Mathematical Problems in Engineering

Volume 2013 (2013), Article ID 485628, 11 pages

http://dx.doi.org/10.1155/2013/485628

Research Article

A Preconditioner for FETI-DP Method of Stokes Problem with Mortar-Type Discretization

Chunmei Wang1,2

¹Jiangsu Key Laboratory for NSLSCS, School of Mathematical Sciences, Nanjing Normal University, Nanjing 210023, China

²Nanjing Normal University Taizhou College, Taizhou 225300, China

Received 19 October 2012; Revised 21 February 2013; Accepted 21 February 2013

Academic Editor: Salvatore Alfonzetti

Copyright © 2013 Chunmei Wang. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

FETI-DP method is developed for incompressible Stokes problem discretized by mortar-type mixed elements. A preconditioner is proposed, and it is proved that the condition number of the preconditioned FETI-DP operator is bounded by , where and are mesh sizes. Finally, the numerical tests are presented to verify the theoretical results.

1. Introduction

FETI-DP method is a nonoverlapping domain decomposition method which has been implemented for large scale engineering applications. It has been developed for elliptic problems with heterogeneous coefficients, Stokes problems, and compressible elasticity problems (cf. [1–4]). Recently, extensions to irregular subdomains and inexact subdomain solvers have been done in [5–7]. Li [4] developed a FETI-DP method for the Stokes problem by adding primal continuity constraints to accelerate the convergence of the method. Li and Widlund [8] developed BDDC methods for incompressible Stokes equations which established a close connection between the BDDC and FETI-DP methods for the Stokes case. Kim et al. [9] developed a BDDC method discretized by mortar finite element methods for geometrically nonconforming partitions, and the BDDC preconditioner was shown to be closely related to the Neumann-Dirichlet version of the FETI-DP method.

Nonconforming discretizations are important for multiphysics simulations, contact-impact problems, generations of meshes and partitions aligned with jumps in diffusion coefficients, -adaptive methods, and special discretizations in the neighborhood of singularities. In this paper, we use the inf-sup stable mixed elements in each subdomain. In the event of applying the mortar condition, it is necessary to know functions on the interface. However, the degrees of freedom (d.o.f) of Crouzeix-Raviart element are associated with the edge midpoints and functions on the interface depend on the nodal values corresponding to interface nodes and some subdomain interior nodes lying closest to the interface. We adopt nonstandard mortar condition introduced in [10] which is only associated with nodal values on the interface.

The inf-sup constant for mortar element space is crucial in the analysis of the approximation order. If the constant is independent of mesh size and subdomain size, then the optimal order of approximation follows independently of the number of subdomains and mesh size as in the case of elliptic problems.

The fundamental idea of this paper is the same as [4], with additional technical complications caused by nonmatching grids across subdomain interfaces and the Crouzeix-Raviart element discretization for velocity space. We propose a preconditioner and analyze the condition number bounded by , where and are mesh sizes.

The paper is organized as follows. In Section 2, a domain decomposition procedure is described for solving incompressible Stokes problem. In Section 3, the matrix form of the mortar condition is introduced. In Section 4, we derive an FETI-DP operator for the Stokes problem and propose a preconditioner for the FETI-DP operator. Section 5 gives several technical lemmas. Section 6 is devoted to the proof of the condition number of the preconditioned FETI-DP operator. The numerical tests are presented to verify the theoretical results in Section 7.

Throughout the paper, the following notations are used: , and mean that there exists positive constants and independent of the subdomain size and mesh size, such that

2. A Domain Decomposition Procedure for Stokes Problems

We consider the following two-dimensional incompressible Stokes problem:

where is a bounded polygonal domain in represents the velocity of fluid, is the pressure, and is the external force.

We first decompose into a nonoverlapping subdomain partition . We consider the subdomains to be geometrically conforming. With each subdomain , we assume that the triangulation is quasiuniform. Let be the mesh size parameter. Let denote the interface between the two neighboring subdomains and , and be the interface skeleton. We denote the sets of the Crouzeix-Raviart element nodes and the P1 conforming element nodes that are contained in , and by  , , ,  and ,  , , , respectively. The resulting triangulation can be nonmatching across the subdomain interface. We note that each interface inherits two independent triangulations. One side of associated with is defined as a mortar side denoted by , and the other side associated with is defined as a nonmortar side denoted by .

For each subdomain , we introduce the following spaces:

Then, the variational form of the problem (2) is as follows: find   , such that

Here, denotes the inner product in for .

