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1 g ' Electronic Transactions on umerical Analysis. Volume 17 pp Copyright SS COVERGECE OF V-CYCE AD F-CYCE MTGRD METHODS FOR THE BHARMOC PROBEM SG THE MOREY EEMET JE ZHAO Abstract. Multigrid V-cycle and F-cycle algorithms for the biharmonic problem using the Morley element are studied in this paper. We show that the contraction numbers can be uniformly improved by increasing the number of smoothing steps. Key words. multigrid nonconforming V-cycle F-cycle biharmonic problem Morley element. AMS subject classifications ntroduction. et be a bounded polygonal domain. Consider the following variational problem for the biharmonic equation with homogeneous Dirichlet boundary conditions: Find such that (1.1 " # where %(' ACB D FEHGJ +*-. % : # <; 0 3 *=> The elliptic regularity of the biharmonic equation (cf. [17 [1 implies that there exists K M#E such that the solution of (1.1 belongs to 57O APQ whenever R ST5>O and %^+a % %^ (1.2 % V :WYXFZ=[\ :b#wdc-xjze[\ where depends only on the shape of f The problem (1.1 can be solved numerically using the Bogner-Fox-Schmit element (cf. [4 the Argyris element (cf. [1 the Hsieh-Clough-Tocher element (cf. [1 the Morley element (cf. [24 and the incomplete biquadratic element (cf. [27. n this paper we will concentrate on the Morley element which is the simplest among all the finite element methods for the biharmonic problem. But of course the analysis of the Morley element is more complicated. Multigrid methods for the Morley element which is nonconforming have been studied in [ [9 [10 [20 [2 [25 and [2. t was shown in [10 that the W-cycle multigrid method converges uniformly if the number of smoothing steps is large enough and that the symmetric variable V-cycle algorithm is an optimal preconditioner without assuming full elliptic regularity. n [11 and [13 an additive theory was developed to study the asymptotic behavior with respect to the number of smoothing steps of the contraction numbers of V-cycle and F-cycle methods for conforming and nonconforming finite elements for second order problems without assuming full elliptic regularity. n this paper we will apply this theory to the biharmonic Received March Accepted for publication January Recommended by Tom Manteuffel. Department of Mathematics niversity of South Carolina Columbia SC 2920.S.A. jzhao000@math.sc.edu. This research was supported in part by the ational Science Foundation under Grant o. DMS

2 V-cycle and F-cycle methods for the biharmonic problem 113 problem. We use the Morley element to illustrate the theory which can also be applied to other elements (cf. [33. et 1 be the contraction number of the -th level symmetric V-cycle algorithm with pre-smoothing and post-smoothing steps. Our main result states that there exists a constant independent of and such that 1 Ö for where the positive integer is also independent of T A similar result also holds for the F-cycle algorithm. The rest of the paper is organized as follows. We discuss the Morley element and its relation with the Hsieh-Clough-Tocher element in Section 2. The relation is important for the analysis of the multigrid methods. We describe multigrid V-cycle and F-cycle algorithms in Section 3. n Section 4 we discuss mesh dependent norms and their properties. Some known results concerning the additive theory are summarized in Section 5. Convergence analysis is then carried out in Section. umerical results are presented in Section The Morley element and the Hsieh-Clough-Tocher element. The Morley finite element is defined on a triangle. ts shape functions are quadratic polynomials on the triangle. ts nodal variables include the evaluations of the shape functions at the vertices of the triangles and the evaluations of the normal derivatives at the midpoints of the edges of the triangles (cf. Figure 2.1(a. (a Morley (b H-C-T FG The Morley element and the H-C-T element The Hsieh-Clough-Tocher macro element is also defined on a triangle. The shape functions are those functions on the triangle whose restriction to each smaller triangle formed by connecting the centroid and two vertices of the triangle is a cubic polynomial. The nodal variables include the evaluations of the shape functions at the vertices of the triangle the evaluations of the gradients at the vertices and of the normal derivatives at the midpoints of the edges of the triangle (cf. Figure 2.1(b. Since the shape functions and the nodal variables of the Morley element are also shape functions and nodal variables of the Hsieh-Clough-Tocher element we call the Hsieh-Clough- Tocher element a relative of the Morley element (cf. [10. is obtained by connecting et be a family of triangulations of f where the midpoints of the edges of the triangles in. We denote the mesh size of by "# %' diam( (* += ote that (2.1 "# -" e et. be the Morley finite element space associated with Then.# if and only if it has the following three properties:

