Optimization Theory, Methods, and Applications in EngineeringView this Special Issue
Mixed Mortar Element Method for Element and Its Multigrid Method for the Incompressible Stokes Problem
We discuss a mortar-type element method for the incompressible Stokes problem. We prove the inf-sup condition and obtain the optimal error estimate. Meanwhile, we propose a -cycle multigrid for solving this discrete problem and prove the optimal convergence of the multigrid method, that is, the convergence rate is independent of the mesh size and mesh level. Finally, numerical experiments are presented to confirm our theoretical results.
As we all know, the application of viscous incompressible flows is of considerable interest. For example, the design of hydraulic turbines, or rheologically complex flows appears in many processes which are involved in plastics and molten metals. Therefore, in recent decades, many engineers and mathematicians have concentrated their efforts on the Stokes problem, especially the problem that can be handled by the finite element methods. In , Girault and Raviart provided a fairly comprehensive treatment of the most recent development in the finite-element method. Some new divergence-free elements were proposed to solve Stokes problem recently (see [2, 3] and others). Due to this development in the finite-element theory, many numerical algorithms were established to solve the Stokes equations. Among these algorithms, multigrid methods and domain decomposition methods for the Stokes equations are very prevalent. In , the authors constructed an efficient smoother. Based on the smoother, the multigrid methods have been greatly developed (see [5, 6]). Meanwhile, a FETI-DP method was extended to the incompressible Stokes equations in [7, 8], a BDDC algorithm for this problem was developed too in  and others.
In the last twenty years, mortar element methods have attracted much attention and it was first introduced in . This method is a nonconforming domain decomposition method with nonoverlapping subdomains. In mortar finite-element methods, the meshes on adjacent subdomains may not match with each other across the interfaces of the subdomains. The coupling of the finite-element functions on adjacent meshes is done by enforcing the so-called mortar condition across the interfaces ( see  for details). There have been considerable researches on the mortar element methods (see [11–13] and others).
In , the author discussed the mortar-type conforming element ( element) method for the Stokes problem, and then Chen and Huang proposed the mortar-type nonconforming element ( element) method for the problem in . It is well known that the rotated element is a rectangle element, and it is not a flexible finite element since it is only suitable for the rectangular or L-shape-bounded domain. Moreover, the rotated element is a quadratic element and is not as convenient as the linear elements in calculating.
In this paper, we apply the mortar element method coupling with nonconforming finite element to the incompressible Stokes problem. The nonconforming finite element is a triangular element and it is suitable for more extensive polygonal domain than the rotated element. Moreover, owing to its linearity, the computational work is less than the rotated element. We prove the so-called inf-sup condition and obtain the optimal error estimate. When solving the discrete problem, we also present a -cycle multigrid algorithm, but the analysis about the convergence of the multigrid is different from . We only prove that the prolong operator satisfies the criterion which proposed in  and we obtain the optional convergence with simpler analysis than that in . Meanwhile, we do some numerical experiments which were realized in . From numerical results, we note that the number of iterations is less than the rotated element method when achieving the same relative error.
The rest of this paper is organized as follows. In Section 2, we review the Stokes problem and introduce the mortar element method for nonconforming element. Section 3 gives verification of the inf-sup condition and error estimate. The multigrid algorithm and the convergence analysis are given in Sections 4 and 5, respectively. The last section presents some numerical experiments. Throughout this paper, we denote by “C” a universal constant which is independent from the mesh size and level, whose values can differ from place to place.
We only consider the incompressible flow problem, the steady-state Stokes problem, so that we can compare the results with those in .
The partial differential equations of the model problem is where is bounded convex polygonal domain in , represents the velocity of fluid, is pressure, and is external force. Define The mixed variational formulation of problem (2.1) is to find such that where the bilinear formulations on , on and the dual parity on are given, respectively, by It is well known that the bilinear form satisfies the inf-sup condition, that is, there exists a positive constant for any such that According to the assumption on and the saddle point theory in , we know that if , then there exists a unique solution satisfying
We now introduce a mortar finite-element method for solving problem (2.3). First, we partition into nonoverlapping polygonal subdomains such that They are arranged, so that the intersection of for is an empty set or an edge, or a vertex; that is, the partition is geometrically conforming. Denote by the common open edge to and , then the interface is broken into a set of disjoint open straight segments , that is, By we denote an edge of called mortar and by an edge of that geometrically occupies the same place called nonmortar.
