COMPUTER METHODS IN APPLIED MECHANICS AND ENGINEERING 77 (1989) 181-212NORTH-HOLLAND

TOWARD A UNIVERSAL hop ADAPTIVE FINITE ELEMENT STRATEGYPART 3. DESIGN OF hop MESHES

W. RACHOWICZ, J.T. ODEN and L. DEMKOWICZTexas Institute for Computational Mechanics, The University of Texas at Austin.

Austin, TX 78712, U.S.A

Received 27 March 1989

In this third paper in our series, the issue of designing hop meshes which are optimal in some sense isaddressed. Criteria for hop meshes are derived which are based on the idea of minimizing the estimatederror over a mesh with a fixed number of degrees-of-freedom. An optimization algorithm is developedbased on these criteria and is applied to several model one-dimensional and two-dimensional ellipticboundary-value problems. Numerical results indicate that the approach can lead to exponentialrates~f~nvergence.

1. Introduction

In previous works in this series [1,2], we developed data structures. finite elementformulations, and a posteriori error estimates for adaptive hop finite element methods forboundary-value problems characterized by general elliptic systems of partial differentialequations. We demonstrated that, by an appropriate choice of parameters, the method couldalso be applied to the numerical solution of time-dependent problems and, in particular, toparabolic systems and to convection-diffusion problems. We shall now address the question ofhow mesh sizes h and spectral orders p can be chosen throughout a finite element mesh.

There are (at least) two basic parameters in any finite element formulation of a boundary-value problem that affect accuracy and convergence: the size h of the mesh and the order p ofthe local polynomial approximations. One can expect that as either of these parameters arevaried independently (h - 0, p - 00) the approximation error should diminish in some sense.Thus, if an adaptive finite element method is designed for which the distributions h of elementsizes and p of polynomial degrees (spectral orders) vary quite generally over the domain, onemight expect it to be possible to obtain accuracies and rates-of-convergence superior to eitherpure h-versions or p-versions of the finite element method by judiciously changing both handp. Indeed, some theoretical results of Babuska and coworkers (e.g. [3,4.5]) suggest thatexponential rates-of-convergence can be attained with the proper combination of h- andp-refinements for special classes of one-dimensional problems, including problems withsingularities, and also for certain two-dimensional cases. However, a systematic approachtoward generating an optimal distribution of II and p for delivering solutions with a presetlevel of accuracy (or preset value of estimated error) is not available.

In the present study, we develop a simple. approximate, hop mesh optimization technique

0045-7825/89/$3.50 © 1989, Elsevier Science Publishers BV (North-HolI"'ld)

182 W. RacJlOwicz el Ill., Toward a universal hop IIdaptil·e strategy. Part 3

that can be used in an attempt to construct hop distributions that produce the best results, asmeasured by error estimates of the type studied in [2]. for the least number of unknowns. Forclarity, we restrict our discussion to model classes of one- and two-dimensional ellipticboundary-value problems. The finite element approximations to these problmes are con-structed on I-irregular meshes and local stiffness matrices are calculated using hierarchicalpolynomial shape functions in the manner discussed in [I]. The scheme is based fully on ourhop adaptive strategy in which the mesh is refined/unrefined and' the spectral order isenriched/unenriched successively (following the rules in [1]), until the error in each element(measured by techniques in [2]) is below a preassigned tolerance, this resulting in a mesh with(approximately) the least possible number of degrees-of-freedom.

We note that the discrete optimization problem described here is a very complex one thatcan be formulated in several ways. It does not seem to be practical or possible at this point toseck a rigorously exact solution to it. Our aim is to offer a formulation of the problem and apractical and, hopefully, efficient approximate scheme that leads to a trajectory in space of hopdistributions that terminates at an hop mesh close to the optimal. We are able to judge thequality of the app~oximate optimal mesh only through numerical experiments, but our hopresults are consistently superior to either the adaptive h- or adaptive p-methods for equalnumbers of degrecs of freedom. Importantly, results suggest that the approach also Icads toexponen~ial convergence rates for the h-p method.

Following this introduction, we review some cssential details on the implementation of thehop adaptive scheme introduced in [11 and we develop some general rules for hop meshoptimization. Section 3 is devoted to an analysis of one-dimcnsional hop mesh optimization forinterpolation of smooth functions. There a precise formulation of an hop optimization problemis given and a therorem is proved for a charactcrization of an optimal mesh under certainconstraints. The interpretation of these results leads us ultimately to a scheme that proves tobe effective in more general problems defined on two-dimensional domains. Scction 4 containsa discussion of local hop refinements. The hop adaptive algorithm is finally presented in Section5. The results of applying the method to several test problems are collectcd and discussed inSection 6.

2. Preliminaries on hop methods and mesh optimization

Review of features of the hop method

The hop finite element method discussed in [1, 2] employs tensor products of hierarchicalpolynomial shape functions which. for the two-dimensional case. are p-unisolvent to degreesof freedom defined at nine nodes: four corncr nodes, four midside nodes, and a node at thecentroid. Generalizations to three dimensions are not discussed here, although the ideas aredirectly extendable to three dimensions and some numerical results on three-dimensional casesare available. The one-dimensional specialization is obvious.

One views the hop adaptive strategy as starting from some initial mesh of elements K whichare images of a master element k = {g = (~l' ~2) 1- 1 ~ ~l' ~2 ~ I} under a quadratic map,

9

~f-+xEn; xi=2:x~Nj(~)' i=I,2,;= 1

(2.1 )

W. Rllchowicz et al .. Toward a IIllil'erSlll h-p adaptive strategy, Part 3 183

where Nj(~) are quadratic Lagrange polynomials associated- with the _usual 9-node Q2element. For purposes of discussion, we consider cases in which the mesh {},,,= U KeJ Kovera partition 22"of {},is topologically equivalent to a uniform rectangular mesh. The pr~cess ofenriching the polynomial basis functions by adding polynomials of larger degree and ofsubdividing elements into four subclements (sons) is referred to as an hop refinement. Forh-refinements, we restrict ourselves to those leading to I-irregular meshes as indicated in Fig.lao Thus, an element is subdivided only if it has smaller or equal neighbors; otherwise, largerneighbors must be first subdivided before the element is refined. For p-enrichments. we adoptthe maximum rule whereby the maximum polynomial degree dominates at the interface of twoelements, i.e., when two neighboring elements are domains of shape functions of differentpolynomial degree, shape functions are added to those of the lower order element so as tomake the global basis functions continuous across the interelement boundary (see Fig. Ib).

