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Applied Numerical Mathematics 43 (2002) 399-421

~ ApPLIED~NUMERICAL

MATHEMATICS

www.elsevier.com/locateJapnum

Abstract

Additive Schwarz algorithms for solving hp-versionfinite element systems on triangular meshes ~

V. Korneeva.b, J.E. Flahertyb.*, J.T. Oden c, J. Fish h

a Sciemilic.Research Im·titllte for Mathematics and Mechanics. St. Petersburg State University.St. Peterslmrl;. Petrodvorets. 198904. Russia

h Scientific Compllta/ion Ro('{/rch Center. Rel/Sse/aer Polytechnic Institute.Troy. NY 12180. USA

c Texas Instilllte for Complllllriollal and Applied Marhemmics, The Universiry (1Texas at Austin,Austin, TX 78712, USA

High]y parallelizable domain decomposition Dirichlet-Dirich]et solvers for hp-version finite element methodson angular quasiuniform triangular meshes are studied under different assumptions on a reference element Theedge coordinate functions of a reference element are allowed to be either nodal with special choices of nodes.or hierarchical polynomials of severa] types. These coordinate functions are defined within elements as beingarbitrary or discrete quasi-harmonic coordinate functions, The latter are obtained from explicit and inexpensiveprolongation operators. In all situations. \ve are able to suggest preconditioncrs which are spectrally equivalentto the global stiffness matrix. which only require e]ement-by-element and edge-by-edge operations. and whichreduce computational cost In this way. elimination is avoided when dealing with the interface problem. Thedomain decomposition algorithms essentially use prolongation operators from the interface boundary inside thesubdomains of the decomposition according to the approach initially used for the hp-version finite element methodswith quadrilateral elements [S.A Ivanov. VG. Korneev. ]zv. Vyssh. Uchebn. Zaved. 395 (1995) 62-8]: TechnischeUniversitat Chemnitz-Zwickau, Preprint SPC 95-35, ]995. 1-15. and Preprint SPC 95-36. 1995, 1-14; Math.Modeling 8 (1996) 63-73].© 2002 IMACS. Published by Elsevier Science BV All rights reserved.

'" Research supported in part by grants from the OrtIee of Naval Research NOOOOI4-97-I-0687, from the Department ofEnergy 8~41495 and 8347883. from the Army Research Office NDAAG55-98-1-0200 and DAAH04-96-1-002. and from theAir Force Office of Scientific Research NF49620-95-1-0407.

" Corresponding author.E-mail addresses: vauim.komeev@pobox.spbu.I'u (V Komecv). flahej@rpi.edu (.I.E. Flaherty). oden@ticam.utcxas.edu

(IT Ouen). tishj@rpi.edu (J. Fish).

0168-9274/02/$22.00 © 20021MACS. Published by Elsevier Science B.V. All rights reserved.PII: SO 16S-9274(02)OO I 04-6

400

1. Introduction

E Kamen' er al. / Applied Numerical Marhematics 43 (2002) 399-421

The /zp-version of the finite element method (FEM) is well suited to efficient computation 111

many practical situations. When used adaptively. it is capable of producing exponential rates ofconvergence 14,27.32]. It also permits the use of highly parallel domain decomposition (DD) solutionprocedures.

Indeed. an important consideration in the ei'llcient implementation of lip-version FEMs is the algebraicsolver and. for this reason, the development of lip-version DD solvers has received a great deal of recentattention. The basic ideas arise from studies of DD algorithms in general and for the h-version FEM.Fundamental developments are due to Bramble et al. [6.7]. Nepomnyaschikh [24]. Widlund [31]. andLions [20]. The first analysis of DD Dirichlet-Dirichlet solution algorithms for the p-version FEM wasdone by Babuska et a1. [3]. who also developed important technical tools for future studies, Assumingorthogonality of the internal and edge coordinate functions of elements. they obtained an O( I + log2 p)estimate of the relative condition number similar to the O((H / /z)2) estimate of Bramble et a!. [6] for theh-version FEM. Further developments of DD Dirichlet-Dirichlet algorithms. which are also our concel11.were aimed at obtaining preconditioners that are spectrally equivalent or almost spectrally equivalent tothe stiffness matrix 1 and providing lower computational cost. Cost was reduced by using less expensivesolvers for the internal and interface subproblems and avoiding direct elimination procedures. Ivanovand Korneev. in [12-] 4] considered hp-version FEyts on square reference elements having productsof integrated Legendre polynomials for coordinate functions. They showed that a spectrally equivalentpreconditioner for internal problems is a matrix of a five-point finite-difference-Iike operator of secondorder in the reference square, The component of the DD preconditioner related to the interface problemwas obtained by an efficient Schur complement preconditioning and the use of explicit prolongationoperators. Some alternative preconditioners were suggested by Korneev and Jensen [18.19]. The globalDD preconditioners of [12-14.18.19J. developed on the basis of the mentioned efficient componentpreconditioning. were proved to have relative condition numbers in the range 0(1) - 0(1 + log3 p).

The problem of inexact solvers on interface smfaces for the hp-version FEM was considered byAinsworth [21. who generalized the result of Bramble et a!. [6J to the hp-version. and Guo and Cao [IS].Their approach relies on a transformation of the basis in the space of polynomial traces on the edge ofthe reference element to the basis of Chebyshev polynomials. This was used for a similar purpose ina different algOlithmic setting by Ivanov and Korneev [] 1-14J and Korneev and Jensen [18]. Iterativesolvers for adaptive lip-version finite element discretizations based on two-level orthogonalization andcoarse grid preconditioning was considered by Oden et al. [25]. Pavarino [26]. Widlund [30]. andCasarin 191. respectively. studied DD preconditions with overlapping regions. for hp-version mortarmodels. and for the nodal hp-version FEM. The approach to preconditioning initiated by Babuskaet at [3J is based on orthogonalization on the reference element and has been developed by Mandel [22](and the references therein). There are also a number of studies on DD solvers for special types of partialdifferential equations. for models based on spectral elements. and for mixed formulations.

The results of this paper. as those ofivanov and Korneev 111-14] and Korneev and Jensen [18.19J. relyupon a specific transformation of the edge and vertex functions to produce efficient preconditionings of

t In this paper. a preconditio/l(,/, for a symmetric nonnegative matrix A is a symmetric nonnegative matrix B such thatkcr(B) c ker(A) and condfB+ A) < cond(A), where condrA) is the ratio of the maximal cigcnvalue to the minimal nonzeroeigcnvaluc of A and B+ is the pseudo-inverse of R. The relalil.'e or g/'n/'raliz.ed conditio/lnulI/bel' of A is cond( B+ A).

V Komeev er ill. / Applied Numerical Ma/hemarics 43 (2002) 399-421 401

r

the global stiffness matrix by using its block-diagonal part with independent blocks related to the interiorunknowns of each subdomain of decomposition and to the interface unknowns. The transfonnationcorresponds to the prolongation of the traces of finite element functions on the inter-element boundaryinside elements as discrete quasi-haJmonic functions. If the edge and vertex unknowns are not splitfurther. then the resulting preconditioner is spectrally equivalent to the global stiffness matrix. Furtherformal splitting of the edge and vertex unknowns may introduce an additional factor of InO + p) inthe relative condition number depending on the definition of the vertex coordinate functions. We furtherreduce computational cost by obtaining preconditioners for the diagonal blocks of the global stiffnessmatrix. This makes it possible to replace expensive direct solutions at each leading iteration of the interiorand interface problems with far less costly direct or secondary iterative procedures. By this, we arriveat DD preconditioners that are either spectrally equivalent or almost spectrally equivalent to the globalstiffness matrix and. at the same time. are easily solved.