3 2 ' ' 114 Jie Zhao 1. < is a quadratic polynomial for all ( 2. is continuous at the vertices of and vanishes at the vertices along 3. The normal derivative is continuous at the midpoints of interelement boundaries and vanishes at the midpoints along. ote that the Morley finite element spaces are nonconforming (i.e.. D and nonnested (i.e..#.. et. be the Hsieh-Clough-Tocher macro element space associated with. Then a function. is on f its restriction to each ( is a piecewise cubic polynomial function and its nodal values along are zero. ote that. D (i.e. conforming. We now discuss the relation between the Morley space and the Hsieh-Clough-Tocher space. We can define an operator <.#.# For each ".# the function " is defined as follows. For any internal vertex and internal midpoint A # # 5 A average of 0 V : or = # and 0 for ( 0 with as a vertex. We can also define an operator.#.# as follows. For each ".# where function in.# satisfying T T for every internal vertex and midpoint (2.2 (2.3 and of A The operators and satisfy the following two properties (cf. [10: %^ "=* %^ % # % X \ % X \ and # ('% W X \ *.# where e* is the identity operator on. (2.4 #T ' 5 <" +. and the bilinear form ; ; on D +. is defined by D Y*=>.# is the ote that the constructions of and and the properties (2.2 and (2.3 rely on the fact that the Morley element and the Hsieh-Clough-Tocher element are relatives. These operators are important for the multigrid analysis (cf. emmas 4.2 and 4.3. We now define the Morley element method and the modified Morley element method for %32 (1.1. f Y10 +*- for a function 54 V then the Morley finite element method for (1.1 is: Find *. such that % 2 (2.5 5A A*= T<*.#

4 V-cycle and F-cycle methods for the biharmonic problem 115 n the case where Q@ V the modified Morley finite element method for (1.1 is: Find TG *. such that (2. G.# The properties of these methods are discussed in [10. The results of this paper can be applied to both of these methods. 3. V-cycle and F-cycle Multigrid methods. n this section we describe the V-cycle and F-cycle multigrid methods for the Morley finite element. et the discrete inner product ; ; on.# be defined by (3.1 " - T + - * *D where is the set of internal vertices of is the set of internal edges of and T (the number of triangles sharing the node as a vertex. We can represent the bilinear form ; ; by the operator.#. defined by (3.2 5 ( 5. Then equations (2.5 and (2. can both be rewritten as (3.3 ( % 2 where S-.# is defined by 5 0 Y*= for all.# for the standard Morley method and for all *. for the modified Morley method. n order to describe multigrid methods we need to define the intergrid transfer operators.. the coarse-to-fine intergrid transfer operator. The We first define. properties of this operator can be found in [9. et *.. We define. by an averaging technique as follows: 1. f is an internal vertex of then (3.4 where 1 then (3.5 A 1 - c ( % (. 2. f is an internal edge of which means that *f * + # ( P X ( for some ( ( *@ We can then define the fine-to-coarse operator....# as follows: " <*. #R. " <*. #R. and the nonconforming Ritz projection operator # A Symmetric V-cycle Multigrid Method (cf. [5 [ [14 [22 [19 [30 and [32. The symmetric V-cycle multigrid algorithm is an iterative solver for equations of the form (3.3.

5 11 Jie Zhao Given ". and an initial guess. the output T of the algorithm is an approximate solution for the equation (3. where is the number of pre-smoothing and post-smoothing steps. For <R- we define For - A we obtain T T in three steps. 1. (Pre-Smoothing For - ;M;:; compute 3 by where is a constant dominating the spectral radius of 2. (Coarse Grid Correction Compute by + a- # V 3. (Post-Smoothing For + ;:;M; + compute 3 by 3 3 Finally we set T T to be +. 3 n this algorithm we use Richardson relaxation as the smoother for simplicity. Other smoothers can also be used (cf. [2 [ and [12. F-cycle Multigrid Method (cf. [23 [32 and [30. The -th level F-cycle algorithm (associated with the symmetric V-cycle algorithm produces an approximate solution T T for (3.. For C- we define - T T For we obtain T in three steps. 1. (Pre-Smoothing For - ;M;:; compute 3 by (Coarse Grid Correction Compute + 3 X and by X a- V V + a- V 3. (Post-Smoothing For + ;:;M; + compute 3 by 3 3 Finally we set T to be +. 3 V V V V X V