Ii-p mesh optimization; general rules

We now turn to the question of optimizing the distribution of mesh sizes h and polynomialdegrees p over a' finite element mesh. We begin with an idealized case in which {}, is arectangle and the elements K C {},'I have diameters h K and shape functions of maximumspectral order PK' For definiteness, we assume that the error indicators for various clementsare of the interpolation error type and are functions of the mesh parameters hand p. Later,more general cases are considered. The distribution of mesh sizes and spectral orders over {},his characterized by functions Ii and p:

(2.2)

The error indicator OK for element K will depend on h K and p K and is represented by a localerror density tpK = tpK(h, p): OK = fK tpK(h. p) dx (see [2]). Thus, an indication of the total

1-irregular(a)

2-irregular

(b)

Fig. 1. Choices made in the hop data structure. (a) Examples of I-irregular and 2-irregular meshes: the hop strategyuses only I-irregular refinements. (b) The maximum rule is used at an interface of elements containing polynomialsof different degrees.

184 W. Rachowicz et Ill.. Toward a ulli~'ersal hop adaptive strategy. Pllrt 3

error over .n is given by the functional

J(h, p) = 2: ( CPK(Iz, p) dx .KE9.hJK

(2.3)

Next, let nK denote the total number of degrees of freedom of element K. We introduce adegree-of-freedom density n = n(h, p) so that the total number of degrees-of-freedom M isgiven by

M = In n(h, p) dx, (2.4 )

The optimal mesh for a fixed number M = Mo degrees-of-freedom will be defined by theh, p-distribution (h*, p*), such that

J(h*, p*) = min J(h, p) .. fnn(h.p)dx~Mo

(2.5)

To resolve this problem by the method of Lagrange multipliers, we introduce the Lag-rangian.

L(Iz, p; A) = J(h, p) - A(L n(h, p) dx - Mn)

and arrive at the optimality conditions

(2.6)

OIpK all _ ()- -A- - .op op (2.7)

Thus, for the entire mesh, the optimal hop distribution results when.. - ..

dlpK I = A = const ,dn p=eonSl

dCPK I = A = const .dn h =eonsl

(2.8)

REMARK 2.1. The optimality conditions (2.7) cannot be fully exploited unless CPK and narcknown as functions of hand p, and this is not possible for general h-p refinements. However,some broad guidelines can be seen. For examplc, for the optimal Iz, p-distribution for fixedMn' (2.8) suggest that the quantity,

Aerror /A ndof

should be equidistributed over .nh• Aerror bcing the change in error uue to a change andof inthe number of degrees-of-freedom. A refinement strategy designed to reduce the error asmuch as possible would make the change in error per change in ndof as large as possible. Itwould seem that if derror IAndofl II aeonSl and Aerror IAndofl p =eonsl are diffcrent, then therefinement resulting in the larger of these two should be adopted in searching for theoptimum.

W. Racholl'icz et al .. Toward a universal hop adaptive strategy. Part 3 185

REMARK 2.2. Some special cases of mesh optImIzation can be obtained from the abovearguments. Consider first only h-refinements on a rectangular mesh:

P = Po = const., 0K(h, Po) = O/«(h) = L f(x) , xJh" dx .

1n(h, Po) = n(h) = h2 '

where f(x) , x2)hrT

, CF > 0 is an error distribution function. Then (2.6) reduces to

CFf(x)hrT-) + 2>..h -3 = 0 ,

and we have (if IK h-2 dx = 1),

r f(x)hrT dx = OK = -2>"/CF = const .JK .

(2.9)

(2.10)

Thus, we obtain the well-known result that the optimal mesh is obtained whenever the error(or error indicators) are equid~stributed throughout the mcsh.

For a hypothetical hop mesh in which (approximately)

r 2"M = In (p the) dx , (2.11)

conditions (2.6) yield the optimality conditions

(2.12)

The last condition suggests an equidistribution of the quantities shown equal to 2>...

In general, the form of the error function 'PK is unknown and, in our h-p scheme, n(h, p)may be a very complicated function of hand p or it may not exist at all. Thus. we can gleanonly general rules for seeking optimal meshes from these results. Onc principle may emergefrom this calculation: choose a sequence of hop refinements wherein the change in error perchange in number of degrees-of-freedom is maximized. We shall attempt to derive a strategybased on this criterion in the next section.

3. An hop refinement strategy for one-dimensional interpolation

We shall now describe an h-p refinement scheme for minimization of intcrpolation errors ofa smooth function u defined on a one-dimensional domain {l = (0, L) C IR. The stratcgy isbased on the following ideas.

(I) We begin with (say) an initial. continuous. piccewise linear (p = 1) approximation of It on

186 W. Rachowicz el al .. Toward a ulliversal hop adaptive Jlmleli.\', ParI 3

a uniform mesh of N elements {lk (hk = LIN), k = 1,2, ... , N. Later we admit the possibilityof more general initial meshes having nonuniform distributions of hand p.

(2) Each of the subdomains {lk (the initial elements) is next subjected to an hop refinement, asubdivision into smaller elemcnts and a possible enrichment of spectral orders. The resultingstructure of the approximation is defined by functions

h = h(x) = mesh distribution, hi K = h K = dia( K) ,

p = p(x) = order distribution, PIK = PK = order of shpae functions over K ,

II = Il(x) = degrees-of-freedom distribution.(3.1)

where K is a subelement of {lk and Ilk is the total numbcr of degrees of freedom of {lk after thehop refinement. Note that, with an appropriate numbering of nodcs, " may be determined as afunction of p/ h, but there are many different choices of h- and p-retincment leading to thesame total number .of degrees of freedom. .

(3) Considcr the local finite element space

. V"'P({lk' Ilk) = the ·finite-dimensional linear space of piccewise polynomialsrcsulting from all possible hop refincments of f2k rcsulting inexactly 11k degrees-of-frccdom . (3.2)

V",P({lk' 2) = @l,([Xk_l, xk]),

V"·P({lk' 3) = @l1([Xk_1' xk - hk/2]) EEl@l.([Xk_1 + hkl2. xk])

EEl{x2- (xk + xk_.)x + XkXk_.} n C"({ld

and Vh,P({lk,5) is the space of piecewise linear, quadratic, cubic and quartic functionscorresponding to the eleven possible refinements of {lJ.. resulting in five degrees-of-freedom,shown in Fig. 2.

(4) We define as an error indicator Ok for subdomain {lk thc local interpolation error in asuitable norm corresponding to Ilk degrees-of-freedom, e.g ..