Our primary concerns are establishing some basic facts related to the general form of DDpreconditioners and analyzing several ways of preconditioning interface problems. We also providesimpler analyses for some basic results on DD preconditioning. Optimizing the preconditioners for theinterior problems on the subdomains of the decomposition is, of course. a much more difficult problemfor triangular elements than for quadrilaterals. Herein. we do not provide an ultimate solution and suggestonly initial steps in this direction.

The procedure for defining edge and vertex coordinate functions by means of prolongation operatorsmay be easily incorporated into existing software, For this reason. Shephard et ai. [291 used a descriptionof edge and face functions based on continuation of the traces through blending functions. In thisconnection. we distinguish two major cases: (i) discrete quasi-harmonic edge coordinate functions, and(ii) arbitrary edge coordinate functions, The underlying ideas of preconditioning are more transparent incase (i) as well as the formulation and analyses of the algorithms. The suggestions that are useful for case(i) may also be used for case (ii). By these reasons. case (i) is paid enough attention. But the authors. asthe estimates of computational work given in this paper. do not imply that among several approaches toDD preconditioning. the transformation to the basis (i) for the preliminary step will be generally. e.g.,for all p. the best choice. To the best of the authors' knowledge. it was never used in DD algorithms forh-version FEMs,

We consider algorithms where elimination of internal unknowns is replaced by less expensiveprocedures; however. elimination of internal unknowns is still used in practice. As we show. all ofour preconditioners that have been designed for the interface sub-problem in the case (i) are equallyapplicable in the general case (ii) to systems obtained upon elimination of internal unknowns.

The paper is organized as follows. In Section 2. we formulate the problem and describe the finiteelement space. Section 3 presents reference elements with the discrete quasi-harmonic edge coordinatefunctions. which are specified in the reference element by means of prolongation operators. In Section 4.we consider several DD preconditioners depending on the type of the edge coordinate functions and theway of handling vertex unknowns. which in a preconditioner mayor may not be split from the rest of thesystem. Preconditioning in the case of arbitrary coordinate functions is studied in Section 5. Sections 4and 5 contain estimates of the generalized condition numbers related to our preconditioners. Technicalproofs are given in Section 6.

402 ~~Kamccl' er al. I Applied Numerical ,'v[athematics 43 (2002) 399~21

2. Finite element space

For definiteness, consider a Dirichlet problem for a second-order elliptic partial differential equation ina two-dimensional domain Q. The generalized formulation of this problem involves determining u E \1such that

aQ(u, v) - (f, v)J? = 0, for all v E \1, (2.1 )

(2.3)

(2.4)

where the bilinear form aJ? (u, v) is symmetric, bounded on \1 x \f, and \f -elliptic with \f being the spaceHri (Q).2 Thus, there exist constants f-i I, {.L2 > 0 such that

.,f-illJvlJj,Q::;; aJ?(v, v). aQ(v, w)::;; 112IIvlll,alJwIJLJ? (2.2)

The problem is discretized into an assemblage of finite elements Tn r = I.2, ... , n, specified bynondegenerate compatible mappings x = x(r)(y): io ---+ ir having positive Jacobians. We assume thatil'l n ir2 E 0. rl #- r2, unless il'l , ir2 have a vertex or an edge in common. Such an assembly of trianglesis called a triangulation and is denoted as S. The reference triangle may either be the unit right triangleTo = {y: 0 < YI. Y2, (YI + Y2) < I} or the unilateral triangle with vertices y = (-1,0), (1,0). (0, J3).When elements have curved edges, the mappings ,.1::'(r)are nonlinear and may be characterized in differentways. For this purpose, we introduce the Lame coefficients

[

2 ] 1/2(1') (1') 2

HI; = 2::)a,.1::'t layd . k = 1. 2.1=1

Also, let h(y) be the unit vector in the x-plane tangent to the line Y3-1; = constant and directed towardsincreasing )'b k = 1.2. We introduce e(r)(y) as the angle between i 1(y) and i2(y). Then the generalizedquasi uniform mesh conditions are expressed as

0< a(l)h ::;; H}")::;; h. 0 < () ::;;e(r) ::;;1T - 2a. a(I), a = constant,

!Dfx(r)(y)1 ::;; chlql• ]::;; Iql ::;;n. y E TO, r = I, 2, ... , n,where h is a mesh spacing parameter. The inequalities (2.4) are needed to analyze convergence, but thestudy of DD preconditioners only requires satisfaction of (2.3). If, however. (2.4) is satisfied for, at least,n = 2, then, with h sufficiently small, (2.3) may be replaced by the geometric conditions [17]

a ::;;PI', I IPr.2. (2.5)

where PI'. 1 and Pr,2 are, respectively, the radii of the largest inscribed and smallest circumscribed circlesrelative to Tr. In this case, (2.3) may also be understood in a simple geometric sense as restrictions on thelengths of edges and angles of the triangle T: having vertices common with the triangle Tr and straightsides. Eqs. (2.3) and (2.5) arc called the conditions of generalized angular quasiuniformity, if h = her).

Lemma 2.1. Let Q be an arbitral)' dOl1/ain with bOllndary aQ E C(t J, t ~ 2, and II > 0 be sufficientlysmall. Then there exists (//I exact triangulation S of Q having cOl1/patible l1/appings x(r) : io -+ Tr andsatisfying (2.3), (2.4) with a(ll =a(I)(Q), e =a(Q), and n = (t - 1).

2 Fundamental notation is summarized in Section A.3.

E Korneel' et ai, I Applied NUlIleriU/1 Mathematic,\' 43 (2002) 399-421

Proof. See Korneev [16. I 71, 0

403

When solving (2.1), polynomial mappings. particularly iso-parametric ones, may be used for X(r1,and S need not exactly coincide with il. However, since convergence of approximate solutions is not ourconcern, only S matters in this study, Thus, unless specifically noted, we assume that Q is the domain ofsome triangulation S and that the generalized quasiuniformity conditions are fulfilled. For simplicity, wealso assume that only one reference element is used for the finite element assembly.

Let the reference element coordinate function be denoted as Pc<. ex= (0'1.0'2.0'3) taking values onsome set W,1, In the case under consideration.

span Pc< = PI'.x.C<Ecv,1

and the coordinate function p~) of an element E, defined on rr satisfies

p~)(x(r)(y») = pc«y). Vy E ro.

These functions induce the following spaces on r, and Q:

(I')H (rr) = span Pc< •«EOJ.1

H(Q)={VEC(Q): vir, EH(rr). r= 1.2 .... ,'R.}.HO(Q) = H(Q) n v.

The system of finite element algebraic equations

Ku = fis equivalent to (2.1) in HO(Q). so that

(2,6)

(2.7)

(Kw. v) =a[J(w, v), (f. v) = (f. v)[J. (2.8)

for any vectors w. v and the corresponding functions w. v E HO(Q).

3. Reference elements

The choice of reference element coordinate functions critically affects the properties of the stiffnessmatrix K. We consider several choices that differ in their edge and vertex coordinate functions. Thetraces of the coordinate functions of one of these choices on each edge of ro are Lagrange interpolation

r polynomials relative to sets of nodes that are the same as those used by Ivanov and Korneev [II] on theedges of square reference elements. We call such coordinate functions "nodal". although they may notbe nodal within To. For other possibilities. the traces of edge coordinate functions on iho are hierarchicalpolynomials and the vertex coordinate functions are linear, We need not select specific internal coordinatefunctions because we neither consider fast solvers for the internal subproblems nor do our main resultsdepend on the specific fonn of the internal coordinate functions; however. different choices may producebetter conditioning of the stiffness matrix [32.8].