6 \ " ( ; V-cycle and F-cycle methods for the biharmonic problem Mesh Dependent orms. One of the main tools for the convergence analysis of multigrid methods is the mesh-dependent norm ; 1 (cf. [3. For each. we define ( t is easy to see that (4.2 ( %^ *. 1 #T '% " #T# X \ T. To avoid the proliferation of constants we use two notations and. The statement means that is bounded by multiplied by a constant which is independent of mesh sizes mesh levels and all arguments in and and means as well as We can also easily see that (cf. [3 and [ " (4.4 1 where : (4.5 (4. The following smoothing properties of can also be easily verified (cf. [19 and [14 where 1e* ( <.# + \ 1 <.# The following lemmas relate the mesh-dependent norms with the Sobolev norms. EMMA 4.1. The following relation holds: Proof. et have ( be a triangle with (4. #T X \ 1 V% X \ 0 4 %^ *.# - Then for all quadratic polynomials on ( we * sing a scaling argument on (4. and by definition (3.1 of the inner product ; ; we have %^ (4.9 V X \ <.# The lemma follows from (4.2 and (4.9. following relation holds: (4.10 EMMA 4.2. et.#" %^ W \. be the operator defined in Section. Then the V1 <.# %^ %^ % Proof. t is known from [10 that the operator is a bounded operator from. ; % X \ to 4 V ; % X \ # and from %.# ; ('% %^ to D V W+X#\ i.e. (4.11 # T# X \ V% X \.#

7 11 Jie Zhao or equivalently (4.12 % % X \ and (4.13 or equivalently (4.14 (4.15 %^ %^ W X \ % W X \ #T('% 1 *. T*. 1 <.# By interpolations of Sobolev spaces and Hilbert scales(cf. [5 and [29 we have W \ % V1 <.# Conversely let 4.# be the 4 projection operator on 54 V the function *. satisfies %^ %^ X \ X \ <. t is known that (cf. [7 (4.1 (4.17 % % X \ % W X \ %^ % %^ X \ W X \ 54 V " #.# i.e. for each We define + 4.# by (" %^ e Then%^ from (2.3 %^ (4.1 and (4.17 we have 1 % % X \ # %^ X \ %^ % X \ 4 # 1 % #'% W X \ W X \ # By interpolations of Sobolev spaces and Hilbert % scales we have (4.1 + V1 W \ V For each *.# we have Then by (2.2 and the definition of ( " From equations (4.1 and (4.19 we have V1 1 %^ W X \ <. e we have (4.20 EMMA 4.3. The following relation holds: -. W \ V1 *.# Proof. t is known that (cf. (E of [10 (4.21 V3 T# X \ %^ " V#'# T<*.#

8 V-cycle and F-cycle methods for the biharmonic problem 119 From (2.4 (4.4 emma 4.2 (4.21 and a standard inverse estimate (cf. [14 [15 we have for all.# -. W \ -. W \ W \ -. WV\ -. %^ %^ W#\ " T #3 T X \ %^ WV\ -. " W+XM\ % W \ " 1 + W#\ V1 Conversely from emma 4.2 (4.21 and a standard inverse estimate we have for all.# V1 %^ W \ W \ -. W \ %^ " V3 X \ " V ' W \ W \ W \ -. W \ -. W \ 5. Some known results for the additive theory. et 1.#. be the error propagation operator of the symmetric V-cycle algorithm applied to the equation (3. i.e. WV\ 1 T V is the exact solution of (3.. The following relations (cf. [5 and [19 are well- where known: (5.1 (5.2 1 V1 B e* + V1 E for 1 From (5.1 and (5.2 the following additive expression for can be derived that is the starting point of the additive theory (cf. [13: ( ;:;:; e* ;M;:; 3