(3.3)

Thus, thc calculation of 0k(llk) has resultcd from the particular h-I' refinement of {lJ.. resultingin the smallest error for nk degrees-of-frecdom, Thc dependence of the error on hand p over{lk has thus been taken care of by the calculation of Ok(1/ k)'

(5) The total error measured by such indicators is given by the functional

N

1(n) = 2: 0k(nk) .k=l

(3.4 )

W. Rachowicz et al .. Toward a ulIh'erslIl h·p adaptil·e strategy, I'art 3 187

'f=1'02'C::sp= 1

Fig. 2. Various combinations of It· and p that result in a five degree-of·freedom clement and form a basis forVh'I'(l1k,5).

Following the plan of the previous section, we seek to minimize 1 subject to the constraint thatthe total number of degrees-of-freedom is constant, i.c., we arc led to the discrete optimiza-tion problem:

find n* (or, equivalently, ,,~, n;, ... ,n~), such that

"l(n*) = min l(n), subject to the constraint L "k = M = const .

nERN k=1

Alternatively, let

,liM = {n E IRn I n = (n" "2' ... , "N)' "; = integer> 0, I ~ i ~ N ,

~ "; = M = const} .

Then (3.5) can also be written:

(3.5)

(3.6)

find ,,* E AlM• such that 1(11*) = min l(n).nE,UM

(3.7)

188 W. Rllchowicz et al.. Toward a ul/iversal hop IIdllptil'e slratcRY, I'lIrt 3

Characterization of the minimizer n *The solution n * of the minimization problem (3.6) can be characterized by an elementaryargument. We know that

J(Il) ~ J(n*) \in E A(\1 '

so that

01(n.)+02(n2)+ ... + O;(n;) + ... +ON(n.V)

~ 81(nn + ... + O;(n~) + .,. + O...(nZ) .

Since Lkllt = M = const, we can set

II. = n r, n2 = n;, .. , , n; = n~+ 1. ' .. , II i.= n; - 1, ... , II", = II ~r ,

so that the constraint is satisfied, But (3.8) yields

Similarly,

Introducing the discrete differences

(3.8)

(3.9)

(3.10)

deCa 8.(n.) = O.(n.) - O(n. - I)- I I " I' , (3.11 )

the relations (3.9) and (3.10) can be written

Let oO;(n;) denote the interval

Then we have the following result on the characterization of Il·.

(3.12)

(3.13)

THEOREM 3.1. A solution n* of the discrete optimization prohli'm (3.7) is sitch that (3.12)holds for any i, j. 1 ~ i, j ~ N. In turn, this conditioll implies Ihat Iheri' l'XiSlS II A E IR. such that

AEolJi(nn Vi,i=1.2,.,.,N, (3.14)

PROOF. Condition (3,12) follows immediately from the calculations preceding it. To prove(3.14), assume, contrary to the assertion, that 3i. j such that

W. Rachowicz et al.. Toward a universal hop IIdaptive strategy, Pllrl3

But then we would have

a+~(nn<a_8i(nn or a+8;(n~)<a_8J(nn

in contradiction to (3.12). 0

189

REM ARK 3.1. If the error indicators 8; are convex functions of 11" then the minimizer n * isunique. In general, uniqueness of ,,* cannot be guaranteed.

REMARK 3.2. The intervals aO;(n;) are regarded as discrete subgradients in analogy with thesubdifferentials of nondifferentiable continuous functions, Since

the discrete subgradients of O;(n;) can be viewed as a sequence of subintervals connected atendpoints.

Some implicatiolls of the characterizatiol/ of 11 *To interpret (3.12) and (3.14). consider two adjacent subdomains n~and fl~. I in the initial

mesh, such as those separated in Fig. 3. Various paths, shown by dashct.llincs in the ligurc,indicate the reduction in error indicators 0A and 04 ~ I due to various choices of hand p-aml

• S2k • S2k+,• •

3 4 5 .•.. 2 3 4 5 ....- n '" • • • • n

I .. n

····..·1... ·· .. ·1

I .. n

Fig. 3. Geometrical interpretation of the variation in 0., i/O, with II.

190 W. Rachowicz et al.. Toward tI universal hop adaplive strategy. Part 3

hence Il. The particular local optimal error reductions are the solid lines Ok (n n, 0k+ I(n Z + I)shown, and the discrete subgradients are the intervals shown in solid lines below the O-graphs.

Now let us plot these intervals versus the number of degrees of freedom f1 for each of the Nsubdomains. For example. for N = 7 the situation is represented in Fig. 4. Consider for thiscase a trial solution Il EAt M of degrees-of-freedom corresponding to some arbitrary distribu-tion of hand p resulting in 0k'S and aok's indicated in the figure. Since this trial degree-of-freedom distribution is not optimal, we will not have AE aOk(n). as' shown. However. theoptimal mesh will be one for which the line AE aOk(n*) passes through all intervals, asindicated in the figure.

It is clear from these observations that to modify a trial mesh so that one achieves anoptimal. requires that one refines those elements with la+(HlIk)1 below A and unrefincs thosefor which la_ok(nk)1 is above A. Or; to avoid unrefinements, we can assume A> la + (}k(llk)\ andrefine these elements with the largest subgradients a + 0k(nk). In short, to achieve a solution to(3,7), we must refine those subdomains for which the anticipated decrease in error is thelargest. If the change in error due to an increase in '!; by unity is unavailable, we employ linearinterpolation, as s\.lggested in Fig. 5. Then the discrete subgradients for these artificial n;'sreduce to points and can be neglected. On the other hand. the aok's for nj's must be scaled bylI(actual increment of n;). In other words, our criterion for optimal refinements is: refineelements. for which the anticipated decrease of the error per unit new degree-of-freedom is thelargest.

o~ S2, S2,

An optimal mesh

_ A trial non-optimal mesh

Fig. 4. Illustration of AE aOk(n·) for an optimal hop mesh.

W. Rachowicz et al .. Toward a universal hop adaptil'e strategy, Part 3

.......'0,

191

n2 3 4 5 6 7 8 9 10

IU(~ I = (0(1) - 0(21l/2a,OH) = (0(7) - Onll/3IU(S) = 0(5) - 0(4) = 0(6) - O(S) = 0.0(.5)

Fig. 5. Definition of 8(n) and a ~ 8(n) for cases of an> 1 by linear interpolation.

REMARK 3.3. The proposed strategy is different from the cquidistribution of errors strategy.However, as we have shown in Section 2, for some .special types of error functions (J(n) thesestrategies coincide ..