404 E Kumec!' er al. / Applied Numerical MllIhell/llIics 43 (2002) 399-421

3.]. Nodal edge and vertex coordinate functiolls

Let V(k) denote the vertices of the reference triangle ordered in a counterclockwise direction, T(k) bethe edge joining vertices V(k-I), V(k), and Sk = Lk - Lk-l, where Lk are barycentric coordinates. Theindices k are understood to be defined modulo 3. With WL\ being the set of indices ex = (ai, a2, (3), let

• ex = (k, O. 0), k = I, 2, 3, identify the vertex coordinate functions,• ex = (k.l, 0), 1= 1,2, ... , P - I, identify the edge coordinate functions associated with edge T(k),

and• ex E Wl1,t := lex: 1 ~ al. a2, (al + (2) ~ p - I, a3 = I} identify the internal coordinate functions.

For definiteness, for the internal coordinate functions we assume

(3.1)

(3.3)k = 1,2, 3, I = I. 2, ... , p - I.

In order to define the vertex and edge coordinate functions, let their traces on 3To be interpolationpolynomials and continue them into TO by a special prolongation operator. To this end, introduce thenodes

sY)=cos(p-/)rrp-l. I=O,I ....• p. (3.2)

on TikI and the con'esponding Lagrange basis of p-degree polynomialsP

A,(/)(, - n (, (/11»)/( (/) (11/»)'f' .I'd - '\k - Sk Sk - Sk '

/11=0, /1/#/

(3.5)

(3.4)

The traces of the edge coordinate functions are

,_{¢I/)(S;). ifi=k,._ _ ')p(k,!.o)h·lI) - 0 ,k - 1.2,3. 1- I, -, ... , P - I., otherwise,

The traces of the vertex coordinate functions are

{

¢(p)(s;). if i = k,p(k.o,odr'i' = ¢(O)(s;), if i = k + t,

O. otherwise.

Suppose that Vo is any function that is continuous on 3To and a polynomial of degree p or less on eachedge T(k), k = 1,2,3. Let IF'r be a linear prolongation operator, such that v = IF'rvo satisfies

I/vl/I.ro ~ clP'l/volll/2.ilrn' Ivll.rn ~ cplvoll/2.ilro' (3.6)

with c]' independent of P and v. With this prolongation, the vertex and edge coordinate functions are

Pet = IF'rPet,ilr' (3.7)

where Prt.ilr are the traces of the coordinate functions Prt on aTo given by (3.4), (3.5). The prolongationoperators that may be used in computation are described in Section 3,3.

3.2. Hierarchical vertex and edge coordinate fUllctions

The vertex coordinate functions for a hierarchical basis are the lil1ear polynomials

pet=L(t!. ex=(al'O,O), al=I,2.3. (3.8)

V. Komeev eT al. / Applied NW!lerica/ MarhemaTics 43 (2002) 399--421 405

(3.9b)

(3, II)

(3.12)

The traces of the edge coordinate functions on TO may be defined by different sets of polynomials. Onechoice is

I _{LI+I(Sd. ifk=a,. k-1231-12 I (39)Pa. TIO - 0 ,- . . . - . , ... , p - , . a, otherwise,

Sk~ f f3jLj =f3j PJ-I(t)dt = 2j -1 [Pj(sd - Pj-2(sd],-I

with Pj(s) being the Legendre polynomial of degree j and fJj being a normalization parameter. As asecond choice. consider the edge coordinate functions with the traces

{(I ~) I-I 'fk.' _ - sk sk • I '= 0'1. '_ _ _Pall(kl- . k-1.2,3,1-1,2, ... ,p I. (3.10)0, otherwise,

Again. we use the prolongations (3.7) to extend traces inside the triangle.

3.3. Prolongation operators

Construction of prolongation operators in polynomial spaces for which (3.6) holds were studied byBabuska and Sun [41. Babuska et al. [3]. Bernardi et at. [5]. and Munoz-Sola [23]. Since we useprolongation operators in our algorithms. we follow Maday [211, Bernardi et al. [5] where these operatorshave a simpler foml for numerical realization,

Consider the equilateral triangle To described in Section 2 with V(I) = (- 1. 0) for definiteness. Todefine edge functions, it suffices to consider one edge and we focus on T(2), Let 1f.r(S2)' S2 E [-1. 1], bea polynomial of degree P or less. If 1f.r(S2) is a trace of an edge function, it has the form

1f.r(S2) = (I - Si)1f.rO(S2).

where 1/fO(S2) is a polynomial of degree p - 2 or less. We set

Pr1/f:= F2,0(1/f)(X) :=4L,L2F2(1/fo(.S2»)(X).

x,+x21J3 I

v'3 f ,If ( X~)F2(x)(x)=- x(t)dt=- X XI+ ;'s ds,2~ 2 v3

.'I-X21,fi -I

Liftings (prolongations) from the other edges are defined similarly: thus. we may wlite the edgecoordinate functions Pa for any k = 1.2.3 as

Pa.(x) = PrPci.iJr := hO(Pa..ilrIT(kI)(X) := Lk-ILkFk((l - s~r'(Pa..ilrITlkl»)(X). (3.13)

,. It follows from (3.11 )-(3,13) that

F2(X(S2»)(x)ITI2i = X (.\'2),

F ('Ir( »)( ',)= {o. if x E ar;,o, \ T(2).2.0 'I-' s2 X ,Ir( , ' 'f T(2)

'I-' .12). I x E .

with similar expressions holding for Fk and Fu). k = 1. 3. Therefore, (PrPa,iJr)lilrll = Pa.ilr' and it iseasy to see that PrPa..iJr E p".x.

406 V. KO/'lleel' et al. / Applied Numerical Mathemlltics 43 (2002) 399-421

With a goal of specifying the prolongations for the vertex coordinate functions. let

Pa.l(s/) = PaITIII,

where the traces are described, e.g .. by (3.5), We introduce the lifting

Fk,{3(1/1 (sd) := 2Lk+ItHl/2Fd(l - {3sd-11/1(Sk»)(X)

for polynomials 1/1(sd that are specified on T(kl and which vanish at one of the ends. i.e .. 1/1({3) = O. and{3 = -lor 1. With ex = (k. O. 0). the function

l[/k(X) = Paor(x) - h-I (Pa.k(sd)(x)

is continuous on aro and only nonzero on side r<k+II. Thus. we may set

~(klPa = lPrPa,iJr := Fk (Pailr)(X).