9 120 Jie Zhao et 1.. be the error propagation operator of the symmetric F-cycle algorithm applied to the equation (3. i.e. 1 T V where is the exact solution of (3.. The following relations are also well-known (cf. [30: (5.4 ( B e* E An additive theory for the convergence analysis of V-cycle and F-cycle multigrid algorithms is developed in [13 based on the expressions (5.3 (5.5. t is shown there that to complete the convergence analysis we only need to verify the following assumptions. Assumptions on.# : 5 V '% Assumptions on and #O + #O 1 + : 1 #O #O % #T X \ %^ " V% X \ T " O Assumptions on and : e* 1 #O e* 1 #O *.# *. T " O T " O " O " O 1 1 5>O 1 <. <. <. 1 >O *. 1 *. M # M # M # t is also shown in [13 that these assumptions can be verified for a specific nonconforming multigrid method by the use of the following framework. First we should establish a relation between the nonconforming finite element space.. n addition to (2.2 and (2.3 the two spaces are also assumed and its conforming relative to satisfy the following properties which have been established in [10. et be related by "D >O P D V *. and. + 5 <*. + 5 <*. %^ ('% " O #W XJZe[ \ % 1 :TO " O W XFZe[ \ % 1 :TO " O WYXFZe[\ Then the following estimates hold: (5. (5.7 (5.

10 G O V-cycle and F-cycle methods for the biharmonic problem 121 where K is the index of elliptic regularity in (1.2 and D. is the Morley interpolation operator defined as follows. For each V the function. satisfies (5.9 A and * *= where and range over the internal vertices and edges of # Secondly we need the following estimates concerning and which have also been established in [10 : 1 (5.10 % 1 %^ T<. (5.11 % X \ " W X \ # (5.12 ' " O #W+XJZe[ %^ >O P V ( #O " O W XFZ=[ \ >O P V Finally the following new estimates are required for relating mesh-dependent norms between two consecutive levels. First of all we have ( " O 1 <*. < : V where the positive constant is mesh-independent. Moreover the operator extended to map +. to. (see the next section for details and we have (5.15 #'# % V#'% < +. (5.1 Q% X \ " #T('% < G " O O 1 <. M V can be (5.17 where the positive constant is mesh-independent. The theory in [13 can be applied to V-cycle and F-cycle multigrid methods using the Morley element once we have (2.2 (2.3 (5. (5. and (5.10 (5.17. We will establish the new estimates (5.14 (5.17 in the next section and complete the convergence analysis.. Convergence Analysis. We first extend the definition of to a larger space. n fact the definition (5.9 can be extended to et a D Secondly the integral 0 then ( First of all the value is well-defined for * is also well-defined for D e n particular if.# * * + X * with (cf. Figure.1. Therefore the linear operator where is well-defined from the larger space into. n particular D +.. and -D +.. are both well-defined. Before we prove the estimates (5.14 (5.17 we give two lemmas. EMMA.1. The following estimate holds: (.2 V % X \ %^ +" #'# " V#'% <" +.

11 122 Jie Zhao FG..1. An edge Proof. et ( nodal interpolant of i.e. SD ( and the quadratic polynomial on ( be the Morley (.3 0 A 0 and * f *= for - and where and are the vertices of ( and and of ( t is well-known that (cf. [14 and [15 (.4 % X \ +" WYXM\ " WYX:\ et +. and ( Then ( and on ( Therefore (.5 V % X \ +" W+X:\ " WYXM\ The estimate (.2 holds because (.5 is valid for all ( EMMA.2. The following equality holds: (. T +. Proof. et D D +.# be arbitrary. The functions and in.# Moreover we have A T for all and *f * + X * * + X * * * for all( where with (cf. Figure.1. Therefore EMMA.3. The estimate 5.1 holds. That is 3Q% X \ % " V#'% <*. are the edges are both

12 ( ( ( V-cycle and F-cycle methods for the biharmonic problem 123 FG..2. A reference triangle divided into 4 triangles and Proof. et ( be divided into 4 triangles ( ( ( and ( in and ( ( " Then (cf. Figure.2. For define "# for ( ote that ( if and only if " ( f then we define to be et. (d be the Morley finite element space associated with ( ( and ote that. (d is the space of functions 4 (d such that is a quadratic polynomial on ( 0 for = and is continuous at and and is continuous at and We can see that. (d is a finite dimensional linear space and 0 47 W+X:\ (d is the space of linear func- defines a norm on the quotient space. (d ( where tions on ( On the other hand X \ defines a semi-norm on. (d (.7 (fv Therefore T X \ 0 47 WX \ A scaling argument on (.7 yields (. Q% X \ " 0 47 WX \ The estimate (5.1 follows. EMMA.4. The estimate 5.15 holds. That is #'% V#'% < +.

