2 ZHANGXIN CHEN AND PETER OSWALD Independently, the NR Q elements have been derived within the framework of mixed nite element methods [, ]. It has be

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1 Series Logo Volume 00, 9xx MULTIGRID AND MULTILEVEL METHODS FOR NONCONFORMING ROTATED Q ELEMENTS ZHANGXIN CHEN AND PETER OSWALD Abstract. In this paper we systematically study multigrid algorithms and multilevel preconditioners for discretizations of second-order elliptic problems using nonconforming rotated Q nite elements. We rst derive optimal results for the W-cycle and variable V-cycle multigrid algorithms we prove that the W-cycle algorithm with a suciently large number of smoothing steps converges in the energy norm at a rate which is independent of grid number levels, and that the variable V-cycle algorithm provides a preconditioner with a condition number which is bounded independently of the number of grid levels. In the case of constant coecients, the optimal convergence property of the W- cycle algorithm is shown with any number of smoothing steps. Then we obtain suboptimal results for multilevel additive and multiplicative Schwarz methods and their related V-cycle multigrid algorithms we show that these methods generate preconditioners with a condition number which can be bounded at least by the number of grid levels. Also, we consider the problem of switching the present discretizations to spectrally equivalent discretizations for which optimal preconditioners already exist. Finally, the numerical experiments carried out here complement these theories.. INTRODUCTION In recent years there has been analyses and applications of the nonconforming rotated (NR) Q nite elements for the numerical solution of partial dierential problems. These nonconforming rectangular elements were rst proposed and analyzed in [23] for numerically solving the Stokes problem they are the simplest divergence-free nonconforming elements on rectangles (respectively, rectangular parallelepipeds). Then they were used to simulate the deformation of martensitic crystals with microstructure [7] due to their simplicity. Conforming nite element methods can be used to approximate the microstructure with layers which are oriented with respect to meshes, while nonconforming nite element methods allow the microstructure to be approximated on meshes which are not aligned with the microstructure (see, e.g., [7] for the references). 99 Mathematics Subject Classication. Primary 65N30, 65N22, 65F0. Key words and phrases. Finite elements, mixed methods, nonconforming methods, multigrid methods, multilevel preconditioners, dierential problems. c0000 American Mathematical Society /00 $.00 + $.25 per page

2 2 ZHANGXIN CHEN AND PETER OSWALD Independently, the NR Q elements have been derived within the framework of mixed nite element methods [, ]. It has been shown that the nonconforming method using these elements is equivalent to the mixed method exploiting the lowest-order Raviart-Thomas mixed elements on rectangles (respectively, rectangular parallelepipeds) [24]. Based on this equivalence theory, both the NR Q and the Raviart-Thomas mixed methods have been applied to model semiconductor devices [] they have been eectively employed to compute the electric potential equation with a doping prole which has a sharp junction. Error estimates of the NR Q elements can be derived by the classical nite element analysis [23, 6]. They can be also obtained from the known results on the mixed method based on the equivalence between these two methods []. It has been shown that the so-called \nonparametric" rotated Q elements produce optimal-order error estimates. As a special case of the nonparametric families, the optimal-order errors can be obtained for partitions into rectangles (respectively, rectangular parallelepipeds) oriented along the coordinate axes. Finally, in the case of cubic triangulations, superconvergence results can be obtained [, 6]. Unlike the simplest triangular nonconforming elements, i.e., the nonconforming P elements, the NR Q elements do not have any reasonable conforming subspace. Consequently, there are dierences between these two types of nonconforming elements. The NR Q elements can be dened on rectangles (respectively, rectangular parallelepipeds) with degrees of freedom given by the values at the midpoints of edges of the rectangles (respectively, the centers of faces of the rectangular parallelepipeds), or by the averages over the edges of the rectangles (respectively, the faces of the rectangular parallelepipeds). While these two versions lead to the same denition for the nonconforming P elements, they can produce very dierent results in terms of implementation for the NR Q elements. With the second version of the NR Q elements, we are able to prove all the theoretical results for the multigrid algorithms and multilevel additive and multiplicative Schwarz methods considered in this paper. However, we are unable to obtain these results with their rst version. In particular, as numerical tests in [22] indicate, the energy norm of the iterates of the usual intergrid transfer operators, which enters both upper and lower bounds for the condition number of preconditioned systems, deteriorates with the number of grid levels for the rst version. But it is bounded independently of the number ofgridlevels for the second version, as shown here. The other major dierence between the nonconforming P and the NR Q elements is that the former contains the conforming P elements, while the latter does not contain any reasonable conforming subspace, as mentioned above. As a result of this, the convergence of the standard V-cycle algorithm for the nonconforming P elements can be shown when the coarse-grid correction steps of this algorithm are established on the conforming P spaces [8, 2]. But this is not the case for the NR Q elements. On the other hand, within the context of the nonconforming methods, i.e., when the coarse-grid correction steps are dened on the nonconforming P spaces themselves, the convergence of the V-cycle algorithm has not been shown, and the W-cycle algorithm has been proven to converge only under the assumption that the number of smoothing steps is suciently large [7, 8, 3, 4, 25,, 2, 4]. However, we are here able to show the convergence of the W-cycle algorithm with anynumber of smoothing steps for the Laplace equation using the NR Q elements. This optimal property cannot be proven for the nonconforming P elements using the present techniques.

3 MULTIGRID AND MULTILEVEL METHODS FOR NONCONFORMING ELEMENTS 3 The multigrid algorithms for the NR Q elements were rst developed and analyzed in [], and further discussed in [2] and [9]. The second version of these elements was used in [] and [2], while their rst version was exploited in [9]. Moreover, the analysis in [9] was given for elliptic boundary value problems which are not required to have full elliptic regularity. However, in all these three papers, only the W-cycle algorithm with a suciently large number of smoothing steps was shown to converge using the standard proof of convergence of multigrid algorithms for conforming nite element methods [2]. We nally mention that the study of the NR Q elements in the context of domain decomposition methods has been given in [3]. In this paper we systematically study multigrid algorithms and multilevel preconditioners for discretizations of second-order elliptic problems using the NR Q elements. We rst consider the convergence of the W-cycle and variable V-cycle algorithms for these nonconforming elements. WeprovethattheW-cycle algorithm with a suciently large number of smoothing steps converges in the energy norm at a rate which is independent of grid number levels, and that the variable V-cycle algorithm provides a preconditioner with a condition number which is bounded independently of the number of grid levels. A main observation in this paper is that the optimal convergence property for the W-cycle algorithm holds with any number of smoothing steps, when the coecient of the dierential problems is constant. Explicit bounds for the convergence rate and condition number are given. The NR Q elements has so far been the rst type of nonconforming elements which are shown to possess this feature for the W-cycle algorithm with any number of smoothing steps. We then study multilevel preconditioners of hierarchical basis and BPX type [5] for the NR Q elements. Wedevelop a convergence theory for the multilevel additive and multiplicative Schwarz methods and their related V-cycle algorithms. We follow the general theory introduced in [22] where the analysis of the hierarchical basis and BPX type for nonconforming discretizations of partial dierential equations was carried out. A key ingredient in the analysis is to control the energy norm growth of the iterated coarse-to-ne grid operators, which enters both upper and lower bounds for the condition number of preconditioned systems as outlined above. So far, the energy norm of the iterated intergrid transfer operators has been shown to be bounded independently of grid levels solely for the nonconforming P elements [9]. In this paper we prove this property forthenrq elements. Based on the present theory,we derive a suboptimal result for the multilevel preconditioners of hierarchical basis and BPX type for the NR Q elements. Finally, we study the problem of switching the NR Q discretization system to a spectrally equivalent discretization system for which optimal preconditioners are already available. This switching strategy has been used in the setting of the multilevel additive Schwarz method see [2] for the references. After we nd a spectrally equivalent reference discretization for the NR Q system, we are able to obtain optimal preconditioner results for the NR Q elements. Thanks to the equivalence between the rotated Q nonconforming method and the lowest-order Raviart-Thomas mixed rectangular method, all the results derived here carry over directly to the latter method [, 2]. For technical reasons, all the results in this paper are shown for partitions into uniform squares (respectively, cubes). They can be extended to triangulations in which the nest triangulation can be mapped to a square (respectively, cubic)