F?\Pa.or)(X):= Fk.-I (Pa.k(sd)(x) + Fk+l.o(lJIkITI1+l)(X).

and it is easy to verify that ~(k)(Pa.iJr)(X) E Ppx and F(k)(Pa,or)(x)liJro = Pa,iJr'Another possible lifting is defined as

1T1l ~(k+l)JrrPa.ilr := Fk (Pa,ilr)(X),

f:(k+1) (PaiJr)(x) := FkH 1 (Pcd-I (sk+d )(X) + hO(lJIk+IITll')(X).

where

(3.14)

(3.15)

(3.17)

l[/k+1 (x) = Pa,iJr(x) - Fk+I.I (Pa.k+1 (Sk+I»)<X),

In order to make the liftings independent of the direction of the vertex indexing, we define them as

I (~(k) , ~(k+I) )lPrPa,iJr := 2 Fk (Pa.i/r)(X) + Fk (Pa,iJr)(x), (3.16)

However. the lifting (3.16) requires twice as many operations as either (3.14) or (3.15).In Section 5, we also need a prolongation operator that may be applied to traces of any polynomial

from Ppx. The space of such traces is denoted as pp.x(aro) and contains functions that are continuouson aro and are polynomials of at most degree P on each edge. To describe the lifting for such traces letvex) E Pp,xearo), choose any side T<k) and define

<l>k= vex) - Fdv)(x)laro'

This function is continuous on (:lTo, zero on side T(k), and is a polynomial on each of the sides T(k-]) andT(k+I). Therefore, any of the operators defined for the vertex coordinate functions in (3.14)-(3.] 6) maybe used to lift it. These liftings will generally give different results, and they are not independent of theedge used in the first step. An invariant lifting may be defined as

I ~[, I (~(k) ~(k+l) )]lPrv:= 3' ~ Fdv)(x) + 2 Fk (CPd(x) + Fk (CPd(x);k=1

however. this requires three times as many operations of the noninvariant lifting. For edge and vertexcoordinate functions (3.17) gives the same result as (3.11) and (3.16), respectively.

V. KOrl/eel' et al. / Applied Numeriwl Mathematics 43 (2002) 399-421 407

(4.4)

(4.1)

The basic operation in the prolongations is F2, which is easily written explicitly [3.5]. Therefore. theexplicit expressions for discrete quasi-harmonic edge and vertex coordinate functions are easily obtainedfor any type of traces described in Sections 3.1 and 3.2.

Remark 3.1. In our algOlithms, prolongations replace discrete solutions of Laplace's equation withinhomogeneous Dirichlet boundary conditions on the reference element. The former is advantageous (seeSection 5 for the operation count) only if the prolongation is less expensive than solving the Dirichletproblem. This is the case for the current state of the art, especially for triangular elements. The situationmay be different for hp-methods with square reference elements. In a forthcoming analysis. we willsuggest a nearly optimal solver of the Dirichlet problem using p-version finite element approximationson square reference elements.

4. DD Preconditioners for discrete quasi-harmonic interface coordinate functions

4.1. Block-stnlctured preconditioners

Let the interface be the set T = (U~ I aT,.) \ a Q. N be the n umber of unknowns. and N" Nil. NIlIbe the number of the internal, edge, and vertex unknowns. respectively. The stiffness matrix K may bewritten in the 2 x 2 or 3 x 3 block forms

(K(/) K(Uf)) (K' Ku, KUfI)

K = K(lU) K(/f) = KfI.1 Kfl KfI.llI•Kllu KIlI.1I Kill

where K(l) = K, and KLare NL x NL, L = 1.1l./lI, sub-matrices corresponding to the internal, sideand vertex unknowns, respectively. We have introduced all interface or edge coordinate functions asdiscrete quasi-harmonic functions by means of explicit prolongation operators. As with square referenceelements [14, I9], this makes it possible to effectively use block diagonal preconditioners of the fonTI

K = diag[K(f), K(lf)]. K" = diag[K,. KII' Klld. (4.2)

where the block diagonal sub-matrices K(l) and K, (which may be identical) and KII are

KI = diag[K" I. KI.2' .... KI.R], KII = diag[KfI, I, K".2. ' ... K".Ql (4.3)

Here Q is the number of element edges T(k) within Q ordered globally. Each block KI.r has dimension11, x 111 with 11, = pep - I)/2 and is related to the internal unknowns on T,.. r = 1. 2, , . , . R. Each blockKII.i has dimension (p - I) x (p - I) and corresponds to the unknowns on the edge in Q having globalnumber i. If coefficients of the elliptic equation are constant. then all blocks K I. •• as well as blocks Krl.i,may be chosen to be identical on an angular quasi-uniform meshes.

Before considering more efficient preconditioners. we consider the simplest choice when the globalstiffness matrix is preconditioned by its block-diagonal part. i.e.,

K=diag[K(I). KllIl], Kl' =diag[K,. K", Kill]'

Lemma 4.1. Suppose that K and Kt, are defined by (4.4) and

(1) the triangulation and mappings X(" are angular quasiunijorm, and

408 E Komeev et al. / Applied Numerical Mathematics 43 (2002) 399-421

(2) the edge and vertex coordinate functions of the reference element are discrete quasi-hal711onic. i.e"they are obtained by means of the lifting operator IPT satisfying the inequalities (3.6).

Then

If the vertex coordinate functions of the reference element are the linear functions (3.8). then

CI,', Kv::::; K::::; 3Kv·1 + logp

The constants CI and Cl.l' depend only 011 (j and IL, and µ2·

Proof. See Section A.l . 0

(4.5)

(4.6)

Remark 4.1. Lemma 4.1 is easily generalized to the case of many spatial variables. We outline ageneralization of (4.5) for a preconditioner with two independent blocks. Let S" be a quasi-uniformtriangulation defined on a polyhedron Q E mm. m ;:: 2. with simplices Tr having, for simplicity. planefaces. We use similar assumptions on the reference simplex TO. In particular, span PO' = pp.x. IfPO' laTo ¢O. and. thus, Pais a vertex. edge or face coordinate function, then it is discrete quasi-harmonic.i,e.. it satisfies inequalities (3.6) with v = Pd and Vo = PO" Under these assumptions, (4.5) remains validwith the blocks K(I). K(!/) corresponding to internal and the remaining unknowns, respectively. Theproof of these inequalities is the same as with In = 2. except that we need to use 3 - d prolongationoperators satisfying (3.6). With m = 3. it is also possible to use preconditioners with three and four blockson diagonaL Inequalities (4.6) hold with a dependence on logk P and k adjusted to the properties of thecoordinate functions. Suppose, that 111 = 3 and the lower indices I. II. III correspond to internal. face. andthe remaining (edge and vertex) unknowns. respectively. Also let T;.o be an edge of TO. i = 1.2 .... , 6.and the edge and veltex coordinate functions satisfy

(,

lip", 1I1f2,dTo ::::; cL II PO' Ilo,1i.o·;=1

Then. assuming (3.6) to be satisfied. we obtain (4,6) without the (1 + log p) factor.

Once preconditioners B = K or B = Kv have been defined. they may be used to solve the system (2.7)by the preconditioned conjugate gradient method. GMRES. or another iterative method. The typicaloperations in a preconditioned iterative method are represented by the simple preconditioned iteration

(4.7)

To implement (4.7) it is not necessary to know B or B-1 explicitly. We have to know the sequence ofoperations necessary to evaluate B-1v for any given vector v. As long as preconditioners B = K. K"are block diagonal. we have to know how to calculate (K<L))-IVIL) or Ki.'VL, where V(L) and VL arethe corresponding restrictions of a vector v E ~W"'. We may also define (K(lI»)-1 v(ll) and K/,) Vll/ bysubsidiary iterative processes with an a priori estimated fixed number of iterations. In this case. we callthe iteration (4.7) the leading iteration process.

