Abstract
We apply a lumped mass finite element to approximate Dirichlet problems for nonsmooth elliptic equations. It is proved that the lumped mass FEM approximation error in energy norm is the same as that of standard piecewise linear finite element approximation. Under the quasi-uniform mesh condition and the maximum angle condition, we show that the operator in the finite element problem is diagonally isotone and off-diagonally antitone. Therefore, some monotone convergent algorithms can be used. As an example, we prove that the nonsmooth Newton-like algorithm is convergent monotonically if Gauss-Seidel iteration is used to solve the Newton's equations iteratively. Some numerical experiments are presented.
1. Introduction
In this paper, we consider a lumped mass finite element method (FEM) to the following Dirichlet problem for a nonsmooth elliptic equation: where is a bounded convex domain with a Lipschitz continuous boundary , is a constant, is a given smooth function, and . The above nonsmooth elliptic problem has many applications. For instance, it can arise from the MHD equilibria, thin stretched membranes problems, or reaction-diffusion problems (see, e.g., [1–4]). In order to solve problem (1), firstly, it is generally discretized by a finite element (volume) method or a finite difference method, and then various numerical algorithms are constructed to solve the corresponding discrete problems (see, e.g., [1, 4–8] and the references therein). In this paper, we apply a lumped mass finite element to approximate problem (1) by introducing the lumping domain for each node to deal with the nonsmooth term. We refer to [9–12] for such schemes of lumped mass type. For nonsmooth problem (1), one advantage of the lumped mass FEM is that one can calculate the nonsmooth terms in the finite element equations very easily and numerical quadrature algorithms are no longer needed. Another advantage of this method is that the operator in the finite element problem is diagonally isotone and off-diagonally antitone and thereby some monotone convergent algorithms can be applied (see, e.g., [13, 14]). In this paper, we will prove that the FEM error in energy norm is the same as that of the standard FEM. Since the finite element problem is a nonsmooth equation, we will apply nonsmooth Newton-like algorithm to solve it and focus our attention on the monotone convergence property of the algorithm for some special inner iterators.
Throughout this paper, we adopt the standard notations for Sobolev spaces on with norm and seminorm . We denote by and and let be the subspace of with vanishing traces on ; that is,
We will use the letter or (with or without subscripts) to denote a general positive constant independent of the mesh size. When it is not important to keep track of these constants, we will conceal the letter or into Xu’s notations “” and “” for convenience. Here
The remainder of the paper is organized as follows. In Section 2, we present a lumped mass finite element to problem (1) and estimate its error in energy norm (or equivalently -norm). In Section 3, we present nonsmooth Newton-like algorithm and analyze its convergence, especially its monotone convergence. In Section 4, we illustrate some numerical results to confirm the theoretical results we obtained and the efficiency of the algorithm. And finally, in the last section, we make a simple conclusion.
2. A Lumped Mass Finite Element Approximation to the Nonsmooth Dirichlet Problem
The weak form of (1) is to find such that where is the inner product defined by .
Let be a triangulation of , where the mesh size denotes the maximum diameter of its triangle elements and let be a polygonal approximation to with a boundary . For simplicity, we assume that . Let denote the triangle element of the triangulation , and let , , denote the interior nodes and , , the boundary nodes, respectively. Let and be defined as follows: We assume that the triangulation satisfies the following quasi-uniform mesh conditions: Associating with , let be the piecewise linear finite element space defined by where is the set of the polynomials of degree 1, and for each , is the basis function corresponding to the node . Let . Then , and the standard piecewise linear FEM approximation of (1) is to find such that
Let be the solution of problem (9). Then, problem (9) leads to the following system of nonsmooth equations: find such that where is a symmetric positive definite (SPD) matrix, , and .
Generally, the operator in problem (10) is not diagonally isotone and off-diagonally antitone. Moreover, from the computational point of view, it is tedious to calculate the nonsmooth term directly and some numerical quadrature algorithms have to be used. Instead of solving the finite element problem (9), the lumped mass finite element problem is to find such that where is defined by Here, with being the characteristic function on the lumping region at node . For any node , its lumping region is the region by joining the centroids of the triangle elements which have as a common vertex to the midpoints of the edges which have as a common extremity [9, 10] (see an example in the shaded part in Figure 1).
In the discrete problem (11), we do not have to calculate the nonsmooth term . Instead, we need only to calculate the following simple term where is the solution of problem (11), denotes the area of . Furthermore, it will be seen in Section 3 that the operator in problem (11) is a diagonally isotone and off-diagonally antitone operator if we assume that the triangulation satisfied the so-called maximum angle condition.
In the sequel, we estimate the error between and . To this end, we introduce two auxiliary problems of finding and , such that respectively. Between the solutions of problems (15) and (16) we have the following results.