4 4 ZHANGXIN CHEN AND PETER OSWALD triangulation in an ane-invariant fashion. Also, the analysis is given for the twodimensional domain an extension to three space dimensions is straightforward for most of the results below. The rest of the paper is organized as follows. In the next section we prove some preliminary results for the intergrid transfer operators. Then the multigrid algorithms and multilevel preconditioners are discussed in x3 and x4, respectively. The problem of switching to a spectrally equivalent discretization is considered in x5. Finally, the numerical results presented in x6 complement the present theories. 2. PRELIMINARY RESULTS For expositional convenience, let = (0 ) 2 be the unit square, and let H s () and L 2 () = H 0 () be the usual Sobolev spaces with the norm jjvjj s = Z X jjs jd vj 2 dx where s is a nonnegative integer. Also, let ( ) denote the L 2 () or (L 2 ()) 2 inner product, as appropriate. The L 2 () norm is indicated by jjjj. Finally, A =2 H 0 () = fv 2 H () : vj ; =0g where ; Let h and E h = E be given, where E h is a partition of into uniform squares with length h 0 and oriented along the coordinate axes. For each integer 2 k K, let h k =2 ;k h and E hk = E k be constructed by connecting the midpoints of the edges of the squares in E k;, and let E h = E K be the nest grid. Also, k be the set of all interior edges in E k. In this and the following sections, we replace subscript h k simply by subscript k. For each k, weintroduce the rotated Q nonconforming space V k = v 2 L 2 () : vj E = a E + a2 E x + a3 E y + a4 E (x2 ; y 2 ) a i E 2 IR 8E 2E k if E and E 2 share an edge e, then and j ;ds =0 Note that V k 6 H0 () and V k; 6 V k, k 2. We introduce the space ^V k = V l V k l= the discrete energy scalar product on ^Vk H 0 () by kx : Z e ds = Z e ds (v w) E k = X E2E k (rv rw) E v w 2 ^Vk H 0 () and the discrete norm on ^Vk H0 () by q jjvjj E k = (v v) E k v 2 ^Vk H 0 ():

5 MULTIGRID AND MULTILEVEL METHODS FOR NONCONFORMING ELEMENTS 5 We introduce two sets of intergrid transfer operators I k : V k;! V k and P k; : V k! V k; as follows. Following [, 2], if v 2 V k; and e is an edge of a square in E k,theni k v 2 V k is dened by Z I k vds = jej e 8 >< >: 0 if Z vds if e for any E 2E k; jej Ze 2jej e (vj E + vj E2 )ds if 2 for some E E 2 2E k; : If v 2 V k and e is an edge of an element in@e k;,thenp k; v 2 V k; is given by Z P k; vds = Z Z vds + vds jej e 2 je j e je 2 j e2 where e and e 2 k form the edge e k;. Note that the denition of P k; automatically preserves the zero average values on boundary edges. Also, it can be seen that (2:) P k; I k v = v v 2 V k; k : That is, P k; I k is the identity operator Id k; on V k;. This relation is not satised when the NR Q elements are dened with degrees of freedom given by thevalues at the midpoints of edges of elements. We also dene the iterates of I k and P k; by R K k = I K I k+ : V k! V K Q K k = P k P K; : V K! V k : Finally, we make the convention on the discrete energy scalar product on the space ^V K : (v w) E =(v w) E K v w 2 ^VK : Obviously, we have the inverse inequality (2:2) jjvjj E C2 k jjvjj v 2 ^Vk k K (here and later, by C, c,... we denote generic positive constants which are independent ofk). In this section we collect some basic properties of the intergrid transfer operators P k; (respectively, I k ) and their iterates Q K k (respectively, RK k ). The crucial results are the boundedness of the operators I k with constant p 2 and the uniform boundedness of the operators Rk K with respect to the discrete energy norm jj jj E. Lemma 2.. It holds that P k; (2 k K) is an orthogonal projection with respect to the energy scalar product i.e., for any v 2 V k, (2:3) (v ; P k; v w) E =0 8w 2 V k; jjvjj 2 E = jjv ; P k;vjj 2 E + jjp k;vjj 2 E : Moreover, there are constants C and c, independent of v, such that the dierence ^v = v ; P k; v 2 ^Vk satises (2:4) c2 k jj^vjj jj^vjj E C2 k jj^vjj:

6 6 ZHANGXIN CHEN AND PETER OSWALD Proof. For any E 2 E k; with the four subsquares E i 2 E k (i = ::: 4, see Figure ), an application of Green's formula implies that (2:5) P 4 (r[v ; P k; v] rw) E = P i= (r[v ; P k;v] rw) Ei 4 P = 4 i= j j E i Ei Re j (v ; P k; v)j Ei ds E i where e j E i are the four edges of E i with the outer unit normals j E i, i = ::: 4. Note that in (2.5) the line integrals over edges interior to E 2 E k; cancel by continuity ofp k; v in the interior of E. Also, if e j E i and e^j E^i form an edge of E, it follows by the denition of P k; that Z Z (v ; P k; v)j Ei ds + (v ; P k; v)j E^ids =0 e j E e^j i E^i j E i e j E i is constant. Then, by (2.5), we see E^i e^j E^i (r[v ; P k; v] rw) E =0: Now, sum on all E 2E k; to derive the orthogonality relations in (2.3). The upper estimate in (2.4) directly follows from (2.2). The lower bound can be easily obtained from a direct calculation of the energy norms of v ; P K; on all E 2E k;. This completes the proof. # e 3 E E 2 E 3 e 2 E e 4 E E E 4 e E Figure. Edges and subsquares of E 2E k;. Before we start with the investigation of the prolongations I k,itwillbeuseful to collect some formulas. For E 2E k; and any v 2 V k;, dene je i E Ze j i E (see Figure for the notation), and set s E = b E + b2 E + b3 E + b4 E vds = b i E 4 E = b3 E ; b E 0 E = b E + b3 E ; b2 E ; b4 E 42 E = b4 E ; b2 E :