V. Komeev et al. I Applied Numerical Matlremarics 43 (2002) 399-421

4.2. Preconditioning of the internal problems

Suppose that A(I) and A(O) are the n x n matrices defined by the identities

409

vT A(l)v = f Y'v, Y'wdx.

ro

vT A(O)v = f vwdx

rn

for any polynomials v, W E pp.x and vectors v. W E mil of the coefTIcients of these polynomials in thebasis (Pet) where n = (p + l)(p + 2)/2. Let A~I) and A~O) denote the restrictions of A(I) and A(O) to theinternal degrees of freedom. The blocks !C'.r of the preconditioner !C, may be defined as either

yo A(l) J2 (OJor ''''.I' = µr , + VI' IrA, ' r = 1. 2.... , R, (4.8)

In the case when coefTIciellts of elliptic equation do not change much. the simplest choice for theparameters µr and Vr is unity. Omitting vrh;A~O) may be justified because A~O) ~ CA~I) with an absoluteconstant c. because Ajl) may be much simpler than AjO) and because h may be small. Unless statedotherwise. we use the first of (4.8) with µr = I.

Let (2.2) and the assumptions of Lemma 4.1 on the mappings x(r I hold. Then K I,r and !C1.r arespectrally equivalent. i.e.,

(4.9)

with the constants depending only on alii. f). µ I. and µ2. These inequalities are direct consequencesof (2.2) and Lemma A, I. The use of !C1.r. r = 1.2, ... , R. as preconditioners for the internal problemscan save computational work even if direct solution procedures are used to solve systems governed bythese matrices. The matrices may be factored once for all elements and the leading iterations. Assumingthis has been done. the computational work to solve R internal problems with kE leading iterations willbe O(kE Rp4) in the worst case of a full A jl).

4.3. Preconditioning of the intelface problem: nodal inteljc/ce/uncrions

Suppose K~I). K:.f.I/J. K~II/). and K~") are the blocks of the element stiffness matrices correspondingto the 2 x 2 block representation of K according to (4,1). The sub-matrix KIll) is the topological sum ofthe sub-matrices K;ll). r = 1,2. , .. ,n. This is represented by the relationship

(4.10)

for all vectors v, WE mN(/ll. where NIIII = NI + Nil. and the vectors Vr, Wr are restrictions of v, W tonodes on a Tr.

410 E KOnJeev ct al. / Applied Numerical Mathematics 43 (2002) 399-421

Let us order the 3p nodes on a TO in counterclockwise order starting from the vertex V( J). By means ofthe mappings x(r). this defines the local numbering of the nodes on aTr' r = 1,2, .... R. We introducethe 3p x 3p matrices

2-I

-I

-I2 -I

-1

-I 2 -1-I 2

1:'2

and. for simplicity. assume µr = VI' = 1. Without changing notation, let us assume that the rows andcolumns corresponding to the nodes belonging to aTr n as? are eliminated from Vr. By assembling thematrices Vr, we may define a matrix VUl) which satisfies a relationship. similar to (4.10). i,e.,

(4.11 )

In Appendix A. we show that 'DUE) and K(II) are spectrally equivalent, but the inversion of V(ll) isstill not simple due to its complex structure. Thus. instead of taking this matrix as the preconditioner. wedefine the inverse (Kul I)- ) of the preconditioner by simple iteration with Chebyshev's parameters Y/:

(K(fl)f 1 ~ [I _ jJ (l- y,V(f") }V(fl)f '. (4.12)

Since we only require that K(II) and Kill) be spectrally equivalent, the number n(8) of iterations maybe determined from the condition that the relative eITor 8 in the nonn induced by matrix 'DU/) does notexceed, say, 0.5. In this case, it is easily verified that

0,5Ku/) ::;;'D(1) ::;; I,5Ku/)' (4.13)

and. consequently. that KUI) so defined is almost as good a preconditioner for K(//) as is 'DUl). Evidently.the vector v"(e) found by the iteration

HI k ('DU/) k A.) (01 0 k 0 I ( ) (4 14'v = v - Yk 'v - '1" V =. =. ., ... 11 C • ' )

is exactly

v"(f.) = (K(/E)rJ cfJ. (4. IS)

Therefore. the product (4.15) may be completed by means of the iteration (4.14).Let us estimate the computational work assuming that (2.2) are satisfied. It is easy to establish that

the condition number of V(//) is O(p). Therefore, in order to guarantee that 8 ::;;0.5 and. consequentlythat (4.13) holds, it is necessary to perfoffi1 11(8) = O(.JP) iterations [28]. At each iteration. themultiplication by VUIl may be done in parallel for each element.

It follows from (4,11) that

(4,16)

where 'Dr and Vr are considered to be extended by zero entries. The eigenvalues and eigenfunctionsof Vr• r = 1. 2 ... " R. are well known and are specified in terms of trigonometric functions. For this

V. KOrtll'ev et af. / Appfied Numerical Mclthemarics 43 (2002J 399-421 411

reason. each multiplication VI' VI' may be performed by the Fast Discrete Fourier Transform (FDFf) inO(p log p) operations. The completion of n (£) multiplications (4.16) requires at most O(Rp.,fP log p)operations. which are the main components of the computational cost of the iteration (4.14). In practicalimplementations. a sufficient number of iterations n(e) may be either determined from a priori estimates.which are easily derived. or numerically as described. e.g .. by Samarskii and Nikolayev [28I-

4.4. Spectral equivalence of the DD preconditioner and the global stiffness matrix

The use of the suggested preconditioners leads. as we have seen. to a reduction of the computationalwork when the number of iterations of the leading iteration process (4.7) does not grow rapidly withincreasing p or h -I. According to Theorem 4.1. which follows. the number of leading iterations doesnot depend on p and h. Use of the preconditioner K does not imply elimination of internal unknowns.However. if this is done. we may use the sub-matrix K(I/) as a spectrally equivalent preconditioner for thesystem obtained after elimination. The matrix of this system is the Schur complement and we introducenotation

K~)~)= K(l[) - K(lI.l) (K(I)rl K(I·/l).

Theorem 4.1. Let the assumptions (~f Lemma 4. I hold and the preconditioner K be of the form (4.2) withK(I) = K" Also, let K, = K1 or the blocks Kl.r he defined as in (4.8), and the sub-matrix K(//) be definedby (4.12). Then, there exist constants CI and C2, depending only on 0'(1). e, ILl, and µ2. such that

C vW) ~ K(II) ~ c v(ll)I'" " ort" 2'" . (4.17)

Proof. See the Appendix, 0

Remark 4.2. The matrix K~'!t) is independent of the continuation of the traces of the vertex and edgefunctions into TO. Thus. the second pair of inequalities (4.17) also hold when the vertex and edgecoordinate functions are not discrete quasi-harmonic.

45 Preconditioning of the edge problems with hierarchical coordinate junctions

Here. we assume that the edge coordinate functions of the reference element are hierarchical, e.g ..as given by the expressions (3.9) or (3.10). We would like to specify the blocks Kif.; in the DDpreconditioner K .. of the form (4.2) and (4.3). where each block corresponds to one of the sides commonto two elements. The main step is to derive the preconditioner for the block of the reference elementstiffness matrix corresponding to one of its sides. After proper scaling. we prove that we can use thispreconditioner for each of the three blocks on the diagonal of the stiffness matrix of any element that isgenerated by the edge coordinate functions, We next specify each block KII.i according to the assemblingprocedure with the preconditioners for the edge sub-matllces given for each element. The cost of solvingthe system with the resulting matrix KII.; is O(p2) arithmetical operations.

