Abstract
The weak Galerkin finite element method is combined with the method of characteristics to treat the convection-diffusion problems on the triangular mesh. The optimal order error estimates in and norms are derived for the corresponding characteristics weak Galerkin finite element procedure. Numerical tests are performed and reported.
1. Introduction
We will consider combining the method of characteristics with weak Galerkin finite element techniques to treat the model problem given by where is a bounded convex polygonal domain with the boundary is an unknown function, , and are known functions, is known bounded vector-valued functions, and is a symmetric, bounded matrix function. Assuming that the matrix satisfies the following condition, there exist positive constants , such that For periodic problems we do away with boundary value condition .
Here and in what follows, we will not write the independent for any functions unless it is necessary.
Let and let the characteristics direction associated with the operator be denoted by , where Then (1) can be put in the form
The weak form for (1) seeks such that , and
Let be a positive integer, , and . The characteristics derivative will be approximated basically in the following manner. Let and note that
We use standard definitions for the Sobolev spaces and their associated inner products , norms , and seminorms for . For example, for any integer , the seminorm is given by with the usual notation The Sobolev norm is given by We use for the norm.
Let be a normed space with the norm . denote the space of the maps of into and define the following norms for and suitable functions : For , the usual modification is made.
In many diffusion processes arising in physical problems, convection essentially dominates diffusion, and it is natural to seek numerical methods for such problems that reflect their almost hyperbolic nature. Convection-diffusion problems have been treated by various numerical methods [1–16]. Some methods of them treating convection-dominated diffusion problems exhibit stability limitation. The goal of this paper is to combine the method of characteristics with the weak Galerkin finite element method [17, 18] for convection-dominated diffusion equation. The principal gains from these procedures appear in time truncation. Approximation of by standard backward differencing leads to errors of the form in suitable norms, while characteristics method will be shown to yield . In problems with significant convection, the solution changes much less rapidly in the characteristics direction than the direction in [19]. Thus, these characteristics schemes will permit the use of larger time steps, with corresponding improvements in efficiency, at no cost in accuracy. We will see that there is no stability limitation on the size of .
An outline of the remainder of this paper is as follows. In Section 2, the characteristics weak Galerkin finite element scheme for the convection-diffusion initial boundary value problem (1) will be formulated. A generalized weak Galerkin elliptic projection is defined in Section 3. Optimal error estimations in both and norms of characteristics weak Galerkin finite element solution are proved in Section 4. The paper is concluded with some numerical experiments to illustrate the theoretical analysis in Section 5.
2. Characteristics Weak Galerkin Finite Element Formulations
In this section, we design the characteristics weak Galerkin finite element schemes for the initial boundary value problem (1). We consider the space of discrete weak functions and the discrete weak operator introduced in [17]. Let be a triangulation partition of the domain with mesh-size . As usual, we assume the triangles to be shape-regular. For each element , denote by and the interior and the boundary of , respectively. The boundary consists of several “sides,” which are edges. Denote by the collection of all edges in . On each , let be the set of polynomials on with degree less than or equal to . Likewise, on each , let be the set of polynomials on with degree no more than (i.e., polynomials of degree on each line segment of ).
A weak function on the region refers to a vector-valued function such that and with . The first component can be understood as the value of in the interior of , and the second component is the value of on the boundary of .
Denote this space by The corresponding finite element space would be defined by patching over all the triangles . In other words, the weak finite element space is given by Denote by the subspace of with vanishing boundary values on ; that is,
Let be a subspace of the set of vector-valued polynomials of degree no more than on and . For each , the discrete weak gradient of on each element is given by the following equation:
To investigate the approximation properties of the discrete weak spaces and , we use a local projection of onto in this paper.
The discrete weak spaces and need to possess some good approximation properties in order to provide an acceptable finite element scheme. In [17], the following two criteria were given as a general guideline for their construction.(P1)For any , if on , then one must have constant on . In other words, constant on .(P2)Let , where , be a smooth function on , and let be the projection/interpolation of in the corresponding finite element space (recall that it is locally defined); then, the discrete weak gradient of provides a good approximation of ; that is, holds true.
Then, the characteristics weak Galerkin finite element method based on the weak Galerkin operator (16) and weak formulation (6) is to find for , satisfying on and ( will be given in Section 3) in and the following equation: where is the weak bilinear form defined by
From the assumption of initial boundary value problem (1), we have
We denote by the approximation of and the evaluation of at the point . Then we obtain the characteristics weak Galerkin finite element scheme for the problem (1): find , such that ( would be introduced in Section 3) and, for , where ; that is, It is obvious that (20) determines uniquely in terms of the data and .