7 MULTIGRID AND MULTILEVEL METHODS FOR NONCONFORMING ELEMENTS 7 Then, with the subscript E omitted, we have the next lemma. Lemma 2.2. It holds that (2:6) and (2:7) jjvjj 2 L2 (E) = h 2 k; ; 6 s f(4 ) 2 +(4 2 ) 2 g + 40 (0 ) 2 jjrvjj 2 L2 (E) =(4 ) 2 +(4 2 ) (0 ) 2 h 2 k; 0 (b ) 2 +(b 2 ) 2 +(b 3 ) 2 +(b 4 ) 2 jjvjj 2 L2 (E) h2 k; 4 (b ) 2 +(b 2 ) 2 +(b 3 ) 2 +(b 4 ) 2 : Proof. Using the ane invariance of the local interpolation problem connecting v with its edge averages b i, it suces to prove (2.6) and (2.7) for the master square E =(; ) 2. A straightforward calculation gives (2:8) v = v(x y) = 4 s x y ; (x 2 ; y 2 ): Now direct integration yields the desired results in (2.6). Also, (2.7) follows from the rst equation of (2.6) by computing the eigenvalues of the symmetric 4 4 matrix T t DT, where D =diag(=6 =2 =2 =40), T stands for the transformation matrix from the vector (b b 3 b 2 b 4 )to(s ), and T t is the transpose of T. These eigenvalues are =0, =6, =6, and =4, which implies (2.7). # Lemma 2.2 is the basis for computing all the discrete energy and L 2 norms needed in the sequel. The formula (2.8) valid for the master square can be used to derive explicit expressions for the edge averages of I k v and I k v ; v. Toward this end, we rst compute the corresponding values for the master square, and then use the invariance of the local interpolation problem for v under ane transformations (for the square triangulations under consideration, these transformations are just dilation and translation) to return to the notation on each E 2E k;. e ;e +e 2 e +e2 e 2 ;e ; e e 2 e 2 +e e 2+e2 e e 2 2+e e ;e e e 2 Figure 2. An illustration for Lemma 2.3. Note that, by the denition of the triangulation E k;, to each E 2 E k; is uniquely assigned a = ( 2 ) such that 0, 2 2 k;. For notational convenience, let b and b2 denote the averages of v 2 V k; over the horizontal and vertical edges e and e2, respectively, k; (see Figure 2, where e 2 2, e2 2+e2, e 2+e2, e 2+e2 ;e k, e =( 0), and e 2 =(0 )). The corresponding quantities for I k v 2 V k are indicated by a j, j =,2. Now, introduce the notation ^ = b + b ;e ; b +e2 ; b +e2 ;e ^ 2 = b2 + b2 ;e2 ; b 2 +e ; b 2 +e ;e 2:

8 8 ZHANGXIN CHEN AND PETER OSWALD With these notation, it follows from the denition of I k that the edge averages of I k v can be written as follows: (2:9) and (2:0) a 2 = b + 8 ^ 2 a 2+e = b ; 8 ^ 2 a 2+e2 = 5 8 b2 + 8 b2 +e + 8 b + 8 b +e2 a 2+e2 +e = 5 8 b2 +e + 8 b2 + 8 b + 8 b +e2 a 2 2 = b2 + 8 ^ a 2 2+e2 = b 2 ; 8 ^ a 2 2+e = 5 8 b + 8 b +e2 + 8 b2 + 8 b2 +e a 2 2+e2 +e = 5 8 b +e2 + 8 b + 8 b2 + 8 b2 +e when the edge average a j is associated with an interior edge k for boundary edges, this value is set to be zero. Note that I k (as well as P k; ) can be extended to the larger spaces ^Vk in a natural way. In order to dene the extension ^Ij : ^Vk! V k, observe that any v 2 ^Vk coincides on each E 2E k with a polynomial from V k j E, so the form of the previous denition for I k remains the same for ^Ij. Clearly, ^Ik j Vk; = I k and ^Ik j Vk =Id k. To express the edge averages of I k v ; v, set = b +e2 ; b + b ;e ; b +e2 ;e 2 = b2 +e ; b 2 + b2 ;e2 ; b 2 +e ;e 2 if e and e2 are interior edges k;. For boundary edges, they need to be modied to give the correct expressions for I k v ; v. If e is a boundary edge, for example, we dene 2 =2(b 2 +e ; b 2 ): With these, we see that (2:) R e 2+e2 ;e (I kv ; v)j ;e ds = R e 2+e2 (I kv ; v)j ds =0 je 2 2 Re j 2 (I k v ; v)j ;e ds = 2 je Re 2 2+e2 j 2 2+e2 (I kv ; v)j ds = ; 8 je 2 2 Re j 2 (I k v ; v)j ds = 2 je Re 2 2+e2 j 2 2+e2 (I kv ; v)j ;e ds = 8 where (I k v ; v)j denotes the restriction of I k v ; v to the element associated with. The averages of I k v ; v on other edges are given similarly. Then, by Green's formula and (2.), we see that (2:2) jji k vjj 2 E ;jjvjj2 E = jji k v ; vjj 2 E v 2 V k;: From (2.9){(2.2) and Lemma 2.2, we immediately have the next lemma. Below the notation stands for two-sided inequalities with constants independent of k. Lemma 2.3. It holds that (2:3) jj^ik vjj q 52 jjvjj 8v 2 ^Vk jji k vjj E p 2jjvjj E 8v 2 V k;