Lemma 1. Let and be the solutions of problems (15) and (16), respectively. Then
Proof. By (15) and (16) we have
Then,
For any real constants and , we have
This combining with (19) implies
Here, we used the following properties of :
Then, we immediately obtain (17) by Poincaré’s inequality.
According to the finite element theory, the following lemma is obvious [15]. We give a simple proof for completion.
Lemma 2. Let and be the solutions of problems (5) and (15), respectively. Then
Proof. We can prove the lemma by following standard steps of estimating the error in energy norm. Setting in (5) and combining with (15), we have and then Therefore, where is the nodal interpolation of on and we assume it in , while the “” comes from the following well known interpolation results: Therefore, by (26) and Poincaré’s inequality, we have , which implies (23).
Noting (11) and (16), we have It then follows that where the first “” is from (20), the second “” is from (22), and the last “” is from (28). Hence, This together with (28) obtains
That is, the following theorem holds.
Theorem 3. Let and be the solutions of problems (5) and (11), respectively. Then we have the estimate .
3. Nonsmooth Newton-Like Method
In this section, we consider some numerical solvers to the discrete problem (11). First, we give a definition [14, 16].
Definition 4. A mapping is said to be -differential at a point if there exists a function called the -derivative of at , which is positively homogeneous of degree 1 (i.e., for all and all ), such that
Let be the solution of problem (11). Then, problem (11) leads to the following system of nonsmooth equations: find such that where is an SPD (Symmetric Positive Definite) matrix, , and . Due to the occurrence of , is not differentiable but is semismooth. So, we use the following nonsmooth Newton-like method to solve problem (34) (see, e.g., [16, 17]).
Algorithm 5 (nonsmooth Newton-like method). Consider the following.
Step 1. Given an initial guess and a precision , let .
Step 2. If , stop. Otherwise, turn to Step 3.
Step 3. Compute the step length , an approximation of the solution of the Newton’s equations such that Here, , ; that is, is a generalized derivative of at , and is a sequence of positive numbers.
Step 4. Let , , and turn to Step 2.
Noting that with being the smallest eigenvalue of matrix , is strongly monotone. On the other hand, noting that is diagonal, that is, is dependent only on the th entries [14], is a diagonal mapping. Assume furthermore that the triangulation satisfies the maximum angle condition, that is, for any , the maximum angle of is less than . The matrix is then a symmetric -matrix. That is to say, the matrix , with positive diagonals and nonpositive off-diagonals, has a nonnegative inverse (see, e.g., [18]). By the definition of , where is the th unit vector in . Hence, for , where we have used the inequalities and (). Equation (39) shows that is diagonally isotone and off-diagonally antitone.
In Algorithm 5, we usually choose , so that the local convergence rate of Algorithm 5 is -superlinear (see, e.g., [19]). Furthermore, According to (39), some monotone convergent algorithms can be constructed. As an example, in the following, we will investigate the monotone convergence of nonsmooth GS-Newton algorithm, in which the subproblem (35) is solved iteratively by Gauss-Seidel iteration. In other words, in nonsmooth GS-Newton algorithm, is generated by -times Gauss-Seidel iteration with zero initial: where .
The following property is important to the monotone convergence.
Lemma 6. Let be an -matrix and satisfy . Then, , where is the solution of nonsmooth equation . Similarly, let satisfies . Then, , where is the solution of (35).
Proof. Let Then , , and , where . Therefore, Since that is an -matrix and is a diagonal matrix, (see, e.g., [14]). Multiplying (42) on the left by , we obtain . The rest part of the lemma can be proved in a similar way.
Remark 7. By Lemma 6, for any satisfying , we have . Therefore, is usually called an upper-solution of problem . Similarly, for any satisfying , we can conclude and thereby is called a lower-solution of problem .
Lemma 8. Let be an -matrix and . Then where () are generated by (40).
Proof. It is easy to check that
By the definition of ,
where is a generalized derivative of semismooth function at point . Noting that
is still an -matrix (see, e.g., [14]). Therefore, . On the other hand, by (40), we have
Hence
We then conclude that (43) holds for . Assume that (43) holds for . By (40), we have
and then
where we used the inequalities and . By the use of (40) and again, we have
On the other hand, it is easy to calculate that
where are defined similar to (41). If , and then . If , by , we have and then . Therefore, by (46), we have . In one word, always holds. This together with (52) and (50) implies that
Combining above inequality with (50) and (51), we complete the proof by the principle of induction.
Theorem 9. Let be an -matrix, and let be generated by nonsmooth GS-Newton algorithm with the initial satisfying . Then where is the solution of nonsmooth equation (34). Moreover, converges to monotonically.