We consider the inf-sup stable elements in each subdomain . Let     is  continuous  at  the  midpoints  of  the edges  of , and is  a  midpoint  on ,

where is a set of polynomials of degree less than or equal to in .

The Lagrange multiplier space corresponding to is defined as follows:

Then we take the Lagrange multiplier space

We introduce the nonstandard mortar condition (cf. [10]):

where is the orthogonal projection defined as . The operator is defined in the following. We can also write (8) as follows:

We use to denote a triangle edge. Let be the space of piecewise linear functions defined on the triangulation , and let be the triangulation obtained as a result of dividing each edge of into two equal segments. Let be the conforming space of piecewise linear continuous functions on the triangulation . The midpoint, left and right endpoints of each edge are denoted by , , and , respectively. The length of is denoted by .

We now define the operator . It is based on the definition of another interpolation operator .

Definition 1. For is defined by the nodal values as

Here, and are the left- and the right-neighboring edges of , respectively. represents a triangle edge of , touching , and is the corresponding neighboring edge.

Each component of the interpolation is done basically by first joining the neighboring edge midpoints using straight lines, and then simply extending the two end straight lines toward the end of the mortar (cf. Figure 1). The operator can now be defined using .

Figure 1: Here each component of is a (discontinuous) piecewise linear function shown (using solid lines) in the lower figure. The dashed lines (in both upper and lower figures) correspond to each component of . The solid lines in the upper figure correspond to each component of .

Definition 2. For is a vector whose components are piecewise linear functions on the edges of defined by its values at the two endpoints of each edge . If is an interior edge of , then

It is easy to see that, if is a boundary edge of , then for .

We define the following spaces:

where and are the spaces for velocity and pressure, respectively.

3. Matrix form of Mortar Conditions

For , where the symbols , , and represent the d.o.f corresponding to the interior nodes, nodes related to , and nodes related to the remaining parts of edges, respectively. We define the spaces , , and which consists of , , and , respectively.

Let be a matrix with entries

where is basis for and is basis for . Let be a matrix with entries

where is basis for and is basis for . Then we write (9) into a matrix form as

Let . The standard discrete Stokes harmonic extension operator   is defined by the following: given a velocity , find , and , such that on each subdomain ,

Define to be an extension operator by zero and for to be a restriction operator, and let . Then (15) becomes

where

Let and be matrices that consist of the columns of corresponding to the d.o.f related to and the d.o.f related to the remaining parts of the edges, respectively. Then, (17) can also be written into

where and , where is the restriction map that gives , for . We will use any of these two expressions (17) and (19) for the sake of convenience.

4. FETI-DP Method

4.1. FETI-DP Operator

In this section, we formulate a FETI-DP operator with the mortar condition (19).

We define the following spaces:

To solve the Stokes problem efficiently and correctly, we will consider the following primal constraints:

Note that (21) holds by replacing or in (9), because constant multipliers belong to the Lagrange multiplier space . These primal constraints were introduced by Li (cf. [4]). The primal constraints enlarge the size of the coarse problem, so that it may lead to a fast convergence of the FETI-DP iteration.

We rewrite (21) as

where the matrix is a Boolean matrix, the number of columns of the matrix equals to twice the number of , and the number of rows of the matrix equals to the d.o.f of the space . For , at each interior nodal point of , has two components corresponding to horizontal and vertical parts of velocity function. For means that for all , the sums of corresponding to the horizontal and vertical parts of velocity function are zero.

Let be the Lagrange multiplier space corresponding to the constraints (22), and for has two components that correspond to the constraints for horizontal and vertical velocity. By introducing Lagrange multipliers and to enforce the constraints (19) and (22), the following is induced from the Galerkin approximation to (4) as follows: find    such that

Here, is a stiffness matrix induced from , is a matrix induced from , and is a matrix induced from . Since is constant and for , we have by the divergence theorem.

Let

where is a primal variable, then (23) can be written as

In Section 5, we will show that the matrix is invertible (cf. Lemma 9). After eliminating , we obtain the following equations for and :

where

The matrix which is a coarse problem in the FETI-DP method is invertible (cf. Lemma 10). Hence by eliminating , we obtain the following equation for

where , and we call it the FETI-DP operator. Since we add the primal constraints to the coarse problem, is not uniquely determined in the space . We define a subspace of as follows:

The matrix is symmetric and positive definite (s.p.d) on , and the solution of (28) is uniquely determined in .

