Raviart-Thomas elements, which we consider, one can perform iterations in a ... A simple choice for M, M = wI, was made in Lu, Allend and Ewing [11], and .... given only in an implicit way); and (c) the "lumped mass" system of (iii) is given in an.
Preconditioning indefinite systems arising from mixed finite element discretization of second-order elliptic problems Richard E. Ewing, Raytcho D. Lazarov*, Peng Lu, Panayot S. Vassilevski* Department of Mathematics, Box 3036, University Station University of Wyoming, Laramie, Wyoming 82071, USA
Abstract We discuss certain preconditioning techniques for solving indefinite linear systems of equations arising from mixed finite element discretizations of elliptic equations of second order. The techniques are based on various approximations of the mass matrix, say, by simply lumping it to be diagonal or by constructing a diagonal matrix assembled of properly scaled lumped element mass matrices. We outline two possible alternatives for preconditioning. One can precondition the original (indefinite) system by some indefinite matrix and hence use either a stationary iterative method or a generalized conjugate gradient type method. Alternatively as in the particular case of rectangular Raviart-Thomas elements, which we consider, one can perform iterations in a subspace, eliminating the velocity unknowns and then considering the corresponding reduced system which is elliptic. So we can use the ordinary preconditioned conjugate gradient method and any known preconditioner (of optimal order, for example, like the multigrid method) for the corresponding finite element discretization of the elliptic problem. Numerical experiments for some of the proposed iterative methods are presented. K e y w o r d s : indefinite system, preconditioning, iterations in subspace, conjugate gradients, mixed finite elements, second order elliptic problems. S u b j e c t Classifications: AMS(MOS) 65F10, 65N20, 65N30.
* On leave from Center for Inforlnatics and Computer Technology, Bulgarian Academy of Sciences, G. Bontchev str., bl. 25-A, 1113 Sofia, Bulgaria
29
1. I n t r o d u c t i o n In this paper, we consider indefinite linear systems of equations that arise in the mixed finite element discretization of second-order elliptic problems. In general, this approximation leads to systems of the form
(1)
AX
( M
0)
N
V
where M is a symmetric, positive definite, and constantly-conditioned matrix and A is nonsingular. In the existing literature, one can find various approaches for solving such systems. For example, in [6], Bramble and Pasciak have constructed a preconditioner for a properly transformed form of the original system where the resulting matrix is definite, and the preconditioned conjugate gradient method is then applied. Another approach was proposed and studied in Bank, Welfert and Yserentant [5], where a parameter-free iterative method (but not of conjugate gradient type) for solving the original system is used. However, at every iteration, one uses a variable-step preconditioner, which involves approximate solving of systems with M and approximate solving (possibly with conjugate gradient-method) of the approximate reduced system (2)
N T . ~ - I N P = F,
where M is an approximation to M. A general approach of parameter-free preconditioned conjugate gradient (PCG) type methods for solving problem (1) was proposed recently in Axelsson and Vassilevski
[4]. For certain applications, the cost of evaluating M -1 applied to a vector is not too expensive. This is true for tensor product Raviart-Thomas elements on a regular rectangular grid, where the action of M -1 involves banded solution (of bandwidth proportional to the degree r of the piece-wise polynomials used in the finite element spaces). For such situations one can consider the (exactly) reduced system (3)
N T M - 1 N P = F,
which has a symmetric and positive definite matrix. In this case in Bramble, Pasciak and Xu [7], an optimal order V-cycle multigrid method has recently been proposed. In the present paper we choose a preconditioner B for the original matrix A of the following form:
where D - N T M - 1 N - C, and the matrices ~ r and C (or 29) have to be specified.