13 124 Jie Zhao Proof. From (.2 (. emma.3 and an inverse estimate we have that for all D +. '% '% 3 T '% + %^ '% " T % X \ + T('% '# V '% Before we prove the estimates (5.14 and (5.17 we state an elementary inequality: (.9 + # T d + : V n the rest of the section we use for a mesh-independent constant. The values of different appearances are not necessarily identical. EMMA.5. The estimate 5.14 holds. That is T " O 1 O for all.# and : # Proof. et.# be arbitrary and". Then by (3.1 we have 1 # " T + (.10 (.11 and (.12 1 " - c - T T + - c - * * where T+ and is the set of triangles sharing as a common vertex. ote that T is independent of T f then the value is well-defined i.e. T T T for all ( sharing as a common vertex. From (3.4 in the definition of we have T. f. Then is the midpoint of some which is the common edge of two triangles (d (fg (cf. Figure.3. After subdivision is the common vertex of triangles in and therefore T = Hence we can write (.13 T + ( c - c Suppose and are the endpoints of (cf. Figure.3. We have T +@ + Then from (.9 we can write B + Q E + + T B S FE at

14 ( ( + V-cycle and F-cycle methods for the biharmonic problem 125 (dg FG..3. A vertex ote that the Mean-Value Theorem and a standard inverse estimate imply that Therefore we have (.15 and similarly (.1 B FE T T Ü T \ Thus from (.14 (.15 and (.1 we have (.17 + #B + E + Taking summation of (.17 over - + c - c + - c Therefore it follows from (.13 that (.1 (.19 - T By the definition of c (cf. (3.5 we have * a gives B + W#\ W \ W \ W#\ T * From the Mean-Value Theorem and a standard inverse estimate we have (.20 * d\ W#\ c c -. c WV\ W#\ W#\ EF WV\

15 O + 12 Jie Zhao for all ( (.21. Therefore from (.19 and (.20 we have * -. - c W#\ By (2.1 (4.4 emma 4.3 (.11 (.12 (.1 and (.21 we have 1 " + - c - c + " T " T " V1 T " O c c W#\ W \ -. c WV\ (.22 EMMA.. The estimate 5.17 holds. That is for all.# and 1 : # T " O Proof. et ".# be arbitrary. t is easy to see from (3.1 that as follows: (.23 1 " where is the set of the vertices of the triangle ( et Then (.24 1 " -. + c - - c O 1 * 1 can be expressed * D By the definition (5.9 of and (.1 we have TA T for all * * + X * * + X * for all (.25 et ( where withf (cf. Figure.1. Therefore 1 " -. c - T + be divided into four triangles ( ( ( labeled as in Figure.4. Then we have " and ( - and * in whose vertices are

16 ( O (.2 - T V-cycle and F-cycle methods for the biharmonic problem 127 FG..4. A Triangle ( ( ( divided into four triangles in. B Q 0 5FE B E T + T 0 47 W \ From (2.1 and (.2 we have (.27 " (.2 (.29 - Summing up over all ( " -. c - " " gives T T " sing a similar argument as in (.19 (.21 we have - * W \ T " WV\ Therefore from (4.4 emma 4.3 (.23 (.25 (.2 and (.29 we have 1 "^ "^ - -. c " T " V1 T " O 1 -. " - WV\ 0 47 W#\ *

17 12 Jie Zhao (a Square domain (b -shaped domain FG The triangulation for We have proved the required estimates (5.14 (5.17 in emmas.3.. The following theorems are then established by the additive theory (cf. [13. THEOREM.7. There exist a positive constant and a positive integer both independent of T such that for all and *. a (.30 T :('% TO5 ('%e where is the exact solution of 3.. THEOREM.. There exist a positive constant and a positive integer both independent of T such that for all and *. (.31 T T :#'# Ö #'%e where is the exact solution of umerical Experiments. n this section we present some experimental results to illustrate Theorem.7 and Theorem.. First let be the unit square : : (cf. Figure 7.1(a. Since the domain is convex we have full elliptic regularity i.e. the index K in (1.2 is = et 1 be the contraction number of the th level V-cycle iteration with pre-smoothing and postsmoothing steps. According to Theorem.7 there is a constant independent of and such that (7.1 1 The numerical results in Table 7.1 are consistent with (7.1. n fact they seem to indicate that could be some number less than 10 and that (7.1 is valid for 1 m=20 m=30 m=40 m=50 m=0 m=70 m=0 k= k= k= k= k= k= TABE 7.1 V-cycle results on the unit square

