hand side term and a -type solution satisfying an analytic-solved fractional equation. Then, the expanded mixed finite element method proposed in [19] can be applied to solve the -dependent fractional equation and obtain good error estimates whatever the regularity of the solution is.
The rest of the paper is organized as follows. In Section 2, we revisit the fractional differential operators and their properties. In Section 3, we split the fractional diffusion equations (1a) and (1b) into two equations: one is an -dependent fractional diffusion equation that permits a fully regular solution, and the other is an analytic-solved fractional equation from which the -type analytic solution is easily solved. By doing so, the solution of (1a) and (1b) is expressed as the sum of the -dependent solution and the -type solution. We are devoted to discretizing the -dependent fractional equation by the expanded mixed finite element procedure in Section 4 and deriving the related error estimates in Section 5. Obviously, the error estimates for the original fractional diffusion equations (1a) and (1b) obey those for the -dependent fractional equation since the -type solution is analytically solved. In Section 6, numerical experiments are performed to confirm our theoretical findings.
2. Fractional Operators and Fractional Sobolev Spaces
In this section, we will recall the definitions and properties of Riemann-Liouville fractional differential operators and fractional Sobolev spaces.
Definition 1 (see [20–22]). Let . The left and right Riemann-Liouville fractional integrals of order are defined by where is Gamma function.
Definition 2 (see [20–22]). Let with . The left and right Riemann-Liouville fractional derivatives of order are
We note that fractional integral operator [20–22] satisfies semigroup property; that is, for , and adjoint property We also observe that the left (right) fractional derivative operator is a left inverse of the left (right) fractional integral operator [20–22]:
We introduce the fractional derivative spaces and present their well-established equivalence to fractional-order Sobolev spaces [9, 19, 23].
Definition 3 (see [9]). For , define the norm with seminorm Let be the left fractional derivative space defined as the closures of with respect to norm .
Definition 4 (see [23]). For , define the norm and let denote the closures of with respect to norm .
The right fractional derivative spaces and and their respective norms are defined analogously.
Definition 5 (see [9]). For , define the norm with seminorm where denotes the Fourier transform of Let be the left fractional derivative space defined as the closures of with respect to norm .
Definition 6 (see [23]). For , define the norm and let denote the closures of with respect to norm .
We then reiterate the equivalence theories established in [9, 23] for the fractional derivative spaces.
Lemma 7 (see [9]). Let and , . Then, , , and the fractional-order Sobolev spaces are equal, with equivalent seminorms and norms.
Lemma 8 (see [23]). Let . Then, , , and the negative fractional-order Sobolev spaces , the dual space of , are equal with equivalent norms.
We now restrict the fractional derivative spaces to .
Definition 9 (see [9]). Let . Define the spaces , , and as the closures of under their respective norms.
Definition 10 (see [23]). Let . Define the spaces , , and as the closures of under their respective norms.
Lemma 11 (see [9]). Let and , . Then, , , and the fractional-order Sobolev spaces are equal, with equivalent seminorms and norms.
Lemma 12 (see [19]). Let . Then, , , and the negative fractional-order Sobolev spaces , the dual space of , are equal with equivalent norms.
For simplicity of presentation, we use and to stand for the norms and seminorms of fractional-order Sobolev spaces and to stand for the norms of negative fractional-order Sobolev spaces , respectively.
3. Decomposition of (1a) and (1b)
In this section, we shall split the fractional diffusion equations (1a) and (1b) into an -dependent fractional diffusion equation that permits a fully regular solution and an analytic-solved fractional equation from which the -type analytic solution is easily solved.
Theorem 13. Let be the solution to the fractional diffusion problem ((1a) and (1b)) with right term . Then,(1) can be expressed as where and are the solutions to the following fractional problems here with ;(2)let and be fractional-order flux. Then, with , , , ; further,
Proof. The solution and the regularity of the fractional diffusion equations (1a) and (1b) are given in [19], and the proof is provided only for completeness. Introducing two intermediate variables and , ((1a) and (1b)) can be rewritten as Solving (19c) to obtain and noting for , we conclude that and .