REMARK 3.4. The assumed property that each function (}k(nlc) depends only on refinementsof the kth element is not satisfied in most problems if errors other than interpolation errorsare considered. However, for a wide class of problems, the influence of refinements is local inthe sense that the refinement of a given element affects accuracy within this element morcthan in neighboring elements.

4. The hop strategy in two dimensions

Some preliminary observations

The two-dimensional case is fundamentally different from the one-dimensional case dis-cussed above. First of all, most of the elementary refinements cannot be restricted to only oneelement: p-enrichment of the midside nodes affects the approximation inside two neighboringelements; h-refinement of a given element may cause subdivision of some of its neighbors inorder to preserve the I-irregularity of the mesh, Since any refinement of an element affectingits boundary affects its neighbor, we cannot formulate the optimization problcm as a sum ofelementary noninterfering problems on initial elements as in (3.7) cven for interpolationerrors [2]. However, the general relations derived in Section 2 and the details of theone-dimensional case in Section 3 suggest a simplified approach for the two-dimensional cascwhich is a straightforward extrapolation of thc one-dimensional strategy: pcrform refinementsfor which the anticipated decreases of the error per new dcgree of frecdom are as large aspossible.

Some technical details of two-dimensional strategies

p-refinementsRecall our maximum convention: if two elements of differcnt degrces havc a common side.

then we supplement the space of shape functions of thc lowcr ordcr clement by the missing

192 W. RacllOwicz 1'1 at .. Toward a lll/iversallt-p adaptive straregy. Part 3

Nodes affected by the refinement:

= -constrained nodes

- - active nodes

p p+l

.. -..

Fig. 6. Possible cases of enriching nodes of neighbors caused by raising p in a given clement.

higher order functions. Since hierarchical shape functions arc used, this means in practice thatthe order of the element is identified with the order of its central node and for two elementswith spectral orders PI and P2' the order of the common boundary node is max (PI' P2)' Thusenriching a given element may affect the boundary nodes of all its neighbors, What is in someway unexpected is that the. affected nodes may not necessarily belong to the common side: inthe case of constrained elements, the order of approximation may be increased along thecommon constrained leg so that a constrained node not belonging to the common sidebecomes enriched (node 1 in Fig. 6). We present all possible cases of enriching nodes ofneighbors caused by raising p in a given element in Fig. 6. Note also that if, according to theresults of Section 3, we attempt to examine changes of the error due to various possiblerefinements, then the trial enrichments of the clement itself and of its neighbors may lead tosome of the situations displayed in Fig. 7,

It-refinementsIn the case of h-refinements, the restriction to I-irregular meshes limits the local character

of refinements: whenever a constrained element is subdivided. we must first bisect itsneighbor. Of course, this procedure may caus'e a sequence of subdivisions of neighbors.

p p,·l

P,~P2+l P"I~P2+1L:J , ...,L:JP, p,.l

=2' =32 pouibilitiee

Fig. 7. All possible ways of enriching an clement.

W. Rachowicz et al., Toward a IIfliversallr-p IIdaplil'e strategy. Part 3

t··.. · ········H H .

193

+

o

•............ ;

......... ]

I--

II

Subdivided Element

Constrained Node

Active Node Created from a Constrained Node

Domain Affected by the the Refinement

Fig. 8. Modifications of the mesh caused by subdivision of a constrained element.

Another modification of elements occurring during this process consists of changingconstrained nodes into active nodes: if a constrained node of a given element belongs to a sideof the larger neighbor which is bisected, then the element-son resulting from the bisection hasthe same size as its previously constrained neighbor, so there is no longer need for constraintsand all the nodes along the common side become active.

The modifications of elements discussed above are illustrated in Fig. 8.Analyzing various types of Iz-refinements, we must keep in mind the objective of determin-

ing changes of errors caused by subdividing elements. From this point of view, subdivision ofan element may be performed in one of the 16 ways shown in Fig. 9a while subdividing itsneighbors may remove constrained nodes, resulting in one of the three situations shown inFig.9b.

EEl , EEl, ...,EE=2~=16 possibilities

o -Conltrolned Node

• -Active Node

(b) llemovin,R cOllstrained oOlh'S:

b,D=3 possibilities

Fig. 9. Possible ways of subdividing an element and removing its conslrained nodes.

194 W. RllChowicz ellll .• Toward a universal hop adaplive stmlegy, Part J

5. An hop mesh refinement algorithm

In constructing the algorithm two important questions arise:

(1) Is it sufficient to examine changes of the error caused only by single elementaryrefinements: raising p by 1, or subdividing the elemcnt only oncc?

(2) Is it necessary to introduce unrefinements in thc proccss of optimization?

That the answer to the flrst question may be negative is suggested by the following example:adding symmetric degrees of freedom may not decrease the error if the solution is antisymmet-ric. Thus. the element may never by adequately refincd if only one stcp forward is employed.

It is also easy to construct examples for which unrefinemcnts are necessary: analyzing aone-dimensional singular solution (such as u(x) = XU), we find that raising p gives a largerdecrease of the interpolation error than subdividing the elemcnt. even though, by consideringa number of combined h- and p-refinements, the best mcsh is found to be that which isgeometrically graded towards the singularity with low p. If we raise p early in the process, itmay be impossible to follow a refinement path toward a true optimal without unrefinement.

These Qbservations suggest that multilevel checking and unrcfincmcnt may be important ina robust optimization algorithm. However. because of the complexity in implementing theseprocedures, we do not consider them in designing the present algorithm, except for theone-dimensional algorithm and for two-dimensional cases in problems with singularities,where limIted versions of multilevel checking are introduced.

In the description of the algorithm below, the following notations are used:

flOh, flOp''flnh, flflp'a) ~ 1,a2,

decreases of the error indicators caused by h- and p-refincments,the corresponding changes in the numbers of degrces of freedom,a parameter characterizing the algorithm.the required accuracy tolerance (the algorithm stops when accuracy

attained).

The algorithm of hop refinements

(1) For all elements in an initial mesh, compute the anticipatcd decreascs of errors for all ofthe refinements displayed in Figs, 7, 9 which may actually occur.

(2) For every element evaluate flOp/ !1np' flOh/flnh. (For interpolation errors, use the proce-dure outlined below.)Set flO/fln = max (AOp/Aflp' A01,lAll/,) and storc for evcry element.

(3) Scan the mesh and identify the largest value of !.i.e/!1n in the mesh: (!10/fl1l)max'(4) Create a list of elements for which /10//111 ~ a, . (~e//1I1)m.u'(5) Perform refinements of elements on the list. The types of refinemcnts corrcsponds to

information stored in (2).(6) Update the solution: solve the problem on the ncw mcsh.(7) Estimate the global error J = Ek Ok' If J ~ a2, stop, otherwise go to (I).