9 MULTIGRID AND MULTILEVEL METHODS FOR NONCONFORMING ELEMENTS 9 and (2:4) 2 k jji k v ; vjj jji k v ; vjj E s X f( )2 +( 2 )2 g 8v 2 V k; : We now prove the following property of the iterated coarse-ne intergrid transfer operators R K k. Lemma 2.4. It holds that (2:5) jjr K k vjj E Cjjvjj E 8v 2 V k k K: Proof. The proof is technical it follows the idea of the proof of an analogous statement for the P nonconforming elements [9]. First, we consider the case of =IR 2. That is, we assume that all our denitions are extended to innite square partitions of IR 2 due to the local character of all constructions, this is easy to do. We keep the same notation for the extended partitions E k,edgese j k, squares E 2E k, etc. In order to guarantee the niteness of all norm expressions, we restrict our attention to functions v 2 V k with nite support. By the construction of I k, this property is preserved when applying the operators I k and Rk K. After the extension to the shift-invariant setting of IR 2, it is clear that it suces to consider the case of k =. Set, for simplicity, R ~ k = R, k k = ::: K. Our main observation from numerical experiments [2] was that the sequence fjj R ~ k v ; R ~ k; vjj 2 E k =2 ::: Kg decays geometrically. What we want toprove next is the mathematical counterpart to this observation. To formulate X the technical result, introduce j = () j 2 j =0 2 2Z2 where the quantities j are determined from the edge averages of v 2 V by the same formulas as above. The corresponding quantities computed for ~v = I 2 v 2 V 2 are denoted by ~ j and ~ j, j =0 2. From (2.4) in Lemma 2.3, we see that + 2 jj~ R 2 v ; vk 2 E and ~ +~ 2 jj~ R 3 v ; ~ R 2 vjj 2 E moreover, we can iterate this construction. Thus, if we can prove that (2:6) ~ c ~ 0 +~ +~ 2 (c ) where 0 < < andc > 0 are constants independent ofv, then,by Lemmas 2.2 and 2.3, (2:7) jjr K vjj E P K jjvjj E + P k=2 jj R ~ k v ; R ~ k; vk E K; p jjvjj E + C k= ( ) kp Cjjvjj E : Since this gives the desired boundedness of Rk K on (2.6). (for IR2 ) via dilation, we concentrate

10 0 ZHANGXIN CHEN AND PETER OSWALD From (2.9) and (2.0) we nd the following formulas for ~ j : ~ 0 2+e = 8 +e ; ~ 0 2 = ; ~ 0 2+e2 = 8 ; 8 2 +e ~ 0 2+e +e 2 = ; 8 +e e ~ 2 = 2 ; 8 (2 + 2 ;e) ; 3 8 (0 ; 0 ;e) ~ 2+e = 4 2 ~ 2+e2 = 2 ; 8 (2 +e2 + 2 ;e +e 2)+ 3 8 (0 ; 0 ;e) ~ 2+e +e 2 = ; 4 2 +e2 ~ 2 2 = 2 2 ; 8 ( + ;e2)+ 3 8 (0 ; 0 ;e2) ~ 2 2+e2 = 4 ~ 2 2+e = 2 2 ; 8 ( +e + ;e2 +e ) ; 3 8 (0 ; 0 ;e2) ~ 2 2+e +e 2 = ; 4 +e: These formulas are used to compute the quantities ~ j. In order to write them in reasonably short X form, we introduce the X notation = j j + = j l + k l =0 2(j 6= l) j 2Z2 jl 2Z2 if 2 Z 2 is the null vector, it is omitted in this notation. With them, we see,by carefully evaluating all squares, that ( ~ 0 2 )2 +( ~ 2+e) 0 2 +( ~ 2+e2) 0 2 +( ~ 2+e 0 +e 2) 2 ~ 0 = P (~ 0 ) 2 = P = ( + 2 ) ; ~ = ; 6 9 e e 2 ; 8 ( e2 + 2 ;e + e2 ;e 2 ) ; {z } 3 ~ 2 = ; 6 9 e e2 ; 8 ( e2 + 2 ;e + e2 ;e {z } 2 ) {z } 32 ( e2 + 2 ;e + e2 ;e 2 ) ; 3 Thus, introducing A = + 2 and ~ A =~ +~ 2,wehave (2:8) where 32 (e 02 + ;e +e 2 02 ; 02 ;e ; e +e 2 02 ) 32 (;e2 0 + e +e 2 0 ; 0 e2 ; e ;e 2 0 ): ~ 0 = ~A = A; e2 )+ 6 (e2 + 2 e ) ; 4 ; 32 3 = ;e2 0 + e +e e 02 + ;e +e 2 02 ; e2 0 ; e ;e 2 0 ; ;e 02 ; e +e 2 02 :

11 so that MULTIGRID AND MULTILEVEL METHODS FOR NONCONFORMING ELEMENTS Next, we simplify and. Note that ; 2 e2 = P ( e2 + 2 ;e +e ;e ; +e2 ; ;e2) = P (0 +e2 ;e + 0 ;e2 ; 0 ;e ;e 2 ; 0 +e2 +2 ) ; 2 e 2 = P 2 ( + ;e2 + +e ;e 2 + +e ; 2 +e ; 2 ;e) = P 2 (0 ;e2 ;e + 0 +e ; 0 +e ;e 2 ; 0 ;e +2 2 ) = e2 + e 2 +2A; 2 : Analogously, we can simplify as follows: = P 0 ; ( +e +e 2 + ;e e + 2 ;e +e 2) ;( +e2 + +e ;e ;e + 2 +e +e 2) = P 0 ; ( 0 +e +e e ;e ;e ;e ;e +e 2) ;2( 0 +e + 0 +e2 + 0 ;e + 0 ;e2)+4 0 ) =2( e +e e ;e 2 0 ) ; 4( e 0 + e2 0 )+4 0 : Substitution of and into (2.8) leads to (2:9) ~ 0 = A; = A + 32 ; 32 e2 + 2 e A ; 6 32 (e2 + 2 e2 )+ 64 e ;e e +e 2 0 ; 20 e ; 2e2 0 where we have used the fact that j j j j j =0 2, which isvalid for arbitrary. With the same argument, we see that ~A = A; e + 0 e2 ; 6 3 e2 + 2 e = 5 (2:20) A + 6 e ;e e +e e + 0 e2 e2 + 2 e A: ; 3 6 Now, set B = c 0 and ~ B = c~0. Then it follows from (2.8) and (2.9) that and ~A c B ~ B c 6 A + 2 B ( A ~ + B) ~ 5 max 8 + c 6 4c + (A + B): 2 Let c = c 3 p 5 ;, so we see that (2.6) holds with = c 6 = 4c + 2 = 3 p 5+9 < : 6