With each , we associate a quasiuniform triangulation made of elements that are triangles. The mesh size is the diameter of largest element in . We define , . Let nodal points be the nonconforming nodal points, that is, the midpoints of the edges of the elements in . Denote the set of nodal points belonging to , and by , and , respectively.
For each triangulation on , the nonconforming element velocity space and piecewise constant pressure space are defined, respectively, as follows: Then the product space is a global nonconforming element space for on .
For any interface , there are two different and independent triangulations and , which produce two sets of nodes belonging to : the midpoints of the elements belonging to and denoted by and , respectively.
In order to introduce the mortar condition across the interfaces , we need the auxiliary test space which is defined by For each nonmortar edge , define the -projection operator: by
Now we can define the mortar-type nonconforming element space as follows: the condition of the equality in (2.12) which the velocity function satisfies is called mortar condition.
The global element pressure space on is defined by
We now establish the discrete system for problem (2.3) based on the mixed finite-element spaces .
We first define the following formulations: Let Then the discrete approximation of problem (2.3) is to find such that In the next section, we prove that the discrete problem (2.16) has a unique solution and we obtain error estimate.
3. Existence, Uniqueness, and Error Estimate of the Discrete Solution
According to the Brezzi theory, the well-posedness of problem (2.16) depends closely on the characteristics of both bilinear forms and . We equip the space with the following norm:
We can find in  that the local space family is div-stable; that is, there exists a constant independent of such that where .
In order to prove that the global space family is div-stable, it is necessary to define the global spaces as
We first prove that the family is div-stable.
Lemma 3.1. The following inf-sup condition holds: where the constant does not depend on .
Proof. We decompose the space by and define a local interpolation operator : as
where is an edge of , is the midpoint of . Then we can define a global interpolation operator as follows:
Define the operator : by
We can deduce that for any , there exists a satisfying In fact, we can set . Obviously and On the other hand by norm equivalence we have where are the midpoints of the edges of , and is the integral average of in , by Hölder inequality, trace theorem, and Friedrichs’ inequality we can get and combining (3.11), we obtain Using norm equivalence we derive From trace theorem and (3.13), it follows that So we only need to estimate . Owing to , we then obtain The bounds in (3.15) and (3.16) lead to which together with (3.13) and (3.17) give Since is div-stable, following (3.8) and (3.18), by Fortin rules, we have completed the proof of Lemma 3.1
Now we recall the following Brezzi theory about the existence, uniqueness, and error estimate for the discrete solution.
Theorem 3.2. The bilinear forms and have the following properties:(i) is continuous and uniformly elliptic on the mortar-type nonconforming space , that is, (ii) is also continuous on the space family , that is, (iii) the family satisfies the inf-sup condition, that is, there exists a constant that does not depend on of triangulation such that so the problem (2.16) has a unique solution, and if one lets , be the solution of (2.3) and (2.16), respectively, where , , , then
Proof. The statements of Brezzi theory are that the properties (3.19)–(3.21) lead to the existence, uniqueness, and error estimate of the discrete solution. In , it is proven that is continuous on and is elliptic with a constant uniformly bounded. Furthermore, it is straightforward that is continuous on . The point that needs verification is a uniform inf-sup condition (3.21), or equivalently that the family is div-stable.
Using local inf-sup condition (3.2) and the above lemma, arguing as the proof in Proposition 5.1 of , we have the global inf-sup condition (3.21).
4. Numerical Algorithm
In this section, we present a numerical algorithm, that is, the -cycle multigrid method for the discrete system (2.16), and we prove the optional convergence of the multigrid method. We use a simpler and more convenient analysis method than that in .
In order to set the multigrid algorithm, we need only to change the index of the partition in Section 2 to be , and let be the coarsest partition. By connecting the opposite midpoints of the edges of the triangle, we split each triangle of into four triangles and we refine the partition into . The partition is quasi-uniform of size . Repeating this process, we get a sequence of the partition , each quasi-uniform of size .
As in Section 2, with the partition , we define the mortar nonconforming element velocity space and element pressure space as and , respectively. We can see that are nonnested, and are nested. Furthermore, we denote the nonconforming element product space on by .