W. Rachowicz et al .. Toward a universal h-p adaptil'e strategy, Part 3 195

The procedure for computing tlfJ/tlnp and tlO,,Itln,, for interpolation error indicators in thetlVo-dimensional case

(2, I)

(2.2)

Subroutine for determining tlfJ/tlnp:

1. Evaluate new orders of nodes of a given element K, and its neighbors.2, Compute tlfJp using errors for K, and its neighbors in the situations shown in Fig. 7.3. Determine the number of the new degrees of frecdom tlnp' and compute tlfJ 1tln .p pSubroutine for determining tJ.fJ,,1 tln/,:1. Evaluate the sequencc of elements that must be subdividcd due to refining the

given element Kj and two-to-one rule (see Fig. 8).2. Determine elements with disappearing constrained nodes (Fig. 8).3. Compute tlfJh using erro~s for all the subdividing clemcnts found in 1 and elcments

listed in 2; appropriate situations from Figs. 9a and 9b must be considered.4. Count the number of new degrees of freedom .In/, and compute!:ifJ/,Itln/,.

6. Numerical exantples

We shall now present several numerical examples designed to illustratc the hop rcfincmentstrategy'proposed in previous sections. In all of these examples, we construct hop approxima-tions of functions which exhibit some typc of irregular behavior duc to singularities or highgradients since, for very smooth functions, our optimization scheme generally leads to onlyp-refinements. All examples arc analyzed using the algorithms presentcd in Section 5. As afirst model error estimate. we usc the interpolation estimates as an crror indicator. assumingthat the intcrpolatcd function is known explicitly. Howcver, wc also consider cases driven byintcrpolation error estimates of solutions of model boundary value problems obtained bypost-processing computed values and gradicnts of the finite element approximations. Ourpost-processing method is based on extraction formulas and a brief discussion of the method isgiven in [2].

In the examples that follow, we attempt to analyze the accuracy of approximations onoptimal meshes and make comparisons of rcsults and rates of convergence obtained by otherrefinement techniques. We also verify numerically that thc optimal ratc of convergence isexponential: Ilell ~ C exp (- f3VN), N being the number of degrces-of-frcedom [6]. Detailsconcerning issues of numerical integration which are crucial in these studies are collected inAppendix A.

EXAMPLE 1. Consider the function

.,-

u=xa, xE(O,l), with a=0.6 .

The initial mesh is uniform and consists of 10 linear elements. Figure 10 shows the optimalmesh obtained after a sequence of refinements. Our algorithm leads to a mcsh geometricallygraded towards the singularity. Only a small number of elcments werc enriched to secondorder and only one element with a node at singularity was enriched to third order.

196 W. Rachowicz et al .. Toward a ulliversal hop adaptive strategy, Part 3

b

I , I l¥i.jtSlip_l 2 3 • 5 " 7 d

c

I : !.;;i"4pot 2 J • 5 6 I 5

Fig. 10. (a) An optimal hop mesh for u = x", a = 0.6, for error indicators obtained by exact interpolation of u, (b)Zoom of (a) centered at the singularity. Magnification 100. (c) Zoom of (a) centered at the singularity.Magnification 109

.

We also considered II = u(x) to be a solution of the boundary value problem

-u"(x) = -a(a - 1)x"-2. a = 0.6, u(O) = 0, u(1) = 1 (6.1 )

and used the post-processed finite element approximation u" of u(x) instead of exact u in theoptimization procedures. The resulting mesh is slightly different than in the first case: it is alsogeometrically graded but no p-enrichmcnts appear.

W. Rachowicz et al .. Toward a universal hop adaplh'e strategy. Pari 3 197

Figure 11 shows comparisons of computed rates-of-convergence for various types ofrefinement strategies. Results of pure h-adaptive refinement are not shown as thcy wercessentially coincident with the h-p refinements in this case. The pure p-adaptivity resulted inraising only the spectral order in the element containing the singularity.

EXAMPLE 2. Next we consider a function which is 'almost singular'. namely a function withsingularity slightly outside the domain: .

u(x) = (x + 0.001)", a = 0.6, x E (0, I) .

The initial mesh is the same used in Example 1. The computed optimal mesh is shown in Fig.12. As in the previous singular solution, the mesh is fIner close to the left endpoint but nowsubdivisions of clcments do not appear extensively in the refinements: beyond five elementspast the origin only p-refinements are chosen and both subdivision and high p-enrichments areused only near the singularity. .

The optimizatio,n results obtained using post-processed flnite element solutions was con-ducted assuming that u is a solution of the boundary value problem,

. -u"(x)=-a(a-l)(x+O.OOl)"-2, a=O.6, u(O) =(0.001)", u(I)=(1.001)".(6.2)

Inllell

-8

........... b

--c___._u d

.----. e

-2.5782.197 ~.369

In(N)

Fig. 11. Rates~f-convergence for various types of refinement strategies obtained for a singular function u = x".a = 0.6. (a) h-uniform, (b) p-uniform, (c) p-adaptive. (d) hop adaptive with exact interpolation. (e) h-p adaptivewith post-processing.

198 W. RacltolVicz et al .. Toward a IIl1iversallt-p adaplive slmtl'gy. Part 3

11I]II!il!CD

Fig, 12, All.opitmal hop mesh for II = (x + 0.001 r. a = 0,6. for error indicators obtained by exact interpolation ofu.

This approach resulted in meshes identical to those produced by interpolation.Rates of convergence are shown in Fig. 13. We observe that the performance of p-adaptive

methods in this problem is as good as the hop scheme. The pure p-scheme, however, producesvery high order p when a small number of elements is used.

EXAM PLE 3. The following two problems illustrate the performance of the hop strategy inthe case of smooth (even analytic) functions with high gradients in some small subdomains.The model function is

u(x) = (1 - x)[tan -I (a(x - xo» + tan -I axo], x E (0, I) . (6.3)

Optimal meshes are computed for the cases a = 50 and a = 200 (a controls the magnitude ofgradient at xo) with Xo = 4/9.

The optimal meshes are presented in Figs. 14 and 15. In both cases, combined subdivisionsand enrichments are observed in the vicinity of xo' In the case a = 200, more h-refinementshave been performed than for a = 50.

Mesh optimization with post-processing was conducted assuming that II(X) satisfies

-(a(x)II'(X»' = [(x) ,

11(0) = 0, li(l) = 0,

a(x) = 1 fa + a(x - Xn)2 ,

[(x) = 1+ [1- 2a(x - xll)(tan-I a(x - xo) + tan-I axo)J/a,

(6.4 )

W. RacJlOlViczet al., Toward a unil'ersllllr-p adaptive strategy. Part 3 199

-0.544\

a

............. b

\\~....'\'\, .