12 2 ZHANGXIN CHEN AND PETER OSWALD It remains to reduce the assertion of Lemma 2.4 to the shift-invariant situation just considered. To this end, starting with any v 2 V k on the unit square, we repeatedly use an odd extension. Namely, set ^v = v on [0 ] 2 and after this, dene ^v(x y) =;^v(;x y) (x y) 2 [; 0) [0 ] ^v(x y) =;^v(x ;y) (x y) 2 [; ] [; 0) and continue this extension process with the unit square replaced by [; ] 2 such that after the next two steps ^v is dened on [; 3] 2. Outside this larger square we continue by zero. Clearly, jj^vjj 2 E =6jjvjj2 E, where the norms for ^v and v are taken with respect to IR 2 and the unit square, respectively. It is not dicult to check by induction that on [0 ] 2 the functions Rk K ^v (obtained by the repeated application of the prolongations dened on IR 2 ) and Rk Kv (as dened above with respect to [0 ] 2 ) coincide. Also, the values of I k+^v on [;2 ;(k+) +2 ;(k+) ] 2 depend solely on the values of ^v on the square [;2 ;k + 2 ;k ] 2, and on this enlarged square I k+^v coincides with its odd extension from [0 ] 2. Finally, the zero edge averages are automatically reproduced along the boundary of [0 ] 2 from the above extension procedure. Therefore, by (2.7) and the dilation argument, we obtain jjr K k vjj 2 E jjr K k ^vjj 2 E Cjj^vjj 2 E =6Cjjvjj 2 E which nishes the proof of Lemma 2.4. # The second inequality in (2.3) is critical for the convergence results of multigrid algorithms developed in the next section, while (2.5) is crucial for the multilevel preconditioner results in x4. 3. MULTIGRID ALGORITHMS In this section and the next section we consider multigrid algorithms and multilevel preconditioners for the numerical solution of the second-order elliptic problem (3:) ;r (Aru) =f in u =0 on ; where IR 2 is a simply connected bounded polygonal domain with the boundary ;, f 2 L 2 (), and the coecient A 2 (L ()) 22 satises (3:2) t t A(x y) 0 t (x y) 2 2 IR 2 with xed constants, 0 > 0. The condition number of preconditioned linear systems to be analyzed later depends on the ratio = 0. Problem (3.) is recast in weak form as follows. The bilinear form a( ) is dened as follows: a(v w) =(Arv rw) v w 2 H (): Then the weak form of (3.) for the solution u 2 H 0 () is (3:3) a(u v) =(f v) 8 v 2 H 0 (): Associated with each V X k,weintroduce a bilinear form on V k H0 () by a k (v w) = (Arv rw) E v w 2 V k H0 (): E2E k

13 MULTIGRID AND MULTILEVEL METHODS FOR NONCONFORMING ELEMENTS 3 The NR Q nite element discretization of (3.) is to nd u K 2 V K such that (3:4) a K (u K v)=(f v) 8 v 2 V K : Let A k : V k! V k be the discretization operator on level k given by (3:5) (A k v w) =a k (v w) 8 w 2 V k : The operator A k is clearly symmetric (in both the a k ( ) and( ) inner products) and positive denite. Also, we dene the operators R k; : V k! V k; and R 0 k; : V k! V k; by a k; (R k; v w) =a k (v I k w) 8 w 2 V k; and ; R 0 k; v w =(v I k w) 8 w 2 V k; : It is easy to see that I k R k; is a symmetric operator wtih respect to the a k form. Note that neither R 0 k nor R k is a projection in the nonconforming case. Finally, let k dominate the spectral radius of A k. The multigrid processes below result in a linear iterative scheme with a reduction operator equal to I ; B K A K, where B K : V K! V K is the multigrid operator to be dened below. Multigrid Algorithm 3.. Let 2 k K and p be a positive integer. Set B = A ;. Assume that B k; has been dened and dene B k g for g 2 V k as follows:. Set x 0 =0andq 0 =0. 2. Dene x l for l = ::: m(k) by x l = x l; + S k (g ; A k x l; ): 3. Dene y m(k) = x m(k) + I k q p, where q i for i = ::: p is dened by q i = q i; + B k; hr 0 k; 4. Dene y l for l = m(k)+ ::: 2m(k) by g ; A k x m(k) ; A k; q i;i : y l = y l; + S k ; g ; Ak y l; : 5. Set B k g = y 2m(k). In Algorithm 3., m(k) gives the number of pre- and post-smoothing iterations and can vary as a function of k. In this section, we set S k = ( k ) ; Id k in the pre- and post-smoothing steps. If p =, wehave av-cycle multigrid algorithm. If p = 2, wehave aw-cycle algorithm. Avariable V-cycle algorithm is one in which the number of smoothings m(k) increase exponentially as k decreases (i.e., p = and m(k) =2 K;k ). We now follow themethodology developed in [6] to state convergence results for Algorithm 3.. The two ingredients in their analysis are the regularity and approximation property and the boundedness of the intergrid transfer operator: (3:6) ja k (v ; I k R k; v v)j C jja kvjj p k p ak (v v) 8 v 2 V k and (3:7) a k (I k v I k v) Ca k; (v v) 8 v 2 V k;

14 4 ZHANGXIN CHEN AND PETER OSWALD for k =2 ::: K, where k is the largest eigenvalue of A k. The proof of (3.6) is standard see the proof of a similar result for the P nonconforming elements in [4]. Inequality (3.7) has been shown in [] using the approximation property of the operator I k. However, here we see that if A = 0 I is a scalar multiple of the two-by-two identity matrix I, by the second inequality in (2.3) in Lemma 2.3, we actually have (3:8) a k (I k v I k v) 2a k; (v v) 8 v 2 V k; : This leads to the following main result of this section. Let the convergence rate for Algorithm 3. on the kth level be measured by the convergence factor k satisfying ja k (v ; B k A k v v)j k a k (v v) 8 v 2 V k : Theorem 3.. (i) Dene B k by p = and m(k) = 2 K;k for k = 2 ::: K in Algorithm 3.. Then there are 0, > 0, independent of k, such that with 0 a k (v v) a k (B k A k v v) a k (v v) 8v 2 V k 0 p m(k)=(c + p m(k)) and (C + p m(k))= p m(k): (ii) Dene B k by p =2and m(k) =m for all k in Algorithm 3.. Then if A = 0 I is constant, there exists C>0, independent of k, such that k C C + p m : The same conclusion holds if the assumption that A = 0 I is replaced byrequiring that m m 0, where m 0 is suciently large, but independent of k. The proof of this theorem follows from (3.6){(3.8) and Theorems 6 and 7 in [6]. From Theorem 3., we have an optimal convergence property of the W-cycle and a uniform condition number estimate for the variable V-cycle preconditioner. 4. MULTILEVEL PRECONDITIONERS In this section we discussmultilevel preconditioners of hierarchical basis and BPX [5] type for (3.4). More precisely, we derive the condition numbers of the additive subspace splittings (4:) fv K ( ) E g = R K fv ( ) E g + Rk K fv k 2 2k ( )g k=2 and (4:2) fv K ( ) E g = R K fv ( ) E g + Rk K f(id k ; I k P k; )V k 2 2k ( )g: k=2 The condition number of (4.) is given by [20] (4:3) = max min where KX jjvjj 2 max = sup E v2v K jjjvjjj 2 jjjvjjj 2 = inf P jjv jj 2 v k 2V k : v= k RK k v E + k k=2 A similar denition can be given for (4.2). ( KX jjvjj 2 min = inf E v2v K jjjvjjj 2 ) KX 2 2k jjv k jj 2 :