30
We concentrate mai.nly on the case D = 0, that is exact solution of the perturbed reduced system (2). Approximate solving of (2) (that is C • N T M - 1 N , or equivalently D / t 0) can be considered in the framework of the papers of Bank, Welfert and Yserentant [5], Axelsson and Vassilevski [4] or Ewing and Wheeler [8]. A simple choice for M , M = w I , was made in Lu, Allend and Ewing [11], and the parameter was chosen globally in order to minimize the norm of the iteration matrix I - B - 1 A in a stationary iterative method. Then, the system with the matrix = w - l N T N was solved using a multigrid method for the Poisson equation. As shown in [11], for rapidly varying coefficients of the differential equation, this iterative method may converge very slowly although being independent of the mesh parameters. In this paper, we advocate another diagonal approximation M to M, which is obtained by assembling properly scaled lumped element mass matrices. Then the resulting reduced system (2) is solved by inner iterations. We give a precise upper bound of 1/2 for the spectral radius of the iteration matrix I - B -1A. We also prove that the eigenvalues of B - 1 A are real and positive and that they are dominated by the eigenvalues of M - 1 M . Hence the generalized conjugate gradient method from Axelsson [2] will have a convergence factor bounded by ( x / ~ - l ) / ( v ~ + l ) , where x is the condition number of B - 1 A . We consider this to be the main result of this paper. The remainder of the paper is organized as follows. In Section 2, we formulate the mixed finite element discretization of second-order elliptic problems using RaviartThomas elements. In the next section, we give some properties of the finite element matrix. In Section 4, the preconditioning technique sketched above is described and the corresponding main results are proved. At the end, we present some numerical experiments using the proposed preconditioner in a stationary iterative procedure.
2. Problem
Formulation
Consider the following equation governing steady state heat transfer in a nonhomogeneous d-dimensional conducting medium: (5)
= / in i,j=l
We suppose that the matrix K = {kij(x)}i,j--1 .....d is symmetric and uniformly positive definite in f~; i.e., there exist constants al, a0 > 0, such that
Therefore, introducing the vector (7)
u = -KVp,
0p
0p
where Vp = ( \ 0xl""'0xd}
T
'
31
equation (5) reads as (8)
- d i v u = f in ~2.
For simplicity, the domain ~ is taken to be the unit hypercube in Ra; i.e. ~ / = [0,1] a. We impose the boundary condition p = 0 on 0~. In the context of fluid of a single phase in a d-dimensional porous medium, u,p, and f represent the Darcy velocity, pressure, and source term, respectively. Problem (7) and (8) can be formulated as a saddle point problem. Let us first introduce the vector space V = {v = ( v l , . . . , v ~ ) : vi • L~(O), div v • L2(a)} and endow it with the norm
Ilvll~
--
~ = Ilvll~=¢n) +lldiv vllL,(~). IlvllH¢div;.) =
For u , v • V and p,w • L 2, we define the bilinear forms (9)
(a)
a ( u , v ) = ( K - l u , v),
(b)
b(u,w) = (div u,w).
Then problem (7) and (8) is equivalent to solving the saddle point problem given by
a ( u , ~ ) - b(~,p) = 0, (10)
w • v,
-b(u,w) = -(f,w),
Vw • L 2.
Let Th be a partitioning of the domain ~ by parallelipiped finite elements and V~ and W~ be the Raviart-Thomas finite element spaces, which are subspaces of V and L 2, respectively. The spaces V[, and W~ are usually referred to as Raviart-Thomas spaces of order r, [12], for v > 0. A Lagrangian basis for these spaces that is useful for implementation is constructed using a tensor product of Gaussian and Lobatto points. An example of sets of such points in the reference element E = [-1,1] d is shown in Figure 1 (r = 0, d = 2 and r = 1, d = 2).
Now, we formulate the mixed finite element approximation to problem (10). Find U E V~ and P E W/~ satisfying: a(U, V) - b(V, P ) = 0,
(11)
- b ( v , w) = - ( f , w),
V E V~
~ • wL
Since The Raviart-Thomas spaces satisfy the Babuska-Brezzi condition [9,12], the finite element solution (U, P) exists and converges in V x L2-norm to the exact solution with a rate O(h"+l), [9].
32
s
~i
s
×
o
D
O - -
r=l
r=O x - U,,
o -P
[] - U 2 ,
Figure 1
Raviart-Thomas elements, r = 0 and r = 1
3. P r o p e r t i e s of t h e F i n i t e E l e m e n t M a t r i x We concentrate our attention on methods for solving the corresponding system of linear equations for the unknown values of U and P at the grid points. We introduce the same notation for these vectors: U, vector of the values of U and P, the values of P. Then, (11) can be presented in the algebraic system (12)
M U + N P = O, N T u = -F,
where the matrices M and N are defined by the bilinear forms a(., .) and b(., .), respectively. We now consider the case d = 2. If we partition the unknown values of U as U = (U1T, uT) T, then the matrix M will have the following 2 × 2 block form
( M11M12 ) (13)
M=
M12
M22
"
In the particular case when kn(x) - k21(x) =- 0, we have
M=(
MnO M220) .
(14)
Then Mii, i = 1,2 will have a block diagonal structure, and one can easily eliminate the velocity vector U from system (12). We obtain (15)
N T M - 1 N P = F.