3. A Weak Galerkin Elliptic Projection
In the study of numerical methods for parabolic problems, an elliptic projection associated with the problem is usually introduced. The following Lemma 1, which is proved in Wang and Ye [17], gives the error estimate for the second order elliptic problem.
Lemma 1. Assume that with and are the solutions of the problems respectively. Let be the projection of in the corresponding finite element space. Then there exists a positive constant independent of such that provided that the mesh-size is sufficiently small.
According to our problem (1), we introduce a weak Galerkin elliptic projection operator defined: find , such that It has been proved in [17] that the solution of (24) is existence and uniqueness; then has been defined. So or in (17) or (20) are well defined.
From Lemma 1, we have the following error estimates for .
Lemma 2. Assume that with and are the solutions of the problems (1) and (37), respectively. Let be the projection of in the corresponding finite element space. Then there exists a positive constant independent of such that provided that the mesh-size is sufficiently small.
Differentiating (24) on , we can prove the following Lemma 3 in the same way.
Lemma 3. Under the assumption of Lemma 2, if , with , then there exists a positive constant independent of such that provided that the mesh-size is sufficiently small.
4. Optimal Order of Error Estimates in and the Discrete
In this section, we will develop the error estimates in the and norms for the characteristics weak Galerkin finite element method.
Assume that , and are the solutions of (1), (20), and (24), respectively. Let be the projection of in the corresponding finite element space, and Then can be handled by applying the results in Lemma 2. So, our main goal here is to bound .
Let be any test function. By testing (5) against the first component , we arrive at the following: Subtracting (28) with from (20), and using (24) with , we have the following error equation: and .
In order to derive error estimates, we give three lemmas which have been proved in [19].
Lemma 4. If and , where , and are bounded, then
Lemma 5. Let , where ; then for all ,
Lemma 6. Let be the projection of in the corresponding finite element space; then
The error estimates for the characteristics weak Galerkin finite element method in and norms are provided in the next two theorems.
Theorem 7. Assume that , (), and are the solutions of (1), (20), and (24), respectively. If , and , with , then there exists a positive constant independent of and such that provided that the mesh-size and are sufficiently small.
Proof. Taking in (29), we arrive at
For , we first estimate the left items of (34) as follows:
where the last step uses Lemma 5.
Next, we estimate the right items of (34). Using Lemma 6, we have that
Write as the sum and use , for all ; then
And using Lemma 4, we have
This completes the treatment of the right-hand side.
The inequalities (35), (36), (37), and (38) can be combined with (34) to give the recursion relation
If (39) is multiplied by and summed in time and if it is noted that , then it follows that
Using the discrete Gronwall inequality and Lemmas 2 and 3, when is sufficiently small, we have
Then, the result of Theorem 7 follows from (41) and . The proof is complete.
Theorem 8. Assume that , (), and are the solutions of (1), (20), and (24), respectively. If , and with , then there exists a positive constant independent of and such that provided that the mesh-size and are sufficiently small.
Proof. For , taking in (29), we arrive at the following:
By Lemma 4, the left-hand side satisfies the inequality
The right-hand side terms can be estimated as below. First
Second,
Combining (44), (45), and (46) with (43), we arrive at the following:
Multiplying and then summing over from to at both sides of (47) and noting that , we obtain
By , for all and the discrete Gronwall inequality and Lemmas 2 and 3, when is sufficiently small, we have
Using Lemma 2, (49), and triangle inequality, we have
This completes the proof.
5. Numerical Experiments
In this section, we present two examples to demonstrate the convergence order of the studied characteristics weak Galerkin finite element method. Let be a quasi-uniform triangulation with mesh-size and let be the time step. In the numerical tests, discrete weak spaces and , with being the lowest order Raviart-Thomas element on the triangle , are considered. We denote the numerical solution of by and the error by .
Example 1. In this example, we take , , the coefficient matrix , the exact solution which satisfies homogeneous Dirichlet boundary condition, and the initial condition . Here are two settings of : (i) , and (ii) . With each selection of , the source term can be obtained by substituting the exact solution into the equation. For a fixed mesh ratio , the norm of and with the options (i) and (ii) of is reported in Tables 1 and 2, respectively. It is observed that the numerical results reflect the convergence order of scheme (20) and support our theoretical conclusions in Section 4.
Example 2. In this example, we study the case for which the exact solution has a sharp front moving with time. The exact solution is given by , where , and . We set , , the coefficient matrix , and . The norm of and for is reported in Table 3. It shows that the characteristics weak Galerkin finite element method can simulate the solution with a sharp front effectively.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
Acknowledgments
The project is supported by the National Natural Science Fund (11171193), the Fund of the Natural Science of Shandong Province (ZR2011MA016), and a Project of Shandong Province Science and Technology Development Program (2012GGB01198).