18 V-cycle and F-cycle methods for the biharmonic problem 129 REMARK 7.1. ote that the condition number of the operator (cf. (3.2 and (4.3 is of order " while the condition number for second order problems is of order " Therefore the effect of smoothing steps for fourth order problems is equivalent to the effect of smoothing steps for second order problems. The key to the improvement of the performance of multigrid methods for fourth order problems is in the design of new smoothing operators. Besides [2 [ and [12 there are also some new composite relaxation schemes that may also apply (cf. [21. et 1 be the contraction number of the th level F-cycle iteration with presmoothing and post-smoothing steps. According to Theorem. there is a constant independent of and such that (7.2 1 The numerical results in Table 7.2 are consistent with (7.2 and seem to indicate that 1 and (7.2 is valid as long as m=10 m=11 m=12 m=13 m=14 m=15 m=1 k= k= k= k= k= k= TABE 7.2 F-cycle results on the unit square n the case of the -shaped domain (cf. Figure 7.1(b the index of elliptic regularity is umerical results for V-cycle and F-cycle algorithms are reported in Table 3 and Table 4 which are also consistent with (7.1 and (7.2. K 1 O m=30 m=40 m=50 m=0 m=70 m=0 m=90 k= k= k= k= k= k= TABE 7.3 V-cycle results on an -shaped domain REMARK 7.2. Even though the asymptotic convergence rate for both algorithms is TO # the performance of the F-cycle algorithm is clearly superior as demonstrated by the numerical results in Table 7.5 and Table 7.. Similar results also hold for the -shaped domain. Compared with the W-cycle algorithm the contraction numbers for F-cycle are larger for small numbers of smoothing steps. n Table 7.7 the contraction numbers of W-cycle

19 130 Jie Zhao O 1 5 m=11 m=12 m=13 m=14 m=15 m=1 m=17 k= k= k= k= k= k= TABE 7.4 F-cycle results on an -shaped domain algorithm are given for n general F-cycle algorithm diverges for these s. However for larger (say for the contraction numbers for both algorithms are almost the same and sometimes the F-cycle algorithm is even better (cf. Table 7. and Table 7.. Considering the fact that the cost for W-cycle is higher we could say that the F-cycle algorithm is even more efficient then the W-cycle algorithm for between 11 and 1. t would be interesting to find a theoretical explanation for the superior performance of the F-cycle algorithm (see also [31. 1 m=34 m=35 m=3 m=37 m=3 m=39 m=40 m=41 k= k= k= k= k= k= TABE 7.5 Contraction numbers for V-cycle algorithms on the unit square 1 m=11 m=12 m=13 m=14 m=15 m=1 k= k= k= k= k= k= TABE 7. Contraction numbers for F-cycle algorithms on the unit square Acknowledgment. The author would like to thank his advisor Professor Susanne C. Brenner for her advice and encouragement.