33 One can now see that the matrix of the reduced system (15) is symmetric and positive definite. In general, this is an approximation of Equation (5) that has quite a strange global stencil (see Figure 2).
@ •
•
•
@
@
•
•
@
Figure 2 Stencil f o r the pressure equation, r = O.
The matrix M is obtained using exact computation (never done in practice for variable coefficients) or quadrature rules that are exact for polynomials of degree 2r + 3 in each variable. If we use Lobatto quadrature in the xl direction with r + 2 nodes and Gauss quadrature in the remaining directions for computing Mii, then we obtain a diagonal matrix Mii; i.e.,
(16)
(]galu1
y l ) r,~
yT~iiVl
In this case, the system (15) is an immediate consequence of our computational procedure. In the case r -- 0, this is a 5-point approximation for the pressure equation. Therefore, we have the following three cases: (i) the general system (12) with a "mass" matrix M in the form (13); (ii) the system (12) with a consistent "mass" matrix M in the form (14) where Mii, i = 1,2, have block diagonal form with banded blocks and are therefore easily (rather effectively) reducible to the system (15); and (iii) the "lumped mass' system, written in the form (17)
A M U + N P = O, NTu
-- - F ,
which yields N T M - 1 N P = F. It is important to mention that: (a) in general, the system of case (i) cannot be reduced to the form (15) in an effective way; (b) the reduction of system (12) with
34 a matrix M of the form (14) can be performed in an effective way (the arithmetric operations are proportional to the number of unknowns but the m a t r i x N T M - 1 N is given only in an implicit way); and (c) the "lumped mass" system of (iii) is given in an explicit way and can be obtained using an element by element assembling procedure. Further, we discuss methods for solving the corresponding system (12), arising from the mixed finite element approximation of the original problem (10). We now discuss various preconditioning techniques for their solution.
4. P r e c o n d i t i o n i n g of t h e M i x e d F i n i t e E l e m e n t Systems We first consider the case of "lumped mass" approximation: the system can be written in the form (17), where the matrix N T _ ~ - I N is known explicitly. For the lowest order finite elements, r = 0, this is a 5-point approximation of the elliptic equation, and here we can use some of the known fast solvers. For example, a multigrid m e t h o d will be suitable even when the coefficients are piecwise constant or piecewise continuous. Therefore, we can assume that problem (17) can be solved in an efficient way. For the elements of higher order (r > 0), there is no obvious fast solver for the reduced system (17). Probably, one can design a multigrid m e t h o d for this system or use some preconditioning m e t h o d with preconditioners based on the approximations of lower order. This approach needs further investigation and computational experiments. Anyway, we may suppose that we have constructed a fast solver for the problem (17), for any r, in a rectangular domain. Further, we use this solver to design iterative methods for the general problem (12). In the general preconditioning matrix B, we can distinguish two cases: (a) M -- M and (b) M ~ M . T h e first case, M = M , is easier to analyze since all iterations are performed in the subspace of vectors satisfying the condition M U + N P -- 0 and where the matrix A is definite. In many cases, the realization of this approach can be much more difficult since we need to evaluate the action of M -1. In our case of tensor product elements on a refular rectangular grid, this evaluation involves banded solvers (of bandwidth proportional to r) along the lines 0 < xi < 1 for the velocity component Ui and therefore can be done with arithmetic work proportional to the number of unknowns. In the case of triangular elements, one might consider iterative evaluation of M -1, which is a well-conditioned matrix. Since A is a symmetric matrix in this subspace, we can use the preconditioned conjugate gradient method. In order to b e t t e r explain the preconditioner in this case, we present matrix A in a factored form: A =
NT
0
=
N T
-NTM-1N
I
"
35 Here C = - N T M - 1 N is the Schur complement of A after elimination of the velocity vector U. Since solving the system with matrix C is a difficult problem (for arbitrary r), we could solve it by preconditioning. This will produce a preconditioner B, where M = M in (4). In this case, we have a variety of possibilities of approximating C by C, or equivalently approximating M -I. For example, we may choose C = N T M - 1 N , where . ~ is the lumped mass matrix M or M is the diagonal part of M. These two choices are particular cases of the general approach of replacing M by some variant of its incomplete factorization [1]. In all cases considered, matrix C is spectrally equivalent to A (in the subspace M U + N P = 0) with constants independent of the problem size. Thus, this preconditioner B is an optimal one. Nevertheless, in this case we still relay on the existence of an effective solver of the problem C P = F where the condition number of is O(h-2). This can be done by a multigrid method [7] or an algebraic multilevel method [3], for instance. The second case, M # M, is much more general, since we are predonditioning the original matrix A and need not exact solving with M. It can be applied for general differential equations and mixed approximations of type (12). Here we must overcome two main difficulties: (i) to choose a matrix M such that it scales M in a proper way and (ii) to perform the convergence analysis of the iterative procedure; in this case, the spectrum of B - ~ A is complex in general (if D # 0) and difficult to analyze. This approach was used by Lu; Allen and Ewing [11] with 2~r = wI and L) = O, where the choice of the iteration parameter w is based on the properties of M. The computations performed in [11] showed that the iterative method slows down substantially when maxftkii(x)/n~n kii(x ) >> 1. This means that I does not scale the matrix M in a proper way. A better choice for M is a diagonal matrix obtained from assembling properly scaled lumped element mass matrices, which we describe later. In the analysis below we treat the equation (5) without mixed derivatives, that is M12 = M21 = 0, and kll(x) = k22(x) = k(x), and consider only the case 5 = 0. We have the following main result: T h e o r e m 1. The eigenvalues of B - 1 A are real and positive and lie in the interval [Ami,, A~x] where Ami,, A~x are the extreme eigenvalues of M - 1 M .