Let be the basis of , and let be the basis of . For any , , we have the corresponding vector and . We introduce the matrice , , and having the entries , and , respectively. Then at level , the problem (2.16) is equivalent to
In the following of this section, we introduce our multigrid method; the key of this method is the intergrid transfer operator.
We first define the intergrid transfer operator on the product space, where , is the partition of , respectively, .
Then we define the intergrid operator on the mortar nonconforming element velocity space, : where is defined as (3.7).
On the element pressure space, we apply the natural injection operator : , that is,
Therefore, our prolongation operator on velocity space and pressure space can be written as
If , compute the directly. If , do the following three steps.
Step 1. Presmoothing: for , solving the following problem: where is a real number which is not smaller than the maximal eigenvalue of .
Step 2. Coarse grid correction: find , such that Compute the approximation by applying iteration steps of the multigrid algorithm applied to the above equations on level with zero starting value. Set
Step 3. Postsmoothing: for solving following problem:
then, is the result of one iteration step.
For convenience, at level the problem (2.16) can be written as followes: find such that
Since is a symmetric bilinear form on , there is a complete set of eigenfunctions , which satisfy
In order to verify that our multigrid algorithm is optimal, we need to define a set of mesh-dependent norms. For each we equip with the norm and define
For our multigrid algorithm, we have the following optional convergence conclusion.
To prove this theorem, we give in the next section two basic properties for convergence analysis of the multigrid, that is, the smoothing property and approximation property.
5. Proof of Theorem 4.1
From the standard multigrid theory, the -cycle yields a -independent convergence rate based on the following two basic properties.
We first show the smoothing property. By [ Theorem 5.1], we have the following.
Lemma 5.1 (smoothing property). Assume that , if the number of smoothing steps is , then
The property has been proved in .
Next, we prove the approximation property. We just apply the following conclusion in , which can simplify the complexity of theoretical analysis.
Lemma 5.2. If the prolongation operator defined in (4.5) satisfies the following criterion, Then, the approximation property in multigrid method holds and the multigrid algorithm converges optimally.(A.1), (A.2), (A.3). where is the solution of (2.3) with the force term and , are the mixed finite element approximation of at levels and , respectively.
This lemma has been proved in .
Lemma 5.3 (approximation property). Let be defined as follows: Then one has
Proof. By Lemma 5.2, we only need to prove our prolongation operator that satisfies (A.1), (A.2), and (A.3).
For any , the inequality (A.1) holds. In fact by Lemma 5.2 in , we can get by norm equivalence, we deduce Using trace theorem and (5.5), we have Owing to , then The second term of the above inequality can be estimated as follows: where is the orthogonal projection onto one-dimensional space which consists of constant functions on an element , and is an edge of which is in the triangulation . Using the scaling argument in , for any constant we have which combining with (5.9) gives For the first term of the right side of (5.8), we have Trace theorem and (5.5) give For , by trace theorem and the approximation of the operator , we have which together with (5.4)–(5.13), gives (A.1).
Obviously, (A.2) naturally holds so we only need to prove (A.3).
By proof of Lemma 5.2 in , we can see that Arguing as (5.6), we obtain By (5.15) and trace theorem, we get that together with (5.15), (A.3) has been proved, and we have completed the proof of Lemma 5.3.
6. Numerical Results
In this section, we present some numerical results to illustrate the theory developed in the earlier sections. The examples are as same as those in , so that we can compare the conclusion with the mortar rotated element method.
Here we deal with . We choose the exact solution of (2.1) as for the velocity and for the pressure.
For simplicity, we decompose into two subdomains: as nonmortar domain and as mortar domain. The sizes of the coarsest grid are denoted by and , respectively (see Figure 1). The test concerns the convergence of the -cycle multigrid algorithm. In what follows, denotes the level, and are the number of the unknowns of the velocity and pressure, the norm is the usual Euclidean norm of a vector which is equivalent to . denotes the number of iterations to achieve the relative error of residue less than , where and are the presmoothing steps, and the postsmoothing steps respectively, and the initial approximative solution for the iteration is zero. The numerical results are presented in Tables 1 and 2.
From Table 2, we can see that the convergence for the -cycle multigrid algorithm is optimal; that is, the number of iterations is independent of the level number . Meanwhile, we note that the number of iterations is less than the rotated element method in  when achieving the same relative error.
The authors would like to express their sincere thanks to the referees for many detailed and constructive comments. The work is supported by the National Natural Science Foundation of China under Grant 11071124.
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