Solving (19a) and applying to both sides of (19b), we deduce Substituting (20) into the above equation and noting the boundary condition , we get Enforcing the boundary condition to the above equation, we obtain , which leads to the solution of (1a) and (1b):Next, we are to solve (16a) and (16b) and (17a) and (17b). Substituting for in (22) and noticing the boundary condition , we get the solution of (16a) and (16b): Similarly, we get the the solution of (17a) and (17b): It follows from (23)–(25) that the solution to the fractional problem ((1a) and (1b)) can be split into the solutions of (16a) and (16b) and (17a) and (17b).
Noting that is a bounded operator from fractional Sobolev space to [3], we get the first term of solution in (23) is in with . The second term with [14]. Combining the regularities of two parts and the boundary conditions, we derive and . Clearly, the regularity of the second term is no less than the regularity of the first term in (24). This implies , , and .
Remark 14. The decomposition of (1a) and (1b) is not unique in Theorem 13; that is, we can choose any other function , provided that the regularity of is no less than the regularity of the first term in (24).
Remark 15. From Theorem 13, the regularity of original problem (1a) and (1b) is only , due to the presence of the singular term , while the regularity to (18) is with . In particular, for the right term , , the regularity can be improved to by choosing a proper decomposition. For example, we can replace by with in the decomposition of the solution when . This is the advantage of the splitting.
Remark 16. Equations (17a) and (17b) is analytic-solvable. Once the regular part is determined, the solution of the source problem ((19a), (19b), and (19c)) (or (1a) and (1b)) has the representation So we focus on -dependent fractional diffusion equations (16a) and (16b) from now on.
4. Saddle-Point Formulation and Mixed Finite Element Method
In this section, we shall formulate a saddle-point variational formulation and establish a mixed finite element formulation to the -dependent fractional diffusion equations (16a) and (16b).
Introduce the fractional-order flux with . Then, we can rewrite (16a) and (16b) into the following mixed form: The mixed variational formulation corresponding to (27a), (27b), and (27c) is to find such that where , , and .
Lemma 17 (see [19]). There exists a unique solution to the problem (28a), (28b), and (28c).
Next, we shall establish a mixed finite element procedure based on the saddle-point variational formulation (28a), (28b), and (28c).
Let be a uniform division of by , with , , and . Let denote Raviart-Thomas space [24–27]: where is the restriction of the set of all polynomials of a degree not bigger than on . Then, the mixed finite element formulation is to find such that Last, we construct an approximate solution to (19a), (19b), and (19c) (or (1a) and (1b)) by
5. Convergence Analysis
In this section, we are devoted to the convergence analysis. In error analysis below, we shall make use of three projection operators. The first operator is defined by The other two operators are the -projections and : Lemmas 18–20 hold in the integer-order Sobolev spaces (if see [19] and if see [24–27]). We can prove that they also hold in the fractional-order spaces by applying standard real interpolation methods [26].
Lemma 18. Let and be any real numbers. Assume that . Then,
Lemma 19. Let and be any real numbers. Assume that . Then,
Lemma 20. Assume that ; then,
Now we proceed to the convergence analysis. Let , , and ; then, from (28a), (28b), and (28c), (30a), (30b), and (30c), (32), and (33), we can get the error equations
Lemma 21. Suppose that satisfies (37c). Then, the following holds:
Proof. Set in (37c) to derive that , which implies . Then, we obtain the desired result.
Lemma 22. There exists a constant , such that
Proof. Set in (37a). Then, noting , we have Set in (37b); then, where , being the extension of by zeros to [23, Lemma 2.6], is used and . Now, we estimate the right-hand side of (41) term by term. For the first term, we get For the second term, applying Green’s formula, we have where is used. In fact, taking in (37a), we get ; that is, . Introduce the -projection operator . Then, the approximation property holds. By the property of the -projection operator , (43) can be rewritten as where is used. Substituting (42) and (44) into (41) yields In conjunction with triangle inequality, we obtain the desired estimate.
Lemma 23. There exists a constant , such that
Proof. Taking in error equations (37a), (37b), and (37c), we have Then, Together with triangle inequality, we finish the proof of Lemma 23.
Now let us apply the duality argument of Douglas and Roberts to obtain an error estimate in the -norm.