Proof. Consider the following generalized eigenvalue problem A(U)=
~B ( U ) ,
(see (4)with D = 0 ) ,
or equivalently, (M -
08)
,M)U +
=
N T u = )~NTU.
36
Hence either A = 1 or N T u = O. If A = 1, then ( M - M ) U = 0; that is A -- 1 is also an eigenvalue of M-aM. Now let N T u = O. Then multiplying the first equation in (18) by U *T (the complex conjugate transpose of U) we get U*r(M
-
+
=
That is u * T ( M -- AM)U = 0 since N T u = 0. Hence A is real and positive and since U*TMU )~ =
~
U*TMU
E ['~min, )~rnax],
•min, ,'~max are the extreme eigenvalues of M-1M, the proof is complete. C o r o l l a r y 1. The spectral radius of I - B-1A is bounded by the spectral radius of I - M - ~ M and cond(B-1A) < cond(M-1M).
Proof. Follows directly from Theorem 1. C o r o l l a r y 2. The generalized conjugate gradient method from Axelsson [2] applied to the preconditioned system
B-1A(U) = B-I(OF), will h a v e a convergence factor bounded by ( v r ~ - l ) / ( v ~ + l ) , where x = cond(B -1 A) _< cond(M-aM).
Proof. Follows directly from [2] and corollary 1. Let us consider the stationary iteration method
(19)
BX(k+ ~) = @+ ( B - A ) X (k)
where ~. p(k) ) ,
q~ =
and X (°) is given.
It is well known that the convergence rate of this algorithm is bounded by the spectral radius of the matrix [ - B - 1 A , denoted by p(I-B-1A). The Corollary 1 tells us that to increase the convergence speed of the iterations, it suffices to make p(I-iW--1M) as small as possible. Taking into account the construction of the matrices M and M, we get (20)
P(I-'M-IM)=p(I-(
~ - 1 0 0 M2~l) ( M110 d 2 2 ) )
= max{p(I-M~XMll ), p(I-M~lM22)}.
37
p(I-'M~' MI ) = p(I-M-~'/2MII M~I/2) (21) ~-
m a x
u,#o
--
UITMllU1
and similarly,
(22)
l ~V~rM~2 V2] 2~
p(X-~lM~2) = max 1 ~o
We start now the construction of the preconditioning matrix based on the local assembly procedure by optimal scaling of the lumped element mass matrices. Let us denote by U~, i = 1, 2 the vector of the values of the velocity function Ui(x) at the points of the element e E Th. Then, 2
2
l u ~ dx = i=1
U~'M~U:. "~
Here, M~i are the element mass matrices. The global mass matrix is an assembly of M~ over all e E Th, i.e.