20 V-cycle and F-cycle methods for the biharmonic problem m=3 m=4 m=5 m= k=7 m= k= k= k= k= k= k= TABE 7.7 Contraction numbers for W-cycle algorithms on the unit square 1 1 m=11 m=12 m=13 m=14 m=15 m=1 k= k= k= k= TABE 7. Contraction numbers for W-cycle algorithms for large s. REFERECES [1 J.H. ARGYRS. FRED AD D.W. SCHARPF The TBA family of plate elements for the matrix displacement method J. Roy. Aero. Soc. 72 (19 pp [2 R.E. BAK AD C.C. DOGAS Sharp estimates for multigrid rates of convergence with general smoothing and acceleration SAM J. umer. Anal. 22 (195 pp [3 R.E. BAK AD T.F. DPOT An optimal order process for solving finite element equations Math. Comp. 3 (191 pp [4 F.K. BOGER R.. FOX AD.A. SCHMT The generation of interelement compatible stiffness and mass matrices by the use of interpolation formulas Proc. Conf. Matrix Methods in Structural Mechanics 195. [5 J.H. BRAMBE Multigrid Methods ongman Scientific Technical Essex [ J.H. BRAMBE AD J.E. PASCAK The analysis of smoothers for multigrid algorithms Math. Comp. 5 (1992 pp [7 J.H. BRAMBE AD J. X Some estimates for weighted projection Math. Comp. 5 (1991 pp [ J.H. BRAMBE AD X. ZHAG The Analysis of Multigrid Methods Handbook of umerical Analysis P.G. Ciarlet and J.. ions eds. orth-holland Amsterdam V (2000 pp [9 S.C. BREER An optimal-order nonconforming multigrid method for the biharmonic equation SAM J. umer. Anal. 2 (199 no. 5 pp [10 S.C. BREER Convergence of nonconforming multigrid methods without full elliptic regularity Math. Comp. (1999 pp [11 S.C. BREER Convergence of the multigrid -cycle algorithm for second-order boundary value problems without full elliptic regularity Math.Comp. 71 (2002 pp [12 S.C. BREER Smoothers mesh dependent norms interpolation and multigrid Appl. umer. Math. 43 (2002 pp [13 S.C. BREER Convergence of nonconforming -cycle and -cycle multigrid algorithms for second order elliptic boundary value problems Math. Comp. 73 (2004 pp [14 S.C. BREER AD.R. SCOTT The Mathematical Theory of Finite Element Methods 2nd ed. Springer- Verlag ew York-Berlin-Heidelberg [15 P.G. CARET The Finite Element Method for Elliptic Problems orth-holland Amsterdam ew York Oxford 197. [1 R.W. COGH AD J.. TOCHER Finite element stiffness matrices for analysis of plates in bending Proceedings of the Conference on Matrix Methods in Structural Mechanics Wright Patterson A.F.B. Ohio 195. [17 M. DAGE Elliptic Boundary Value Problems on Corner Domains ecture otes in Mathematics no. 1341

21 132 Jie Zhao Springer-Verlag Berlin 19. [1 P. GRSVARD Elliptic Problems in onsmooth Domains Pitman ondon 195. [19 W. HACKBSCH Multigrid Methods and Applications Springer-Verlag Berlin 195. [20 M.R. HASCH Multigrid preconditioning for the biharmonic Dirichlet problem SAM J. umer. Anal. 30 (1993 pp [21 O.E. VE AD A. BRADT ocal mode analysis of multicolor and composite relaxation schemes Comput. Math. Appl. in press [22 S. MCCORMCK ed. Multigrid Methods SAM Frontiers in Applied Mathematics 3 SAM Philadelphia 197. [23 J. MADE AD S. PARTER On the multigrid F-cycle Appl. Math. Comput. 37 (1990 pp [24.S.D. MOREY The triangular equilibrium problem in the solution for plate bending problems Aero. Quart. 19 (19 pp [25 P. PESKER AD D. BRAESS A conjugate gradient method and a multigrid algorithm for Morley s finite element approximation of the biharmonic equation umer. Math. 50 (197 pp [2 P. PESKER W. RST AD E. STE terative solution methods for plate bending problems: multigrid and preconditioned cg algorithm SAM J. umer. Anal. 27 (1990 pp [27 Z. SH On the convergence of the incomplete biquadratic nonconforming plate element Math. umer. Sinica (19 pp [2 R. STEVESO An analysis of onconforming multi-grid methods leading to an improved method for the Morley element Math. Comp. 72 (2003 pp [29 H. TREBE nterpolation Theory Function Spaces Differential Operators orth-holland Amsterdam 197. [30. TROTTEBERG C. OOSTEREE AD A. SCHÜER Multigrid Academic Press San Diego [31 S. TREK Efficient Solvers for ncompressible Flow Problems ecture otes in Computational Science and Engineering Springer-Verlag [32 P. WESSEG An ntroduction to Multigrid Methods Wiley Chichester [33 J. ZHAO Multigrid Methods for Fourth Order Problems Dissertation niversity of South Carolina 2004.

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