\\,.\.~'\ ."'............\\ ......................

\ \ ....................\ '" ..........\ ,

\ '., "-

" "" \", \ --- c

'\ \ -.--- d\. --\.\ ------- e

\ ' - (\ ----\

Inllell

-6.252.197 4.369

In(N)

Fig. 13. Rates-of-convergence for various typcs of refinement strategies obtaincd for" function u = (x + 0.(01)",a = 0.6. (a) h-uniform, (b) p-uniform, (c) h-adaptivc, (d) p.adaptive. (e) hop i1daptive with exact intcrpoliltion. (f)hop adaptive with post-proccssing .

.. - ..

! I I l·ilmIp.l 2 3 4 5 6 I ij

Fig. 14. An optimal hop mesh for u = (1 - x)[tan -I (a(x - xu» + tan -I axo)' a = 50, Xo = ~/9, for error indicatorsobtained by exact interpolation of u.

200 W. Rachowicz et al., Toward II universal hop adaptive strategy. Part 3

8 b

I I I I @Eap., 2 3 • 5 6 7 8

[ ~'. : I·~ h·llmI ES"p., 2 3 • 5 6 7 8

Fig. 15. (a)-An optimal hop mesh for u=(l-x)[tan-'a(x-xn)+tan-'axn]. a =200. xn=4/9. for errorindicators obtained by exact interpolation of u. (b) Zoom of (a) centered at Xn = 4/9. Magnification 100.

.The resulting meshes (Figs. 16 and 17) are not exactly thc same as those obtaincd from

pure intcrpolation; however, in general they have very similar features. Also. the accuracy forthese meshes is only slightly inferior to interpolation (Figs. 18 and 19) .

I i I leQp., 2 3 • 5 6 7 8

Fig. 16. An optimal hop mesh for u = (1 - x)(tan -I a(x - xo) + tan -I axn), a = 50. Xo = 4/9, for error estimatesobtained using extraction procedures.

W. Rachowicz et al., Toward a uflil'ersal hop adaplive strClfegy, Part 3 201

a b

[ iDil ! I. r I I ti?~1M

p~ 3 • 5 6 1 8, I I I g@

p.l 2 3 • 5 6 7 8

Fig. 17. (a) An optimal hop mcsh for u=(I-x)(tan-1a(x-xn)+tan-'axo). a =200, xo=4/9, for crrorestimales,oblained using eXlraclion formulas, (b) Zoom of (a) ccntered ;It Xo = 4/9. Magnification 100.

,,,'\

a

...... b

---c

-.-.-.- d

.------ e

\"~,\

~'\",\\ ....................

'<:'>" ....................--o::~.~ __. ..........- \ \ ---\ .\ \\

\ \\ \\ \..'........~.

'~.",.\',\,

0.967

Inllell

.-.-.- f

-6.752.197

\

4.:l34

InN

Fig. 18. Ratcs-of-convergence for various types of refinement strategies for u = (1 - x)(tan - 1 a(x - xo) +tan-1 ax,,), a = 50, Xn = 4/9. (a) II-uniform, (b) p-uniform, (c) II-adaptive. (d) p-adaptive. (e) h-p adaptive withexact interpolation, (f) hop adaptive with post-processing.

202 W. Racllowicz el ul., Toward a IIlliversal hop adllp/ivl.' strategy, Part 3

a

... b

-.-.-.- d

------. e

-.-.-- (

---c

~'-.';-\ -.-.--.~"" \ .~

\ \"-""," \ ............ --....

'\ -.......,~. ,....., .

""''-.

" "" \, ,,,\.,

" " ,,,,\,

\,,'\

2.131

Inllell

-0.7532.197 4.36\

InN

Fig. 19. Rates-of-convergence for various types of refinement slr,llcgics for u = ( I - x)( Ian - I a(x - XII) +tan-l CUll)' a: = 200, xn = 4/9. (a) Ir-uniform, (b) p·uniform. (c) Ir-adaptivc. (d) p-adaplivc. (e) hop adaptive withexact interpolation. (f) h-p adaptive with post-processing.

EXAMPLE 4. A two-dimensionallz-p optimization problem is now considered. We approxi-mate the function

3 2/3 . 2() ( 2)( 2)u(x, y) = 2 r SIn 3' 1 - x 1 - Y ,

r=Vx2+y2, O=tan-I(-xly), (x,y)EfL

where .n is the L-shaped domain shown in Fig. 20.

It turns out that the algorithm introduced in Section 6 fails in the case of the singularfunctions. One finds that after the first sequence of p-enrichmcnts of elements surrounding the

y

-1 "

-t

Fig. 20. L-shaped domain.

W. Rachowicz et al., Toward a universal hop adllPtil'e strate!?y. Part 3 203

singularity, these elements are not subsequently refined any more. The explanation IS asfollows,

Unlike the one-dimensional case. the two-dimensional interpolation operator cannot bedefined by a minimization principle that produces a function from the space of shape functionswhich minimizes the error in the H I-seminorm (because its boundary values must be definedonly by the boundary values of a given function to guarantee continuity of the approximation).For the two-dimensional case, wc are not guaranteed that expanding the space of shapefunctions implies a decrease of the error, though in most cases this happens. It turns out thatfor functions like ,2/3 sin ~ 0, raising p from 2 to higher orders causes an increase ofinterpolation error (at least for the kinds of quadratures we use. see Appendix A). Moreover,a single subdivision of only one of the .elements surrounding the corner leaves its boundarystiff (i.e .• introduces only constrained nodes) and this causes only an insignificant decrease ofthe error. As a result, the algorithm avoids refinements in zones where they are most needed.

To avoid these difficulties, we propose the following modifications of the algorithm.

(1) We exclude corner elements from p-refinements; this corresponds to the techniques ofadaptivity'. around singularities suggested by Babuska and Guo [6\, where corner elements arekept at fixed low order.

(2) We consider thc combincd bisection of all thrcc corner clcments together; this does notleave stiff.element boundaries and results in a significant dccrease of the interpolation error.

Tests suggest that. with these modifications, the pcrformance of the two-dimensionalalgorithm is quite similar to the one-dimensional case.

Figure 21 shows the optimal mesh obtained with thc modificd algorithm. Like theone-dimensional case, the size of elements is graded geometrically towards the singularity.However, in elements far from the corner, more p-enrichments can be obscrved.