U T M U -- ~ ae(U,U). eETh The lumped mass approximation M to M is obtained, when instead of exact evaluation of the integrals in (23), one uses a trapezoidal rule in xl and a midpoint rule in x2. Let x = (xl,x2) be the center of the finite element e E Th. Then by definition (24)
(25)
eta"
"
I
"v,'
~
~
t~2"2 (xl,x2
U1 MIIU, - 2k~1 I(xl -1/2h'x2)-{--~"
U~ M~2U~ - 2k"2
(xl + l/2h, x2)
h2u2]
- 1/2h) 4- ~-~ 2 (xl,x2 + 1/2h)
i
Then the matrix M is defined by scaling the matrix M element by element as follows: (26)
uT~ U = ~
~ ~ . -M~IU1 -o o H-w2U ~ ~~T---o ~ (wlU1 M22U~),
eETh e ~ e e i.e. M~I = wlMll and M~2 = w2M~2 where the scaling parameters w~ and w~ have to be chosen in such way that p ( I - i ~ - ~ M ) is minimal.
38 Now, we show how to choose these parameters. In order to simplify our notations, we consider the choice of w~ and denote M~I by Me, M~I by Mr, M~I by Mr, U~ by U~, and w[ by w~. Then by (24),
. . ( o H o) (27)
h2 2k(xl-1/2h, z2)'
h2
t~22 =
2k(x1+l/2h, x2)'
and by (23),
Me.=_ (m,l
m12"~.
m22 f
\m21
Now taking into account the presentation of Mll by (23) and Mll by (26) (remember that below Mr1 is denoted by Mr and Mr1 by Me), we have
(28)
I
UTMllU1I
1
uT~I--------~IU1 =
1 Ee UTMeU¢
< max
1
~, UTM~..Ue - eeTh
e
VrMeUe UTMeUe
Combining (21) and (28), we conclude that
(29)
p(I-.~IMll)
< maxp(I-M~-iMe).
-- eETh
From (29), we see that if we choose the local parameter we such that p(I-M~-lMe) reaches minimum, then p(I-M~'-'l-llMll) will also be near its minimum. We can use the same technique to optimize the spectral radius p ( I - . ~ l M 2 2 ) . Thus we have the following result. T h e o r e m 2. Let 0 < A~ < A~ be the eigenvalues of matrix M[-1Me for each element e. If we choose local parameter we on the element e to be
(30)
A~ _ A~ + A~ 2 '
then the iteration method (19) is convergent and has a convergence rate independent of the mesh size h. The convergence speed is dominated by the inequality (31)
p ( I - B - 1 A ) < max A~ - A~ -
e
+
Proof. By the assumption, _~[1M e has eigenvalues A~ and A~, then, the matrix I-M~-I M, = I-wT1M[1Me has eigenvalues 1-xA We
and
We
39
SO
P(I-M[lMe)=max( 1-A~we '11- ~'~we)~" When we = xi+x~2, we have
(32)
m < l .
Now we show that, for any other values,
~ + ~" If not, there exists an w > 0 such that
(33)
~ + A~
and (34) Equation (33) implies
(35)
2
1
2A~
oJ
(A~+~Xo)A o
.
..
Similarly, (34) implies 2A~
(36)
1
2
AI(A 1 . *+A~) < w-- < ~'A~+A~
From (35) and (36), we get
~ +~
Ai + ~
2
2
which is a contradiction. So (29) and (32) imply that (31) holds; thus, (19) is convergent. Finally, we show that the convergence rate is independent of the mesh size h. It is easy to show that A~ and A~ are the roots of the characteristic equation (Me are symmetric),
A2- (mn + - ~ \ trt, 11
m 2 2 ,]
A + mnm22^ Fr~llm22
^m~12^ -~ 0, m l 1 ~'~22
40
which are
+
^
-
m22
+
\roll
m22]
"4- m 1 1 m 2 2 ]
m22/
+ ~11---~22/
,
(37)
a;=g Lm~-~+ m22 ^
+
\~tll
+
so, (30), (31), and (37) imply that
we
=
-~ \ rnll
m22/
and
(3s)
( ( m l ' _ ~ n : : ) 2 . t _ 4 m b ~1/2 p(i_~[_lMe ) = \,,m,, mum,,/ ml 1
t'n22 ]
If we use the Simpson quadrature rule for evaluating the integrals in (23), we get the following approximate values for the elements of matrix M~ (for e E Th with center x = (x~, x2)), which for smooth k(x) are very close to the exactly computed ones:
(39)
) h2 m12 = m21 --
1
6 ~(Xl, X2)
Substituting (39) into (38) and taking into account the definition of ~ 1 (27), we get
(40)
and ~22 by
k(Xl - 1/2h, x2) + k(xi + 1/2h, x2) p it i _ , ,~, - l ~,r wej~ = k l ( x l - 1 / 2 h , x2) + 2k(xl,X2) + k(xl + l / 2 h , x2)