Lemma 24. There exists a constant , such that
Proof. Let and , , such that Then, the regularity estimate holds. Taking in (37a), we have First, we are to estimate the first term of (51). Using adjoint property (6), we have We use Lemma 19 to estimate : The estimate of is the result of (37c), Lemmas 18 and 19: Together with above two inequalities, we get the estimate of : Thanks to the adjoint property of fractional integral operator, the relationship of -projection operator and operator , and adjoint problem (50a) and (50b), we get the estimate of : where is used. In fact, taking in (37a), we derive ; that is, . Substituting (55) and (56) into (51), and noticing the regularity of the adjoint problem (50a) and (50b), we obtain By triangle inequality, we get the desired result.
By Lemmas 21–24, the properties of Raviart-Thomas operator , projections , and , we obtain the following.
Theorem 25. Let . Suppose is the solution of (27a), (27b), and (27c) and satisfies (30a), (30b), and (30c). Then, we can get the following error estimates:
Proof. Let . By the regularity of problem (16a) and (16b), we know , , and . Recalling the approximation properties in Lemmas 19 and 20, and combining with Lemmas 21 and 22, we can get the estimates of and : Analogously, combining the above estimate of and the estimate of , we get from Lemma 23The estimate of is a result of Lemmas 21–24 and the following estimate:
By (26), (31), and Theorem 25, we can derive error estimates for the approximation .
Theorem 26. Let . Suppose is the solution of (19a), (19b), and (19c) (or (1a) and (1b)) and satisfies (31). Then, we can get the following error estimates:
Remark 27. From Theorem 26, the estimates and are optimal, and the estimates of and are suboptimal, even though the regularity of the solution remains only in for . Hence, in comparison with the standard mixed finite element methods [19], the error rates for , , and are only ; in spite of the enough smoothness of the right term , the new approach does yield a higher-order convergence rate.
Remark 28. The estimates of and are only suboptimal, due to the structure of the error equations (37a), (37b), and (37c) and the low global regularity of the adjoint problem (50a) and (50b), used in Nitsches trick. The error equations (37a), (37b), and (37c) involve , and as a whole, so the estimates for and heavily depend on . This may weaken the orders of error estimates.
6. Numerical Experiments
In this part, we present some numerical experiments to verify the theoretically proven convergence results. In all the following examples, we let . For each example, we consider three different values, that is, , , and , and present the estimates and the convergence orders of , , and , where the number in the bracket under the column order is the theoretical convergence rate derived in Theorem 26.
We consider four examples with the analytic solution of different regularity. The analytic solution of Example 1 is sufficiently smooth, while the others only belong to with , due to the appearance of singular term . So we split (1a) and (1b) into a -dependent fractional diffusion equations (16a) and (16b) and an analytic-solved fractional equation (17a) and (17b) by Theorem 13. Spaces choose to be (29) with and for each example.
Example 1. Let ; source term . Then, the analytic solution . As is very smooth, the result of Theorem 26 predicts almost , , and for , and , respectively. Tables 1 and 2 include numerical results for and , respectively. From Tables 1 and 2, we can see that they are much higher than the theoretical prediction of Theorem 26.
Example 2. Let source term for , and . Then, the analytic solution only belongs to with , despite the smoothness of the right term . So we split (1a) and (1b) into an -depended fractional diffusion equation with right term and analytic solution , and an analytic-solved equation. The numerical results are shown in Tables 3 and 4. The convergence rates are identical with that for Example 1. This is attributed to the good regularity of the analytic solution .
Example 3. Let source term for , and constants . The analytic solution belongs to with . We split (1a) and (1b) into (16a) and (16b) and (17a) and (17b). We are to solve (16a) and (16b) with right term . The analytic solution is very smooth. The numerical results are shown in Tables 5 and 6. The same convergence rates are observed, which are much higher than the theoretical prediction of Theorem 26.
Example 4. Let source term , which is singular at the origin and belongs to for . The analytic solution only belongs to with and . We split (1a) and (1b) into two equations (16a) and (16b) and (17a) and (17b). We are to solve (16a) and (16b) with right term and analytic solution . The numerical results are shown in Tables 7 and 8.
Remark 5. From the above four examples, we can see that the error rates for are identical to the predicted rates of convergence in Theorem 26. However, numerical convergence rates for and are much higher than the theoretical prediction of Theorem 26.
Competing Interests
The authors declare that they have no competing interests.
Acknowledgments
This work is supported in part by the National Natural Science Foundation of China under Grant nos. 10971254, 11301311, 11471196, 2014ZRB01849, and 61503214.