Let T; be the common side of triangles Trl and Tr,. In general. we may set

(4,18)

412 V Korneel' elal. / Applied Numerical Mathematics 43 (2002 ) 399~21

with KuJ.(» the same for all edges and µ(il = V(il = 2 as the simplest choice, which is done below, Tospecify KuJ.(»). let us introduce the bases

Icos¢

(cos ¢)2 l{I(2) =

Icos¢cos2¢

cos p¢(cos¢)"7 ,,-2(I-sk)sk

Let C, and C 2 be the matrices of the transformations lJtl = C ,lJt( I) and lJt2 = C 2lJt(2), and let A be thediagonal matrix

A = Bol + diag[O. I ..... pl.where Bo is chosen to satisfy 0 ~ BII ~ (I+ log p / II) -I. It is important that C I and C 2 are lower-triangularso that solving the systems Ckv = f and C{v = f. k = 1,2, requires no more than O(p2) operations.Indeed, C I is bi-diagonal so that the cost of solving these systems is only O(p). Deleting the first tworows and columns of

(4.20)

Kg" = CIC2A(C2)T(Cj)T (4,19)

we obtain Kul.o). Although K~/f) is singular when eo = O. Kul.o) is not For any eo pointed out above, themattices Kwo) satisfying (4.19) are spectrally equivalent uniformly in p and the minor so-perturbationin (4.18) for KII.i may generally be neglected. We use this fact in what follows.

Suppose eo ~ O. From the factored representation (4,19). we may conclude that solving the systems

K~I/)v = f. f E ~H("+').V' - f f E \»)(P-l)('-(/I.II)V(lI.O) - (11,0)' (/1.11) ./1 •

requires at most O(p2) operations. Since Kuf.()) does not have such a representation. we first solve (4,20)for v = (vo. VI •. ' .• v,JT using f = Uo. fl ..... f,,)T as a right-hanel side. The two components fa. flof f are considered as unknown parameters which we determine from the system of two equationsthat result when Va = 0 and VI = O. Finally. we correct v so that it becomes the solution of the secondof (4.20). The cost of this algotithm is at most O(p2) operations. Computations are easily adapted to thesituation when Bo =O.

A preconditioner Kul.o). having the same asymptotic cost and effectiveness, is defined in a similarfashion when the traces of the edge coordinate functions are the integrated Legendre polynomials (3.9).One choice is Kul.OJ = So. where S;) was defined by Korneev and Jensen [19]. Two factored forms of thecorresponding matrix K~I) are given by (3.6) and Remark 3.3 of Korneev and Jensen [19]. From theseforms it follows that. once again. the computational cost is at most O(p2).

4.6. Vertex problem. energy inequalities for the three-block DD precollditioner

We discuss a three-block preconditioner with independent blocks related to the vertex. edge. andinternal unknowns and with the vertex coordinate functions being the linear polynomials (3.8). Onechoice is Kill = K /II. There are several possible simplifications when defining Kill such that the spectra

V Komeev et al. / Applied Numerical Mathematics 43 (2002) 399-421 41)

of Kw and K 11/ are equivalent. We only present one example with Kill = K 11/. where K 11/ is assembledfrom the artificial element stiffness matrices

or - (I) 2 (0)Kr.1lI = µrAw + v"h .411/'

The matrices A~)/ and A~;~)are the blocks of A(I) and A(O) corresponding to the vertex unknowns. Thevalues of /1" and v, are either set to unity or chosen to provide a closer approximation of the matrix K '.w.

If the number of elements is large. solving the system

(4.21)

may still be difficult. However, these matrices have the typical propeJ1ies of h-version FEMs and,assuming that (2.2) hold. a variety of iterative methods may be applied. e.g .. multigrid. domaindecomposition. and fictitious components. Since we use such processes as secondary iterations, it is moreefficient to solve (4.2 I) \vithin reasonable accuracy. provided that the spectral equivalence inequalities

O,5Kw :::;;K 11/ :::;;I.SKII/ (4.22)

hold, where Kill is now defined by the iterative procedure for solving (4.21). If (2.2) hold. the number ofiterations sufficient to satisfy (4.22) may be determined from well-known a priori convergence estimates.If. for example, the multigrid method is used. the number of iterations is independent of hand p and thecomputational work to solve (4.21) is 0(h-2).

A preconditioner Kl' of the form (4.2) and (4.3) with KL,i. L = I.ll. as defined in previous sections.and !Cw. as defined here, is less efficient than K. This is due to the formal splitting of the vertex unknownsand the choice of KII.i '

Theorem 4.2. Let the assllmptions of Lemma 4. I hold. the traces of the edge coordinate functionssatisfy (3.10). and the preconditione/' Kl' be of the f01111 (4.2) and (4.3). Let liS also assume that eitherK, = K, or the blocks Ku are defined by (4.8). and the blocks KJ/ and Kill are chosen according to thesuggestions (~fSections 4,S and 4.6. Then there exists constants c, and C2 depending only on 0'(1), e, µ I,

and µ2 stich thatCI

J Kv:::;;K:S;c2(l+logp)KI" (4.23)1+ log- p

The same inequalities hold (fthe traces of the edge coordinate fimctions are given by (3.9) and the matrixK(/!'(lI in the blocks of the preconditionel' KII is defined as described in Sectioll 4.5.

Proof. The proof is similar to that of Theorem 5.2 in Korneev and Jensen I19]. 0

Remark 4.3. As with Theorem 4.1. the inequalities (4.23) remain valid if the edge Galerkin coordinatefunctions are orthogonal to the internal Galerkin coordinate functions for each element. Thus. the two-block preconditioners

KT = diag [KII• Kw Imay be used for solving the system obtained by elimination of the internal unknowns from the rowsand columns corresponding to the edge unknowns. The edge coordinate functions may not be discretequasi-harmonic functions.

414 V. Komeev et al. / Applied Numerical Mathematics 43 (2002) 399-421

5. Arbitrary interface coordinate functions and DD preconditioning

In existing computer codes based on the hp-version FEM. edge and vertex coordinate functions of areference element. as a mle. are not discrete quasi-harmonic functions, However, the preconditioners ofSection 4 may also be used in this case after adjustment by either transforn1ing the algebraic system ormodifying the preconditioners, We discuss both possibilities. In order to distinguish between (2,7) andthe algebraic system generated from a basis having nondiscrete quasi-harmonic interface functions. wewrite the latter as

Ku = j. (5.1)

The notation j( and feu is used for preconditioners that are the counterparts of I( and 1(". and Pa for thecOlTesponding reference element coordinate functions. We let K = K t' and I = Iv when only the edgecoordinate functions are discrete harmonic.

5.1. Transformation of the algebraic system

We transform (5.1) into a system of either the fonn (2.7) or Kuu = I" by setting

Cll = ii.C"u=u.

(5.2)

where C and C I" respectively, represent transfonnations of the edge and vertex and edge coordinatefunctions Prt of To onto the discrete quasi-harmonic coordinate functions

(5.3)

in the manner of (3,7). Thus. Care C" are defined explicitly through the liftings introduced in Section 3.The transformations (5.2) may be performed on the element level in parallel for each element and withoutassembling the matrices K. K, or K". since the iteration procedure only requires multiplication by K orK", Therefore. if K,.. K,.. and f,. and Ir are element stiffness matrices and load vectors correspondingto (5,1), we have

where C e is the restriction of C to an element Similar relations are used when only the edge coordinatefunctions are transformed. Transformations on the element level reduce the computational cost. especiallywhen the coefficients of the elliptic equation are constant on all or some of the elements.