We next compare the accuracy of approximations obtained with our algorithm with thoseproduced on meshes geometrically graded with p varying linearly with the number of layers ofelements surrounding the comer. Figure 22 suggests that in both cases the convergence isbetter than polynomial order; however, we are unable to verify that the error isO(exp «(NN».EXAMPLE 5. In this case, we consider an almost singular function on an L-shaped domainobtained by shifting the singularity outside the domain to the point (E, E). £ = n.OO!:

3 2/3 . 20 V 2 'u(x, y) = 2" SIn 3"' , = (x:- e) + (y - e)- .

In this case, we use the basic algorithm with only the s~cond of modifications introduced forsingularities: we subdivide three corner elements but allow them to acquire separate pcnrichments. The mesh resulting from the optimization procedure is shown in Fig. 23. As inthe one-dimensional case, a number of II-refinements of corner elcments are performed andthen p-enrichments follow and dominate the process. resulting in vcry high-order cornerelements.

The accuracy of approximation is again compared with that of geometrically graded mesheswith increasing p, Fig. 24. In this case, the asymptotic rate-of-convergence (O(cxp«N» iseasily verified.

204 W. RllcilOwicz el al .. Toward a universal hop adaptive slrategy, ParI 3

I! 1_

~~.

1'~Tp.

I

.-1 J._

""''1;'',' ·'I',"~.··..J:.',,~I\<'"

l. .....',I'''~·\~.::t,... ',' '..- .--.~.. '''','.''

I

!.'-'

.:-~-' .... ,~• ... • I

c

Fig. 21. (a) An optimal hop mcsh for u = , r1/J sin ~ (1 - x1)(1 - y1). (b) Zoom of (a) centered at the singularity.Magnification 16. (c) Zoom of (a) centercd at the singularity. Magnification 256.

HI. Racllowicz et III.. ]'()I'ard II III/il'ersal hop ad/lptive str(/{eg.1'. "/lrt 3 2115

-0,058

I nll.1I

-8

......... b

-6.543.045

InN

6.74

Fig. 22. Rates-of-convergence for II = jr~'J sin 1j(I- x~)(1 - y") for two kinds of meshes: (a) h-p adaptive, (11)geometrically graded with increasing p,

.,,~ ""..-- ". I

,;

,. - .., ..... ,; ..,r'.b!

~1 - ~-~~.1 J ,.~--_. ,.I •

Fig. 23. Optimal hop mesh for a function II = ~r",J sin '"\ , r=Vlx - f"): + (I' - Fl:. F '" OIKll.

206 W. Rachowicz et al., TOl1'lIrd a universal hop adaptil'e slrateg\', Part 3

-0.074 .

Inllell

-6.t22.08

InN, "I' .'J \r·' "Fig. 24. Rales-of-convergence for If = ~r- . Sill of, r = V (x - f)" + (Y - f )-. f = 11.001. for two t~'pes of meshes: (a)

h-p adaptivc,.(b) geometrically graded with increasing p.

EXAMPLE 6. In this calculation. the performancc of the optimization algorithm for the caseof a smooth function with large conccntrated gradients was tcsted. The function approximatedin the refinement process is

u(x, y) = tan -1 (a(r - ro» ,, 2 2

r-=(x-xo) +(Y-Yo)' (x,y)Efl=(O,I)X(O,I),

Xu = 1.25, Yo = -0.25, 0'=60, ro = 1.0.

This function is very flat throughout most of the domain hut exhibits a rapid jump along thccircle r = r(l'

The optimal mcshes observed with hop, h- and p-adaptivc proccdurcs arc shown in Fig, 25.In the hop approach, both h- and p-refincments were chosen by the optimization algorithm.Most of the II-refinements occurred in the beginning of the adaptivc pnKCss, then p-cnrichments dominated.

A comparison of accuracy of the thrcc adaptive stratcgies and 01 rt:sults obtained byuniform hand p refincments are prcscntcd in Fig, 26. There tht: supniority 01 the h-papproach is seen and results confirm that an exponential asymptotic behavior 01 the error isattained, At some stage of the h-p optimization process, we observe a lktcrioriation of theconvergence rate for several stcps, after which it rapidly improves. This phenomenon isanalogous to that discussed in Example 4 and is attributcd to the effcct of the process delayingrefincments in somc elements duc to a momentary incrcase in the local interpolation errorcaused by the constrained rcfincmcnt strategy. Fortunatcly, rcfincments of neighboring

W. R/lchowic: et /II.. Towart! II uni,'/'rslIl hop adapti"1' strategy. ParI J 207

--I""

~-~ -I

8 b

"1

-- 1-,

j_'~' ' __ '.0_ •. _

____I:;'.{i;j'lIHJpal.-\

c

pal • •7

Fig. 25, (a) An optimal hop mesh for a function II = tan -, a(r - rll)' r: = (x - XII): + (y - .\'nf XII '= \.25.,v1l = 11.25.rll = \.0. a = bU. for error estimates obtained by exact interpolation. (b) An optimal h-aoaptivc mcsh ( p "" I) for afunction II = tan -, a(r - rll). a = 60, for error estimates obtained by exact intcrpolation. (c) An optimal p-adaptivemesh for a function /I = tan -I a(r - rll}, a = 60. for crror estimatcs obtaincd by cxact intcrpnl<ttion.

208 11'. Rllclroll'ic: <'t Ill .. TOll'lIrl1 a IIIlil,t'r.wl h·p (/(/lIllIil'<' S1Tt1lcgy. Pllrt 3

~.096

Inllell

a

... b

---c

-.-.-.-. d

------. e

-0.2836.t89

InN

6.438

Fig. 26. Rali:s.of-convcrgencc for a function If = tan-I a(r - r,,). a = 1111.for \';IrI11llS IYp.:, III r<:lincrn.:nt 'Ir'llI:gics.(a) "·uniform. (b) p-uniform. (c) "·adaptive. (d) p-adaptivc. (c) "." aUapll\·':.

elemcnts ultimately effect the element itself. and this finall\' leads to its refinement with theresult bcing a very rapid decrease of the local error.

Thc performance of the h-adaptive approach appears to be only slightly infcrior to the hopmethod. Asymptotically. however, the h-method can Icad to a global ratc-of-convcrgence nobetter than that of uniform refinements [7].

EXAMPLE 7. We next consider the case of a function possessing hoth a point singularity anda zonc of large but finite gradients:

3 :.' . 18 _I (l/(x, v) = 2 r Sin 3" + tan (0- ~- f,,))·

~ = (x + y)/\I2, a = 311. fll = -0)5. (x .. ,.) E f}.