15 MULTIGRID AND MULTILEVEL METHODS FOR NONCONFORMING ELEMENTS 5 Theorem 4.. There are positive constants c and C, independent of K, such that (4:4) c jjvjj2 E jjjvjjj 2 CK and (4:5) c jjvjj2 E kkvkk 2 CK where KX 8v 2 V K 8v 2 V K kkvkk 2 = jjq K vjj 2 E + 2 2k jj(id k ; I k P k; )Q K k vjj 2 : k=2 That is, the condition numbers of the additive subspace splittings (4.) and (4.2) are bounded byo(k) as K!. Proof. For k =2 ::: K,itfollows from the denitions of I k, ^Ik,andQ K k, (2.4), and the rst inequality of (2.3) that 2 2k jj(id k ; I k P k; )Q K k vjj2 =2 2k jj^ik (Id k ; P k; )Q K k vjj k jj(id k ; P k; )Q K k vjj2 Cjj(Id k ; I k P k; )Q K k vjj2 E = CjjQ K k v ; QK k; vjj2 : Summing on j and using the orthogonality relations in (2.3), we see that inf vk 2V k : v= P k RK k v k n jjv jj 2 E + P K k=2 22k jjv k jj 2o jjq K vk2 E + P K k=2 22k jj(id k ; I k P k; )Q K k vjj2 Cjjvjj 2 E which implies the lower bounds in (4.4) and (4.5). P K For the upper bounds, we consider an arbitrary decomposition v = k= RK k v k with v k 2 V k. Then we see,by Lemma 2.4, that! X K 2 X K jjvjj 2 E jjrk K v kjj E K jjrk K v kjj 2 E CK jjv k jj 2 E : k= k= k= Consequently, by (2.2), wehave! KX jjvjj 2 E CK jjv jj 2 E + 2 2k jjv k jj 2 : k=2 Now, taking the inmum with respect to all decompositions, we obtain jjvjj 2 E KX n P jjv jj 2 E + K k=2 22k jjv k jj 2o P jjq K vjj2 E + K k=2 22k jj(id k ; I k P k; )Q Kvjj2 k CK inf vk 2V k : v= P k RK k v k CK which nishes the proof of the theorem. # We now discuss the algorithmical consequences for the splittings (4.) and (4.2). Theoretically, Theorem 4. already produces suitable preconditioners for the matrix A K using (4.) and (4.2). However, they are still complicated since they involve L 2 -projections onto V k, < k < K, which means to solve large linear systems within each preconditioning step. To get more practicable algorithms, we replace

16 6 ZHANGXIN CHEN AND PETER OSWALD the L 2 norms in V k and W k = (Id k ; I k P k; )V k V k, k = 2 ::: K, by their suitable discrete counterparts. We rst consider the splitting (4.) (4.2) will be discussed later. Let f j k g be the basis functions of V k such that the edge average of j k equals one at e j k and zero at all other edges. Then each v 2 V k has the representation v = j= 2X X a j j k : Thus, by the uniform L 2 -stability of the bases, which follows from (2.7) in Lemma 2.2, we see that (4:6) 5 2;2k 2X X (a j ) 2 jjvjj 2 2X X 2 2;2k j= Note that (with the same argument as in Lemma 2.2) j= (a j ) 2 : (4:7) 2 2k jj j k jj2 = 4 20 a k( j k j k ) jjj k jj2 E =5 so (4.6) can be interpreted as the two-sided inequality associated with the stability of any of the splittings (4:8) fv k 2 2k ( )g = j= 2X X (4:9) fv k 2 2k ( )g = j= and (4:0) fv k 2 2k ( )g = j= 2X X 2X X fv j k 22k ( )g fv j k ( ) Eg fv j k a k( )g into the direct sum of one-dimensional subspaces V j k spanned by the basis functions j k. Any of the splittings (4.8){(4.0) can be used to rene (4.). As we will see below, the dierence is just in a diagonal scaling (i.e., a multiplication by a diagonal matrix) in the nal algorithms. As example, we consider the splitting (4.0) in detail the other two cases can be analyzed in the same fashion. With (4.) and (4.0), we have the splitting (4:) fv K a K ( )g = R K fv a ( )g + k=2 j= KX 2X X R K k fv j k a k( )g: It follows from (4.4), (4.6), and (4.7) that the condition number for(4.)still behaves like O(K). Now, associated with this splitting we can explicitly state the additive Schwarz operator KX (4:2) P K = R K T + k=2 j= 2X X R K k T j k

17 where MULTIGRID AND MULTILEVEL METHODS FOR NONCONFORMING ELEMENTS 7 and T v 2 V solves the elliptic problem T j k v = a K(v R K k j k ) a k ( j k j k ) j k a (T v w) =a K (v R K w) 8 w 2 V : Thus the matrix representations of all operators with respect to the bases of the respective V k are T k = j= 2X X T j k = S k(r K k ) t A K S k =diag(a j ( j k j k ); ) for 2 k K, and T = A ; (RK ) t A K where for convenience the same notation is used for operators and matrices. Hence it follows from (4.2) that P K = R K A; (RK ) t + KX! Rk K S k(rk K ) t A K C K A K k=2 which, together with the denition of Rk K = I K I k+, leads to the typical recursive structure for the preconditioner C K (4:3) C k = I k C k; I t k + S k k = K ::: 2 S = C A ; : Note that with these choices for S k,themultiplication of a vector by C K is formally a special case of Algorithm 3. if one sets m(k) =, p =, removes the postsmoothing step, and replaces A k by a zero matrix for all k 2. From (4.3) and the denitions of I k and S k,we see that a multiplication by C K only involves O(n K +:::+n 2 +n 3 )=O(n K ) arithmetical operations, where n k 2 2k is the dimension of V k. This, together with (4.4), yields suboptimal work estimates for a preconditioned conjugate gradient method for (3.4) with the preconditioner C K. That is, an error reduction by a factor p in the preconditioned conjugate gradient algorithm can be achieved by O(n K log nk log( ; )) operations. We now turn to the discussion of the algorithmical consequences for the splitting (4.2). Todothis,we need to construct basis functions in W k, k =2 ::: K. Starting with the bases f j k g in V k, to each interior edge e j k; k;, we replace the two associated basis functions j 2 k j 2+e j k with their linear combinations j 2 k = j 2 k + j 2+e j k j 2+e j k = j 2 k ; j 2+e j k j = 2 where e j 2 k and ej 2+e j k k form the edge e j k;. For all other interior edges e j k, which do not belong to any edge k;, weset j k = j k : The new bases f j k g in V k is still L 2 -stable i.e., they satisfy an analogous inequality to (4.6). Moreover, if v = j= 2X X b j j k