In order to describe the stmcture of C" induced by the liftings of Section 3.3. let M = (M(/). M(J/)and M = (MU). M(//) cOlTespond to the rows containing functions fi'om the bases Prt and Pa. where theindices I and I I identify the internal and boundary (edge and vertex) coordinate functions. respectively.Assuming that the internal coordinate functions do not change. we write (5.3) in the matrix form ~\..1(1l) =M(Pr? with Pr = (PrI.l). where I is the 3p x 3p identity matrix and PrJ is a 3p x pep - 1)/2matrix. The matrix Ce is the same as in the equality iVl = A1C;:: therefore.

CT = (I 0)e PI'r, I

E Kflmeev et III. I Applied Numerical Mathematics 43 (2002) 399-421 415

(5.4)vk = CTZkr e'

In iterative procedures such as (4,7). the matrix K appears only in the multiplications vk = K uk,

This may be performed element-wise in parallel, i.e.. by multiplying v; = K I'll;. These, in turn, may beperformed by the sequence

k C k k K~ kY = ellr' Z = rY,

The multiplication of a vector by K would ordinarily be performed; thus. the additional operationsneeded to calculate yk = C "ll; and v; = C;'Zk are at most O(p~) for each element. For R elementsand ke iterations, this amounts to O(Rke pl). Since. by Theorem 4.1. the ke is independent of p and h.this estimate is asymptotically less than O(Rp4). which was obtained for the transformation (5.2) of theglobal stiffness matrix. The cost of the transformations (5.4) is also less than the cost of multiplicationsZk = Kr/ which. at complete fill in, is O(Rk,p4).

If only the edge coordinate functions are discrete harmonic, the prolongation and transformationmatrices, respectively. have the form

5.2. A((justments of preconditioners

If the edge coordinate functions are not discrete quasi-harmonic. we may introduce preconditioners Kand Kr for the global stiffness matrix K that are counterparts of the preconditioners K. and Kv for K, Wedefine them through their inverses as

(5.5)K~-l = K' + (p(S»)T K-1 pis) + K-", II lIf'

~).

ie-I = (K(/)t + pT(K(I/)rJ p,

where

o 0) (0 0 0)o O. Klil = 0 0 0 .o 0 0 0 K0

and P = (PI. 1) is the prolongation operator whose restriction to every element is the matrix P r

introduced in Section 5.1. Again, we need not assemble P and pCI) because multiplications by thesematrices and their transposes may be done in an element-by-element manner. These preconditioners arejustified by the following proposition,

Proposition 5.1. The inequalities

respectively. are equivalent to

K~-J CTK-ICv - l' 1.7 V·

Proof. Using the structures of the transformations C and C I' and that of ie-I and j(~ J • we verify that (5.5)may be rewritten as

R;-l = CTK.-IC.

416 V Kart/eel' et af. / Applied NrunericaJ Marhell/atics 43 (2002) 399-421

From these relationships. it follows that K = CTKC and K" = C~Kl,Cl" Since K, K. and K. Kv arerelated by identical transforms, this completes the proof. 0

This proposition allows us to reformulate Theorems 4, I and 4.2 for K and its preconditioners K andKv. Thus. instead of (4.7). we can use one of the iteration processes

iik+l =iik -O'B-I(Kiik -f). llk+1 =li -O'B-I(CTKCllk -CTf). k= 1.2 ... ,. (5.6)

with the preconditioners B = K. K". The rate of convergence is the same as for the iteration (4.7)with the preconditioners K, Kl: (see Theorems 4.1 and 4,2). Comparing the iterations (4,7) and (5.6).we note first that the transformations of the global stiffness matrix required for (4.7), as established,may be more costly than transformations in the manner of (5.4). Additionally. the matrix K maybe denser. The iteration (5.6) appears to have the same asymptotic computational cost. The cost oftransformations similar to those used in the second iteration of (5.6) are p-times less expensive than thecost of multiplying by K or K. This multiplication is one of the two major operations in these iterativeprocedures,

Appendix A

A.1. Proof of Lemma 4.1

Proof. The right sides of (4.5) and (4.6) are Cauchy-Schwarz inequalities. Let us prove the left inequal-ity (4.5) under the additional II . II u? in Eq. (2.2) may be replaced by I . I u? The proof changes slightlyunder initial assumption. Let us represent D(y) = v(x(rl(y». V E H(r,), in the form

and use the inequality., ., ") )

IDII>I- + Iv(//)I- ~ 21DI- + 31/;(11\1- .I .Tn I .Tn I .TO I .TO

(Al)

(A2)

Since the polynomial D(I/) is discrete quasi-harmonic. i.e,. vun = JP> I' vun I UTn' we can use (3.6) and thetrace theorem [1] to write

. , , , ,I Dl/l} 17. Tn ~ CIP IWfI

) 117/2,iJTn = CII' II V 117n.:JTn ~ CCIl' II V 117. TO

with an absolute constant c. Now. by a Bramble-Hilbert Lemma argument, we attain

IvUll I ~ clvlI.TO I,l'l)

with an absolute constant c. Combining (A2) and (A4). we have

Applying Lemma A I (which foIlows) we conclude that

C3(Q'lllsille)(lvUII~ +IV(/III~ )~Ivl~ ~c4(Q'(')sine)-llvI2 .I .T( I .1', I . I'll I.1',

(A.3)

(A.4)

(A.5)

(A6)

where v = v(t) + villi is a representation of the polynomial v similar to (A2). Using (2.2). (2.8).and (A6), we establish the left inequality (4,5).

V. Komeev et al. / Applied Numerical Matl1elllatio' 43 (2002) 399-421

We prove the left inequality (4,6) by using Lemma A.2 (which folJows). 0

417

(A.7)

Lemma A.I. Let the angular generalized quasiunifonnity conditions hold so that (2.3) hold withh = her), Then, for every wE H(ro), the function v(X(')(y)) = w(y) E HI (rr) (lmlfor some absolutecOl/stants c). Cz

ct(al')sin8)lvI2 ::::;Iwlz ::::;cz(all'sin8)-llvI2.), T, I.TO ,t. r,~

(sine)(a(llht ::::;d,::::; h2,

where d = d,(y) is the Jacobian of the mapping X<'l(y).

Proof. This lemma is a particular case of Lemma 7, I in Korneev 117J, 0

Remark A.I. In the case of affine mappings, we have

PI'.I 2Jd,.T1Vll.rr ::::;~::::; Pr.zlvll,T,.. (A,8)

If the mapping is nonlinear, we can introduce a triangle T, with straight sides that has common verticeswith rr. Suppose inequalities (2.3) and (2.4) hold with 11 ~ 2. Then v.'e can consider f} in (A.7) as thesmallest angle at the vertices of Tr and a(l) may be omitted. Similarly. Prj, PI'.Z may be taken for T,.However. the factors (1 - O(h)) and (I + O(h)) appear. respectively, on the right and the left sidesof (A.7) and (A.8) with the O(h) term depending on the constant in (2,4).