4.2. Preconditioner

For , we define by

where the subscript for a matrix denotes the submatrix corresponding to subdomain . Since the upper left matrix

represents the local Stokes problem with a Dirichlet boundary condition, it is invertible, so that the Schur complementary is well defined. We then assemble local Schur complementary matrices and define

and it can be seen easily that is an s.p.d operator on . Hence, we define

as a norm for . Here, denotes the -inner product of vectors.

We define the following spaces:

For a function , let be the zero extension of into . For we define an extension by

and define a norm on by

We introduce the following subspaces with the norms induced from the spaces and :

where denotes the outward unit normal vector on the subdomain boundary .

Recall the definition of in (29), and let be a duality pairing between and defined as

Then we define a dual norm for by

Now, we will find an operator which gives

and propose as a preconditioner for in (28). Define as a restriction operator and as an extension operator by zero. Then for ,

Let . Moreover, we have

where and is a matrix obtained from after deleting the columns corresponding to the d.o.f. of . Note that is invertible. We define the following -orthogonal projections:

For and , we have

where

Then it can be shown that the operators are invertible and is s.p.d on . Hence, using (46), the maximum in (41) occurs when satisfies . Therefore, we have

Let

be the preconditioner for the .

Define the -orthogonal projections

Then the projection operators and are composed of diagonal blocks of and , respectively. Moreover, it can be shown easily that

is invertible. Hence, it follows that

where , and we obtain

5. Technical Tools

In this section, we state and prove a few technical lemmas necessary for the proof of Theorem 17 of Section 6.

Let be the conforming space of piecewise linear continuous functions on the triangulation which is constructed by joining the midpoints of the edges of the elements of , and let be the subspace of consisting of functions with zero traces on .

For each open edge , we introduce a local equivalence map (isomorphism)    (cf. [11]), where is a subspace of formed by all vectors whose component are zero in . This mapping is a slightly modified version of a similar mapping introduced in [12].

Definition 3. For given , we introduce by defining the values of at the nodal points of the triangulation .

(i)For , let .(ii)For , then , where the sum is taken over all triangles with the common vertex , and is the number of elements with as an vertex.(iii)For , let .(iv)For , then , where are the left- and right-neighboring CR nodal points of and is the length of the segment with as its endpoints.

Note that each component of is piecewise linear between the CR nodes of , and on . The mapping has the following properties (cf. [11]):

Lemma 4 (see [13]). Let and in for a slave edge . Then we have

where is the discrete harmonic extension of .

Lemma 5 (see [11]). Let be a slave, then for any , we have

where is a discrete harmonic function taking the same vector values as those of at CR nodal points on and is equal to zero vector at the remaining CR nodal points on and .

We also need a special mortar operator defined over trace spaces: , where is a set of vectors whose components are continuous and equal to zero at the endpoints of and are piecewise linear over all segments that have their ends in . Each is uniquely determined by its values in .

Definition 6 (see [11]). Let be defined by

The and stability of is stated in the next lemma.

Lemma 7 (see [11]). For defined previously, we have

Lemma 8 (see [10]). Let be a discrete harmonic vector in , with at   . Then we have

Lemma 9 (see [14]). The matrix is invertible on .

Lemma 10 (see [14]). Assume that the domain has the triangulation to satisfy that

for the solution . Then the coarse problem matrix is invertible.

Remark 11. Most triangulations of the domain satisfy the assumption of Lemma 10 because the number of velocity unknowns and is usually greater than the number of unknowns and .

Lemma 12 (see [14]). We have .

Lemma 13 (see [14]). For , we have

Lemma 14 (see [15]). For any , we have

Proof. The proof follows from the definitions of and , (40), and a standard algebraic argument; see [16].

Definition 15 (see [17]). Given , we define an operator as follows:

(i)for , let ;(ii)for , let ;(iii) for , for any slave side , or if the edge midpoint is on or on any remaining master or slave side, let .

We define a projection operator by

where and are, respectively, the restriction of to the mortar side and the slave side of an interface .

Lemma 16. For all , we have

where and .

Proof. Let , and we have . Next we will estimate the term .