18 8 ZHANGXIN CHEN AND PETER OSWALD we have P k; v = and 2X X j= b j 2 j k; 2X X (Id k ; I k P k; )v = c j j k j= 6=2 since j 2 k ; I k j k; can be completely expressed by the functions k l 6= 2 only. More precisely, wehave with c 2+e = b 2+e ; 8 (b2 2 + b2 2(;e2 ) ; b 2 2(+e ) ; b 2 2(+e ;e 2 ) ) c 2+e2 = b 2+e2 ; 8 (5b2 2 + b 2 + b2 2(+e ) + b 2(+e2 ) ) c 2+e +e 2 = b 2+e2 ; 8 (5b2 2(+e ) + b 2 + b2 2 + b 2(+e2 ) ) and similar relations hold for j = 2. Hence any function from W k has a unique representation by linear combinations of f j k : 6= 2g, and this basis system is L 2 -stable. With this basis system, as in (4.), we have the corresponding splitting (4:4) fv K a K ( )g = R K fv a ( )g + Rk K fw j k a k( )g k=2 j= 6=2 KX 2X X into a direct sum of R K V and one-dimensional spaces Rk KW j k spanned by the basis j k. Then, with the same argument as for (4.3), we derive an additive preconditioner ^CK for A K recursively dened by (4:5) ^Ck = I k ^Ck; I t k + ^Ik ^Sk ^It k k = K ::: 2 ^C = ^S A ; where ^S k =diag a k ( j k j k ); 6= 2 j = 2 are diagonal matrices and ^Ik is the rectangular matrix corresponding to the natural embedding W k V k with respect to the bases f j k g in W k and f j k g in V k (one may use the bases f j k g for all V k, which would change the I k representations, but keep ^Ik maximally simple). (4.5) has the same arithmetical complexity as before. We now summarize the results in Theorem 4. and the above discussion in the next theorem. Theorem 4.2. The symmetric preconditioners C K and ^CK dened in(4.3) and (4.5) and associated with the multilevel splittings (4.) and (4.4), respectively, have an O(n K ) operation count per matrix-vector multiplication and produce the following the condition numbers: (4:6) (C K A K ) CK ( ^CK A K ) CK K : The splitting (4.) can be viewed as the nodal basis preconditioner of BPX type [5], while the splitting (4.4) is analogous to the hierarchical basis preconditioner.

19 MULTIGRID AND MULTILEVEL METHODS FOR NONCONFORMING ELEMENTS 9 We now consider multiplicative algorithms for (3.4). One iteration step of a multiplicative algorithm corresponding to the splitting (4.) takes the form (4:7) y 0 = x j K y l+ = y l ;!R K K;l S K;l(R K K;l )t (A K y l ; f K ) x j+ K = yk l =0 ::: K; where! is a suitable relaxation parameter (the range of relaxation parameters for which the algorithm in (4.7) converges is determined mainly by the constant in the inverse inequality (2.2) [27, 26, 5]. The method (4.7) corresponds to a V-cycle algorithm in Algorithm 3. with A k replaced by ~ Ak =(R K k )t A K R K k, one pre-smoothing and no post-smoothing steps. The iteration matrix M K! in (4.7) is given by M K! = (Id K ;!E ) (Id K ;!E K; )(Id K ;!E K ) E k R K k S k (R K k ) t A K : An analogous multiplicative algorithm for (3.4) corresponding to the splitting (4.4) can be dened. From the general theory on multiplicative algorithms [27] and by the same argument as for Theorem 4.2, we can show the following result. Theorem 4.3. For properly chosen relaxation parameter! the multiplicative schemes corresponding to the splittings (4.) and (4.4) possess the following upper bounds for the convergence rate: (4:8) inf! jjm K!jj E ; C K inf! jj ^MK! jj E ; C K K! where M K! and ^MK! denote the iteration matrices associated with (4.) and (4.4), respectively. We end with two remarks. First, one example for the choice of! is that! K ;,which leads to the upper bounds in (4.8). Second, the diagonal matrices S k and ^Sk in (4.3) and (4.5) can be replaced by any other spectrally equivalent symmetric matrices of their respective dimension. 5. EQUIVALENT DISCRETIZATIONS To improve the estimates in Theorems 4.2 and 4.3, we now consider the problem of switching the NR Q discretization system (3.4) to a spectrally equivalent discretization system for which optimal preconditioners are already available. This switching strategy, as mentioned in the introduction, has been used in the context of the multilevel additive Schwarz method see [2] for the references. The most natural candidate for a switching procedure is the space of conforming bilinear elements U K = 2 C 0 () : j E 2 Q (E) 8E 2E k and j ; =0 on the same partition. We introduce two linear operators Y K : U K! V K and ^Y K : V K! U K as follows. If 2 U K and e is an edge of an element ine K, then Y K 2 V K is given by (5:) Z e Y K ds = Z e ds

20 20 ZHANGXIN CHEN AND PETER OSWALD which preserves the zero average values on the boundary edges. If v 2 V K, we dene ^YK v 2 U K by (5:2) ( ^YK v)(z) =0 for all boundary vertices z in E K ( ^YK v)(z) =average of v j (z) for all internal vertices z in E K where v j = vj Ej and E j 2E K contains z as a vertex. Another choice for U K is the space of conforming P elements U K = 2 C 0 () : j E 2 P (E) 8E 2 ~ EK and j ; =0 where ~ EK is the triangulation of generated by connecting the two opposite vertices of the squares in E K. The two linear operators Y K : U K! V K and ^YK : V K! U K are dened as in (5.) and (5.2), respectively. Moreover, for both the conforming bilinear elements and the conforming P elements, it can be easily shown that there is a constant C, independent of K, such that (5:3) 2 K jj ; Y K jj Cjjjj E 8 2 U K 2 K jjv ; ^YK vjj Cjjvjj E 8v 2 V K : Since optimal preconditioners exist for the discretization system A K generated by the conforming bilinear elements (respectively, the conforming P elements), the next result follows from (5.3) and the general switching theory in [2]. Theorem 5.. Let C K be any optimal symmetric preconditioner for A K i.e., we assume that a matrix-vector multiplication by C K can be performed in O(n K ) arithmetical operations, and that (C K A K ) C, withconstant independent of K. Then (5:4) C K = S K + Y K C K (Y K ) t is an optimal symmetric preconditioner for A K. 6. NUMERICAL EXPERIMENTS In this section we present the results of numerical examples to illustrate the theories developed in the earlier sections. These numerical examples deal with the Laplace equation on the unit square: (6:) ;u = f in = (0 ) 2 u =0 on ; where f 2 L 2. The NR Q nite element method (3.4) is used to solve (6.) with fe k g K k= being a sequence of dyadically, uniformly rened partitions of into squares. The coarsest grid is of size h ==2. The rst test concerns the convergence of Algorithm 3.. The analysis of the third section guarantees the convergence of the W-cycle algorithm with anynumber of smoothing steps and the uniform condition number property for the variable V- cycle algorithm, but does not give any indication for the convergence of the standard V-cycle algorithm, i.e., Algorithm 3. with p = and m(k) = for all k. The rst two rows of Table show the results for levels K = 3 ::: 7 for this symmetric V-cycle, where ( v v ) denote the condition number for the system B K A K and the reduction factor for the system Id K ; B K A K as a function of the mesh size on the nest grid h K. While there is no complete theory for this V-cycle algorithm, it is of practical interest that the condition numbers for this cycle remain relatively small.