Here,. and {)arc as defincd previously and n is the L·~haped dlll1l~llI) ...hll\\11 III Fig, 211. We uscthe vcrsion of the algorithm with modifications fpr "illgularitil's (sce Lxampk -l).

The mesh resulting from the optimization scheme is shown in Fig. 27. As coull! beexpected. it exhibits featurcs of the optimal meshes determined carlicr for bnth problems withsingularities and with shock-like zones. Near the corner, we nhserve strong domination ofII-refinements. whilc close to the shock, comhined II- and p-rctincments arc chl)scn, With alimited level of II-refinement. and a high order of p-cnrichments, Fi~urc ~S "Ill'\\" thc

W. RacllOwicz et al.. Toward a unil'ersal hop adaptive strategy, Part 3

..,

209

Fig. 27. An optimal hop mesh for a function II = ir2IJ sin ~ + tan -I a( ~ - ~n)' ~ = (x + y) 1V2, a = 30. ~Il =-0.75. for error estimates obtained by exact interpolation.

1.676

Inllell

-2.1244.06 7.152

InN

Fig. 28. Rates-of-convergence for" = ~r~13 sin ~ + tan-I a( ~ - ~Il)' ~ = (x + y)/V2. ~Il = -11.75. a = ]0. for 1wokinds of meshes: (a) hop adaptive, (b) geometrically graded with increasing p.

210 W. Rachowicz et al .. Toward a universlIllz-p adaptive slrategy, Pllrt 3

computed accuracy of the approximation on optimal meshes compared with that obtained forgeometrically graded meshes with p increasing with the numbcr of layers of elemcnts. Thisresult again determines the superior performance of the Iz-p strategy even though we are ableto detect the exponential (exp( J3-\fN» behavior of the error only for the geomctrically gradedmeshes.

EXAMPLE 8. As a final numerical example, we consider a model two-dimensional boundaryvalue problem

-~u = I in n = (0,1) x (0,1), u=O onan, (6.5)

where f= I(x, y) is chosen to correspond to the exact solution.

-I[ (x+V )]u(x, y) = x( 1 - x)y(1 - y) tan 20 vi - 0,80 .

The problem is solved on an initial mesh and the optimization algorithm is applied usingcstimates .~ased on post-processed values of u and Vll in computing the intcrpolation error.The finite elemcnt solution is then updated after a sequencc of rcfincments.

Figure 29 presents the computed optimal mesh. Comparing it with thc mesh corrcspondingto the same global accuracy but obtained using cxact valucs of II and Vll in the optimizationprocess (Fig. 30). we find that the results in terms of numbcr of degreesoof-frccdom areessentially the same.

..,Fig. 21). An optimal hop mesh for a function lI=x(1-x)y(l-y)tan-lu(~-~ .. ), ~=(x+y)/V2. {,,=o.R.

'ct = ~O. for error estimates obtained using extraction procedures.

W. Rachowicz et al.. Toward a unil'ersallJ-p adaptive strategy. Part 3-- ---ul_J

j

--·_+-l~-.~l~tlltid....' l J <II ~ iii

211

Fig. 30. An optimal !t-p mesh for a function 1/ = x( 1 - x)y( 1 - y) tan -I cr( ~ - ~n)' ~ = (x + )') IV2. ~,,= 0.8.(r = 20, for error eSlimates obtained by exact interpolation.

Appendix A. Numerical integration

Calculation of interpolation errors (defined in [2]) involves the crucial dcsign of anappropriate numerical integration scheme. To compare the accuracies of interpolants fordifferent spaces of interpolating functions (on an enriched or subdivided element). we mustensurc that the discrete representation of an interpolated function-its values at integrationpoints-is the same for all cases considered. For this reason, intcgration over any element oforder p is performed by subdividing it into subdomains-sons resulting from anticipateddivisions. and using for each of them a p + 2-point Gaussian quadrature. In practice. in theonc-dimcnsional case such an approach guarantecs that the intcrpolation error decreases whenthe space of shape functions is extendcd; in the two-dimcnsional case. this property is notguaranteed, but increases of the error seem to occur only in rare circumstances,

The problem of computing integrals of singular functions is rcsolved hcrcin as follows:

(l) In the one-dimensional casc. special Gaussian quadrature schemes arc used that intcgratecxactly products xOf(x). f(x) being a polynomiaL a > - 1.

(2) In the two-dimensional case, functions f(x, y) with an r"-type singularity at a corner of arectangular element are considered. The integration proccdure consists of subdividing thcclement into two triangles and introducing new coordinatcs, r = x. tJ = yl.r, with JacobianJ = x. This transformation lcads to the integration of the nonsingular function f' J ovcr arectangular domain and can be carricd out by standard Gaussian 4uadratun: procedures.

212

Acknowledgement

W. Rachowicz I!t al., TOII'IIrd a 11llivl!Tsallr·p (/(laptil't' stTllte/lY, Pari .'

The support of the Office of Naval Research is gratefully acknowledged.

References

1. L. Dcmkowicz. J.T. Oden, W. Rachowicz and O. Hardy, Toward a univeral hop adaptive finite elementstrategy, Part 1. Constrained approximation and data structure, Comput. Methods Appl. Mcch. Engrg. 77(1989) 79-112.

2. J.T. Oden, L. Demkowicz, T.A. Westerman and W. Rachowicz, Toward a universal h·p adaptive flllite clementstrategy. ParI 2. A posteriori error estimates, Comput. Methods Appl. Mech. Engr. 77 (19H9) 11J-1HO.

3. I3. Guo and L Babuska, The hop version of the finite element method, Parts 1 and 2, Compu\. Mcch. I !IIJR6)21-41. 203-220.

4. 1. Babuska and M. Suri, The hop version of the finite element method with quasiuniform meshcs, RAIRO Math.Mod. and Numer. Anal. 21 (2) (1987) 199-238.

5. W. Gui and 1. Babuska, The h. p and h-p versions of the finite element met\lOd in one dimension, Parts 1.2,3.Numer. Math. 49 (1986) 577-683.

6. 1. Babuska and B. Guo, The I,-p vcrsion of thc finite elcment method for domains with the eurvcd boundaries,Tech. Note BN-1057, Institute for Physical Science and Technology. Univ. of Maryland, IlJH6.

7. L. Demkowicz, Ph. Devloo and J.T. Oden, On an l/-lype mesh relincment strategy based on minimization ofinterpolation errors, Compu\. Methods App/. Mech. Engrg. 53 (1985) 67-89.