Note that can be nonzero only over and the neighboring subdomain that share a common edge with , such that the slave side is associated with . Let denote the set of indices of such subdomains. Thus

Then considering , we have with being a discrete harmonic function which equals to at the CR nodes of and to zero at all remaining nodes of . Thus, by Lemma 5,

Next we consider those for which , consequently the slave side of the edge is an edge of , and the mortar side is an edge of . Then by Lemma 4, we get

We also need to consider the function . From Definitions 3 and 15, we have

Thus we get

The pervious second term can be estimated as

by Lemmas 5 and 7 and the trace theorem. The first term is bounded using an inverse inequality, (68) and Definition 6 as follows:

Next, by using the fact that is an orthogonal projection, we get

Finally, the fact that yields

Applying the property of to the first term and Lemma 3.2 from [10] to the second term, Lemma 5 and the quasiuniform triangulation, we get

Finally, summing over all edges in and then over all subdomain ends the proof.

6. Condition Number Estimate

In this section, we give the condition number estimate of the preconditioned operator in the following main theorem of this paper.

Theorem 17. For any , it holds that

where and .

Proof. In the proof of this theorem we use results from this section and the algebraic arguments from [16, Section 6.4].

Lower Bound. From Lemma 13, the zero extension for , and (41), we obtain

Upper Bound. For any ,

Hence by Lemmas 13, 14, and 16, we conclude that

7. Numerical Tests

Consider the following Stokes problem:

where and is given by the exact solution of the problem (79) as follows:

Let denote the number of subdomains. We only consider the uniform partition of means that is partitioned into square subdomains. For all subdomains, we take the same number of nodes , including endpoints, in horizontal and vertical edges with for some positive integer . We solve (79) on both matching and nonmatching grids. For matching grids, we make uniform triangulations in each subdomain with nodes on horizontal and vertical edges of subdomain. For nonmatching grids, we take random quasi-uniform nodes on each horizontal and vertical edges of subdomain and generate nonuniform structured triangulations.

Now, we solve the FETI-DP operator with and without preconditioner varying and on both matching and nonmatching triangulations. The Conjugate Gradient (CG) iteration is stopped when the relative residual is less than .

In Tables 1–3, the number of CG iterations and the corresponding condition numbers are shown with varying and . In Table 1, and increase by double. On both matching and nonmatching grids, the preconditioner performs well and the condition numbers seem to behave -growth as increases. Especially on nonmatching grids, the preconditioner is much more efficient. In Tables 2 and 3, the CG iteration becomes stable as increases with and .

Table 1: CG iterations (condition numbers) when .

Matching
Nonmatching

Table 2: CG iterations (condition numbers) when .

Matching
Nonmatching

Table 3: CG iterations (condition numbers) when .

Matching
Nonmatching

Moreover, we have observed the convergence behaviors of the approximated solution. The and -errors for velocity and pressure are examined. and denote the approximated solutions for the velocity and pressure, and means the square root of . The errors are shown in Table 4 for various and with matching grids. Three cases are considered: when increases by double with , when increases by double in both edges of with and . For all cases, we can see that the -error for velocity -error for pressure reduce by half and -error for velocity reduces by quarter. For the finite elements , these convergence behaviors are optimal.

Table 4: - and -errors on matching grids.

For the case of nonmatching grids, the errors are shown in Tables 5–7 with various and . In Table 5, we observe that the error and reduce by half and the error reduces by quarter as increases by double with . When and , as increases, the errors also show the optimal convergence behaviors in Tables 6 and 7.

Table 5: - and -errors on nonmatching grids: .

Table 6: - and -errors on nonmatching grids: .

Table 7: - and -errors on nonmatching grids: .

If the inf-sup constant for the space is independent of and , then the optimality of approximation can be shown. Let and be the inf-sup constants for the space and the finite elements, respectively, and be the inf-sup constant for the space . Then the constant depends on and from the trick conceived by Boland and Nicolaides [18]. Hence, if the constant is independent of and , then the same holds for . In [19], for which is obtained from the Hood-Taylor finite elements, it was shown that the constant is independent of , but not shown for . Following the proofs in [19], we can obtain the same results for the space of the finite elements. We have no proof that is independent of . Instead, we compute the constant numerically as increases. The results are given in Table 8 both for matching and nonmatching grids when and . We observe that the constant becomes stable as increases. Table 9 gives the constant as increases with . This confirms that the constant is independent of .

Table 8: Inf-sup constant when and .

Nonmatching
Matching
Nonmatching
Matching

0.6613

Table 9: Inf-sup constant when .