21 MULTIGRID AND MULTILEVEL METHODS FOR NONCONFORMING ELEMENTS 2 =h K v v m Table. Numerical results for the multiplicative V-cycles. For comparison, we run the same example by a symmetrized multilevel multiplicative Schwarz method corresponding to (4.7). One step of the symmetric version consists of two substeps, the rst coinciding with (4.7) and the second repeating (4.7) in reverse order. The condition numbers m for MK! ~ A K with! K ; are presented in the third row oftable, where MK! ~ = MK! t M K! is now symmetric. The results are better than expected from the upper bounds of Theorem 4.3 which seem to be only suboptimal. In the second test we treat the above multigrid algorithm and symmetrized multilevel multiplicative method as preconditioners for the conjugate gradient method. In this test the problem (6.) is assumed to have the exact solution u(x y) =x( ; x)y( ; y)e xy : Table 2 shows the number of iterations required to achieve the error reduction 0 ;6, where the starting vector for the iteration is zero. The iteration numbers (iter v iter m ) correspond to Algorithm 3. with p =andm(k) =forallk and the symmetrized multiplicative algorithm (4.7), respectively. Note that iter v and iter m remain almost constant when the step size increases. =h K iter v iter m Table 2. Iteration numbers for the pcg-iteration. In the nal test we report analogous numerical results (condition numbers and pcg-iteration count) for the additive preconditioner C K associated with the splitting (4.) (subscript a), and the preconditioner C K (subscript s) which uses the switch from the system arising from (3.4) to the spectrally equivalent system generated by the conforming bilinear elements via the operators in (5.) and (5.2). We have implemented the standard BPX-preconditioner [5], with diagonal scaling, as C K.

22 22 ZHANGXIN CHEN AND PETER OSWALD These results are shown in Table 3. The numbers show the slight growth, which is typical for most of the additive preconditioners and level numbers K < 0. The condition numbers s for the switching procedure are practically identical to the condition numbers for C K A K characterizing the BPX-preconditioner [5] in the conforming bilinear case. The switching procedure is clearly favorable as can be expected from the theoretical bounds of Theorems 4.2 and 5. however, the computations do not indicate whether the upper bound (4.6) is sharp or could be further improved. =h K a iter a s iter s Table 3. Results for the preconditioners C K and C K. References [] T. Arbogast and Zhangxin Chen, On the implementation of mixed methods as nonconforming methods for second order elliptic problems, Math. Comp. 64 (995), 943{972. [2] R. Bank and T. Dupont, An optimal order process for solving nite element equations, Math. Comp. 36 (98), 35{5. [3] D. Braess and R. Verfurth, Multigrid methods for nonconforming nite element methods, SIAM J. Numer. Anal. 27 (990), 979{986. [4] J. Bramble, Multigrid Methods, Pitman Research Notes in Math., vol. 294, Longman, London, 993. [5] J. Bramble, J. Pasciak, and J. Xu, Parallel multilevel preconditioners, Math. Comp. 55 (99), -22. [6] J. Bramble, J. Pasciak, and J. Xu, The analysis of multigrid algorithms with non-nested spaces or non-inherited quadratic forms, Math. Comp. 56 (99), {34. [7] S. Brenner, An optimal-order multigrid method for P nonconforming nite elements, Math. Comp. 52 (989), {5. [8] S. Brenner, Multigrid methods for nonconforming nite elements, Proceedings of Fourth Copper Mountain Conference on Multigrid Methods, J. Mandel, et al., eds., SIAM, Philadelphia, 989, pp. 54{65. [9] S. Brenner, Convergence of nonconforming multigrid methods without full elliptic regularity, Preprint, 995. [0] Zhangxin Chen, Analysis of mixed methods using conforming and nonconforming nite element methods, RAIRO Model. Math. Anal. Numer. 27 (993), 9{34. [] Zhangxin Chen, Projection nite element methods for semiconductor device equations, Computers Math. Applic. 25 (993), 8{88.

23 MULTIGRID AND MULTILEVEL METHODS FOR NONCONFORMING ELEMENTS 23 [2] Zhangxin Chen, Equivalence between and multigrid algorithms for nonconforming and mixed methods for second order elliptic problems, IMA Preprint Series #28, 994, East-West J. Numer. Math. 4 (996), to appear. [3] Zhangxin Chen, R. Ewing, and R. Lazarov, Domain decomposition algorithms for mixed methods for second order elliptic problems, Math. Comp. 65 (996), to appear. [4] Zhangxin Chen, D. Y. Kwak, and Y. J. Yon, Multigrid algorithms for nonconforming and mixed methods for symmetric and nonsymmetric problems, IMA Preprint Series #277, 994. [5] M. Griebel and P. Oswald, On the abstract theory of additive and multiplicative Schwarz algorithms, Numer. Math. 70 (995), 63{80. [6] P. Kloucek, B. Li, and M. Luskin, Analysis of a class of nonconforming nite elements for crystalline microstructures, Math. Comp. (996), to appear. [7] P. Kloucek and M. Luskin, The computation of the dynamics of martensitic microstructure, Continuum Mech. Thermodyn. 6 (994), 209{240. [8] C. Lee, A nonconforming multigrid method using conforming subspaces, Proceedings of the Sixth Copper Mountain Conference on Multigrid Methods, N. Melson et al., eds., NASA Conference Publication, vol. 3224, 993, pp. 37{330. [9] P. Oswald, On a hierarchical basis multilevel method with nonconforming P elements, Numer. Math. 62 (992), 89{22. [20] P. Oswald, Multilevel Finite Element Approximation : Theory and Application, Teubner Skripten zur Numerik, Teubner, Stuttgart, 994. [2] P. Oswald, Preconditioners for nonconforming elements, Math. Comp. (996), to appear. [22] P. Oswald, Intergrid transfer operators and multilevel preconditioners for nonconforming discretizations, Preprint, 995. [23] R. Rannacher and S. Turek, Simple nonconforming quadrilateral Stokes element, Numer. Meth. Partial Di. Equations 8 (992), 97{. [24] P. Raviart and J. Thomas, A mixed nite element method for second order elliptic problems, Mathematical aspects of the FEM, Lecture Notes in Mathematics, 606, Springer-Verlag, Berlin & New York (977), pp. 292{35. [25] M. Wang, The W-cycle multigrid method for nite elements with nonnested spaces, Adv. in Math. 23 (994), 238{250. [26] H. Yserentant, Old and new convergence proofs for multigrid methods, Acta Numerica, Cambr. Univ. Press, Cambridge, 993, pp. 285{236. [27] J. Xu, Iterative methods by space decomposition and subspace correction, SIAM Review 34 (992), 58{63. Department of Mathematics, Box 56, Southern Methodist University, Dallas, Texas 75275{056. address: zchen@dragon.math.smu.edu Department of Mathematics, Texas A&M University, College Station, Texas 77843{ address: poswald@math.tamu.edu