Proof. Equation (54) can be obtained directly by Lemmas 6 and 8. Hence, exists. On the other hand, it follows from (43) that holds for each and . Therefore,
which implies
By (45) and (46), we have
and then by (40)
where is the strictly upper-triangular part of . Then, we conclude , which implies . The proof is then completed.
Similar to Theorem 9, we have the following conclusion according to Remark 7.
Proposition 10. Let be an -matrix, and let be generated by nonsmooth GS-Newton algorithm with the initial satisfying . Then is monotonically increasing and convergent to , where is the solution of nonsmooth equation (34).
Remark 11. If we solve (35) by SOR iteration, that is, in the inner iteration, we use the following iteration scheme: Algorithm 5 becomes nonsmooth SOR-Newton algorithm, where is a relaxation factor. It is not difficult to verify that the monotone convergence of the algorithm can be derived similar to the proof of Theorem 9.
4. Some Numerical Examples
In this section, we carry out some numerical experiments to confirm the theoretical results we have obtained. The experiments are performed under Windows XP and MATLAB v7.10 (R2010a) running on a personal computer with an Intel Core 2 Duo CPU at 2.20 GHz and 1.00 GB of memory.
In our numerical examples, we consider problem (1) with the following data (see [5]): and
It is easy to verify that the exact solution of (1) is .
It is shown in Table 1 that the convergent error order in energy norm is , which is consistent with Theorem 3.
The monotone convergence of nonsmooth SOR-Newton algorithm can be investigated in the experiments if we choose the initial suitably. But we do not list the details here. In Table 2, we listed CPU times spent in nonsmooth SOR-Newton algorithm for , , and , respectively. In the experiments, we let , , , and . As a comparison, we also present corresponding results in Table 3 for the active set method (ASM), which is presented in [5] based on primal-dual active set method for constrained optimal control problem (in Table 3, “—” indicates that the corresponding CPU time is more than 6000 seconds).
It follows from Tables 2 and 3 that nonsmooth SOR-Newton algorithm performs much better than ASM. The reason may be that a pair of discrete PDEs have to be solved at each step for ASM.
It can be seen in Tables 2 and 3 that the smaller is, the more the CPU time becomes. Indeed, by (46), we have for any that where the second equality is from for and the “” is from (22) and the inverse inequality, while the “” is from the following equivalent property between the -norm and the mesh-dependent norm under the condition of quasi-uniform grid (see, e.g., [20]): Estimate (46) together with (61) implies that is an SPD matrix and satisfies Therefore, the condition number of satisfies . This indicates that the conditioner of the linear system (35) deteriorates as the mesh becomes finer. Roughly speaking, for the large scale problem, much time may be spent on solving linear subproblem (35). This stimulates us to construct numerical solvers combining nonsmooth Newton with multigrid technique. A simple idea is to apply linear multigrid directly to the linear systems produced in Newton-like method for the correction term at each iteration. We refer to [21, 22] and the references therein for this kind of methods. In Table 4, we present some numerical results for this kind of nonsmooth multigrid-Newton algorithm, in which V-cycle multigrid is used to solve Newton’s equations (35) and Gauss-Seidel iteration is taken as the smoother (with one time presmooth and two times postsmooth at each cycle).
It follows from the Tables 2–4 that compared with nonsmooth SOR-Newton algorithm and ASM, nonsmooth multigrid-Newton algorithm performs excellently for small .
5. Concluding Remarks
In this paper, we apply a lumped mass finite element to approximate Dirichlet problems for nonsmooth elliptic equations. The operator in the relating discrete system of nonsmooth equations is diagonal, which leads to a diagonal generalized gradient matrix for the nonsmooth term. It is proved that the FEM error in energy norm is , which is optimal. For the discrete finite element problem, we use nonsmooth Newton-like algorithm to solve it and locally -superlinear convergence of the algorithm can be obtained directly by the existing literatures. Especially, since the operator in the discrete system of nonsmooth equations is diagonally isotone and off-diagonally antitone, the monotone convergence is obtained for the algorithm if Gauss-Seidel iteration (or more generally SOR ) is used to solve the subproblem iteratively. In other words, the iterate generated by the algorithm converges to the solution monotonically from an upper-solution or a lower-solution of the problem. In the numerical experiments, we adopt SOR or multigrid as the inner iterator. Numerical results indicate that the algorithm is efficient.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
Acknowledgments
The authors would like to thank the referees for their constructive comments leading to an improved presentation of this paper. This work has been partially supported by the National Science Foundation of China under Grants 11201197, 11271069, and 71301067, the Youth Foundation of Nanchang Institute of Technology under Grant 2012KJ025, and the Science and Technology of Jiangxi Provincial Department of Education under Grant GJJ13743.