Nonmatching
Matching

Acknowledgments

The author of this paper would like to express her gratitude to Professor Jinru Chen, for his guidance and encouragement, and to the reviewers, for valuable comments and suggestions. This work was supported by the National Science Foundation (NSF) of China (Grants No. 11071124 and 11226334) and the Program of Natural Science Research of Jiangsu Higher Education Institutions of China (Grant No. 12KJB110013).

References

H. H. Kim, “A FETI-DP formulation of three dimensional elasticity problems with mortar discretization,” SIAM Journal on Numerical Analysis, vol. 46, no. 5, pp. 2346–2370, 2008.

A. Klawonn, O. B. Widlund, and M. Dryja, “Dual-primal FETI methods for three-dimensional elliptic problems with heterogeneous coefficients,” SIAM Journal on Numerical Analysis, vol. 40, no. 1, pp. 159–179, 2002.

A. Klawonn and O. B. Widlund, “Dual-primal FETI methods for linear elasticity,” Communications on Pure and Applied Mathematics, vol. 59, no. 11, pp. 1523–1572, 2006.

J. Li, “A dual-primal FETI method for incompressible Stokes equations,” Numerische Mathematik, vol. 102, no. 2, pp. 257–275, 2005.

A. Klawonn and O. Rheinbach, “Inexact FETI-DP methods,” International Journal for Numerical Methods in Engineering, vol. 69, no. 2, pp. 284–307, 2007.

A. Klawonn, O. Rheinbach, and O. B. Widlund, “An analysis of a FETI-DP algorithm on irregular subdomains in the plane,” SIAM Journal on Numerical Analysis, vol. 46, no. 5, pp. 2484–2504, 2008.

J. Li and O. B. Widlund, “On the use of inexact subdomain solvers for BDDC algorithms,” Computer Methods in Applied Mechanics and Engineering, vol. 196, no. 8, pp. 1415–1428, 2007.

J. Li and O. Widlund, “BDDC algorithms for incompressible Stokes equations,” SIAM Journal on Numerical Analysis, vol. 44, no. 6, pp. 2432–2455, 2006.

H. H. Kim, M. Dryja, and O. B. Widlund, “A BDDC method for mortar discretizations using a transformation of basis,” SIAM Journal on Numerical Analysis, vol. 47, no. 1, pp. 136–157, 2008/09.

T. Rahman, P. Bjørstad, and X. J. Xu, “The Crouzeix-Raviart Finite element on nonmatching grids with an approximate mortar condition,” SIAM Journal on Numerical Analysis, vol. 46, no. 1, pp. 496–516, 2008.

L. Marcinkowski, “The mortar element method with locally nonconforming elements,” BIT Numerical Mathematics, vol. 39, no. 4, pp. 716–739, 1999.

M. Sarkis, “Nonstandard coarse spaces and Schwarz methods for elliptic problems with discontinuous coefficients using non-conforming elements,” Numerische Mathematik, vol. 77, no. 3, pp. 383–406, 1997.

L. Marcinkowski, “Additive Schwarz method for mortar discretization of elliptic problems with nonconforming finite elements,” BIT Numerical Mathematics, vol. 45, no. 2, pp. 375–394, 2005.

H. H. Kim and C.-O. Lee, “A Neumann-Dirichlet preconditioner for a FETI-DP formulation of the two-dimensional Stokes problem with mortar methods,” SIAM Journal on Scientific Computing, vol. 28, no. 3, pp. 1133–1152, 2006.

L. Marcinkowski, “A preconditioner for a FETI-DP method for mortar element discretization of a 4th order problem in 2D,” Electronic Transactions on Numerical Analysis, vol. 38, pp. 1–16, 2011.

A. Toselli and O. B. Widlund, Domain Decomposition Methodsalgorithms and Theory, vol. 34 of Computational Mathematics, Springer, Berlin, Germany, 2005.

L. Marcinkowski and T. Rahman, “Neumann-Neumann algorithms for a mortar Crouzeix-Raviart element for 2nd order elliptic problems,” BIT Numerical Mathematics, vol. 48, no. 3, pp. 607–626, 2008.

J. M. Boland and R. A. Nicolaides, “Stability of finite elements under divergence constraints,” SIAM Journal on Numerical Analysis, vol. 20, no. 4, pp. 722–731, 1983.

F. B. Belgacem, “The mixed mortar finite element method for the incompressible Stokes problem: convergence analysis,” SIAM Journal on Numerical Analysis, vol. 37, no. 4, pp. 1085–1100, 2000.