Lemma A.2. Let any polynomialu E ptl ..\' be represented in the form u = w + v, where W E span(~\11 UMil) and v E iI/till. Let the bilinear form a(, .. ) be

a(cp.VF)= f Vcp·V1/Jdx or a(cp,1/J)= !(VCP.V1/J+CP1/J)dx.I'll

Then. there exists absolute constalll Ct such that

~[a(w. w) + a(v. v)]::::; a(u. II) ::::;2[a(w. w) + a(v. v)].I +Iogp

Proof. The proof is exactly the same as that of Lemma 2.2 in Komeev and Jensen [19] for squarereference elements. except that the norms and the trace theorems for the triangle ro arc used. 0

A. 2. Proof of Theorem 4.]

Proof. First, we use Lemma 4.1 to conclude that the left pair of inequalities (4.17) can be established byprovmg

cllCl ::::; K I ::::;cZ1C1. cllCu/l ::::; KIll) ::::;czlC(//). (A.9)

When ICI =I=- K I, the first pair of inequalities (A.9) follows from (4.9), Thus. it remains to prove thespectra] equivalence of K(//) and 1C11I1• By (4.13) this is reduced to the inequalities c) DUll::::; KU/I ::::;

Cz D(//) and. further. due to (4.10) and (4, II) to the inequalities

C1D;/I) ::::;K :lIJ ::::;Cz D;/I). (A. 10)

418 l~ Komeev et (II. I Applied NUII/erical MlIlhell/lItics 43 (2(){)2)399-421

If the bilinear form ag(l.'. w) corresponds to the Laplace operator, these inequalities are satisfied withconstants c, and C2 depending on Q' ( I) and () as a consequence of Corollary A.I to Lemma A3 (whichfollows) and Lemma AI, For a more general bilinear form. it would also be necessary to use (2.2). Inthis case, the constants in (AI 0) and, consequently. in (A9) also depend on µ I and µ2. The proof of theinequalities (4.17) involving K~~) follows similar lines. Since Galerkin coordinate functions generatingK(//) and K~:!t) possess similar properties (some are discrete quasi-harmonic and others are discreteharmonic functions). we establish that

(I/) K(lil K(/I)C, K on ~ ~ C2 ort·

Using this with (A9) completes the proof. 0

Let ~Ci be the notation for the nodes on the boundary aro described in Section 3.1. where a =(k, I. 0), k = 1.2. 3. I = I, 2..... p - I. Let qCi with the same indices a represent nodes distributeduniformly on aro such that each edge is divided into p equal parts. The positions of nodes ~Ci E T(k) arespecified by their coordinates Sk = s~/) given by (3.2). By L1 (iho). we denote the space of continuousfunctions on aro that are linear between the nodes qa .

Lemma A.3. For lillY v E Pp.x(aro) and WELl (arl) slich that

v(~a) = w(qa). Vqa Earl).

the norms II . IlljVrn and quasinorms I ' 11j2,iJroare equivalent. i.e..

Ilwlllj2.ilro X IIvlllj2.iJro' Iwll!2,;lrn X Ivll/2.ilro· (A 11)

Proof. Similar inequalities were established by Korneev and Jensen [19. Lemma 6.1] for square refer-ence elements with a tensor-product polynomial space. Although the geometric form and the space of theelement under consideration are different. the proof follows the same lines. 0

The stitlness matrix of the reference element may be represented in a block form as

(A(/) A (1.11) ) (A A )A - . , A UiJ _ 1/ 1/,111

- A(/l.1) Ail/l' - AIl/.ll Am '

and factored in the product

(I 0) (A(l)

A = AUl.l1(A (/)-1 I 0

SUIl = AUI) - AUI·l)(A(/)rl A(/·1li.

Corollary A.I. Suppose A = A (I) and the edge and vertex coordinate functions are discrete quasi-harmonic polynomials as described in Section 3. I, Then

D x S(ll) x A (II). (A.12)

Proof. For each edge. we define the mapping f'(k) --i> f'(k) in the form Sk = - cos IUk + I), Sb tk E

[-1. 1]. These mappings define the transformation aro --i> a,(}. which is denoted as x = Y(y). It is

~~K01'lleel' et ul. I Applied Numerical Mathematics 43 (2002) 399-421 419

continuous on tho and preserves the norm II . 111/2.01'0'The latter is proved by a manner similar to (6.10)of Komeev and Jensen [19]. Thus. if vex) E P",x(3ro) and vcos(Y) = v(Y(y», then

IIvlll/2,iJroX IIvcnslll/2.iITo'

Equivalence in the spectrum of the matrices V and SUI) follows from (A.13) and

(A.13)

Iv 11/2.iJToX (ViII)) TS II/) V(II).TIwll/2.ilTIiX (VU/)) Dvll/) , (A.14)

The left relationship (A. 14) follows from the fact that the equivalent definition of the quasi-nann j·ll/2,iJTIIas

Ivll/2.iJTO= infl¢II/2.iITo· (A.lS)

where the infimum is taken over all ¢ E HI (ro) satisfying the boundary condition ¢(x) = v(x), X E aro.and the fact that this equivalence is retained for the polynomial space. The right relationship (A.14) isknown from the studies of domain decomposition techniques for the h-version FEM. see. e.g .. Dryja[10]. It is obtained by interpolation between the spaces HI(oro) n£l(aro) and L2(aro) n£l(aTO).

The spectral equivalence of S(ll) and A (If) follows from (A. 15). the trace theorem. and the definitionof the interface coordinate functions as discrete quasi-hannonic polynomials. i.e .. from (3.5) for 'If v E

Pi!.~= span (.IV/(I/). 0

A.3, Notation

The notation used throughout this work follows.

• pp.x is the space of polynomials of total degree not exceeding p ~ 1; PI'. x (aro) is the space of tracesof these polynomials on oro.

• ('. ').a • II . 1I.a = II . lIo ..a are the scalar product and the norm in L2(D).• 1·le.a. II ·lIk ..a are the semi-norm and the norm in the Hilbert space Hk(D), i.e ..

Ivl~,.a = L 1(D~/vf dy.1<i1=k .a

• HJ (D) is the subspace of functions from H I (D) having zero traces on aD ,• 11·111/2.1,011, 111/21with 1= (a.b) are norms in the space H1/2(1) and the subspace oHlj2(1) C

HI/2(1) of functions having zero values at x = a. b [I]. These norms are defined as

/1 " ~, If (V(X) - V(V»)"11.'11/2.1:= x _ y . d\'dy.

1')~(' fh ~~ ~ v- x) v-ex)0111.'111/2.1:=111.'111/2.1 +2 -d, +2 -ely.

x-a b-xa a

420 ~~Komee\! et al. / Applied NlIlIlericall ....lmlrematics 43 (2002) 39lJ-421

• The norm II· 1I1/2.TIO. where T(kl is an edge of TO joining the vertices V(k-l) and V(k) is definedanalogously to II . 111/2,/, For example, for the edge T<21 on the line X2 = 0, we have

I I '), ff(V(t, 0) - vCr. 0»)-Ivl'/2.7I~) := t _ r dtdT.

-1-1

• We also need the normI, ~ ') ~ f (v-,dt) - v+dt))2

IIV l/i/2.iJro = ~ IIvll'/Hck' + ~ . . dt.k=1 k=1 0

where V-.k and V+,k denote the values of von the sides T(kl and T(k+ll. respectively. at a distance tfrom the vertex V(k). We set Ivel/') 'Ir := IIv1l21/, ~r - IIvllo2 iJr . The norm and semi-norm defined in

_.f. I) .....11 (I • (I

this way for the space H~/2(aTO) m'e equivalent to IIv 1I1/2,oT[j := inf IIw III. Til and IVII/2.iJTO := inf ltV Il.rowith intimae taken over tV E H I (TO) for which tV = V on oro.

• eli) is the class of boundaries a Q composed of a finite number of I-times continuously differentiablepieces which meet at angles distinct from 0 and n.

• A+ is the pseudo-inverse of a matrix A,• The symbols -<. >-, x denote one sided and two sided inequalities, which hold with some absolute

constants omitted.

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