PROGRESS ON

ADAPTIVE FINITE ELEMENT METHODS

FOR COMPLEX PROBLEMS IN SOLID

AND FLUID MECHANICS

J. T. ODEN

TICOM, The University of Texas at Austin

Keynote Lecture

FEMSA '85

The University of Ste11enboschSte11enbosch, R. of South Africa

January 1985

26(

PROGRESS ON ADAPTIVE FINITE ELEMENT METHODS FOR COMPLEX PROBLEMS

IN SOLID AND FLUID MECHANICS

J. T. Oden

Texas Institute for Computational Mechanics

The University of Texas at Austin

Abstract: This paper addresses the general topic of adaptive methods for auto-

matically enhancing the quality of numerical solutions to linear and nonlinear

boundary-value problems in solid and fluid mechanics, and reviews some of the

recent work of the author and his collaborators on this subject. Particular

attention is given to the problem of a-posteriori error estimation and to the

construction of mesh refinement, mesh enrichment, and moving mesh schemes.

Applications to several representative problems in fluid and solid mechanics

are discussed.

I. INTRODUCTION

In this paper, several new methods and results on adaptive finite element

methods are reviewed. These results were developed over the last 18 months

by the author and L. Dembowicz with the help of graduate students T. Strouboulis

and .Ph. Devloo. More detailed accounts of our work on this subject can be found

in a series of furthcoming papers [1 - 4].

The basic objective of an adaptive finite element method is to improve the

quality of an initial finite element approximation by automatically changing

the model: refining the mesh, moving mesh nodal points, enriching the local

order of approximation, etc. Thus, all adaptive methods must attempt to resolve

two basic issues: 1) how is the quality of the approximate solution to be

measured? and 2) how does one adapt the model to improve the quality of the

approximation?

2

The first question is generally resolved by attempting to measure the local

approximation error in some appropriate norm. The error, of course, is the

difference between the exact solution u and a finite element approximation uh

of u on a given mesh. Since u is not known, the problem of assessing the quality

of an approximation reduces to one of a-posteriori error estimation: the deter-

mination of estimatis of the error using computed finite element solutions. A

number of important papers on various schemes for a-posteriori error estimation

has been contributed by Babuska and his collaborators (e.g. [5-8]).

Once an estimate of the distribution of the error is available, the difficult

question of how to best modify the model to improve accuracy arises. There are

three general approaches:

h-Methods. Here the mesh is refined; the mesh size h is reduced and the

number of elements in the mesh is increased in regions of large error.

p-Methods. Here the mesh is fixed but the degree p of the polynomial shape

functions is increased over elements in which a high error is indicated.

Moving Mesh Methods. In these methods, the number of nodes and the type

of finite element remains constant during the adaptive process and the nodal

points are moved to regions' of high error.

Of course, one can also employ combinations of these strategies. But the

correct strategy for use of combined methods is apparently a delicate issue

and one in which much additional study needs to be done.

We shall describe here two methods for error estimation and show how these

can be implemented in each of the three adaptive schemes listed above.

A-POSTERIORI ERROR ESTIMATES

We describe two classes of a-posteriori error estimation, One based on

the computation of element residuals and the other based on interpolation

error estimates. The former class of methods was introduced by Babuska and

Rheinboldt and others [5-8] and the latter was first advocated by Diaz,

Kikuchi, and Taylor [9.]. Variatiants of both of these techniques have been

studied, expanded, and implemented by Oden and Demkowicz et al al [1-4].

3

Residual Methods. Consider the abstract boundary-value problem,

Find u in V such that

where

for all v in V (1)

A = a (possibly nonlinear) operator from a reflexive Banach space

of admissible functions V into its dual V

v = an arbitrary test function in V

*f = given data in V* = duality pairing on V x V

*This problem is equivalent to the abstract problem: Au = f in V "A Galerkin approximation of (1) consists of seeking a function u

hin a

finite dimensional subspace Vh

of V such that

(2)

The residual rh is the degree with which the approximation uhfails to

satisfy the original conditions on the solution:

r = Au - fh h i 0, G V* (3)

*Since the residual belongs to the dual space V and not necessarily V, its*magnitude must be measured with respict to the norm 11.11* on V :

sup ,

VGV~

= sup Ilvll ~1

(4)

where I 1·1 I is the norm in V

In some of the error estimators that we have developed, we use the

following procedure to approximate the supremum in (4):

. The original finite element approximation uh is computed in a space1 ..

Vh

= Vh

of spanned by low-order (say, linear) piecewise polynomial

shape functions.

4

• The full space V is approximated by a higher-order finite element space

V~, p > 1, spanned by piecewise polynomials of degree p.

An approximation r~ of the residual is constructed according to

1cllvO - v~ II II rh II * ~ + sup (5)h h"v~"~1where C is a constant, vOis an element of V and v~ is an arbitrary

element of V~. If h is the mesh size (i.e. for a partition Th·of

elements K,

h diameter (K)

we generally have

so that it makes sense asymptotically (as h --> 0) to approximate1 p

sup< rh,v> by sup< rh,vh>

As an example of how this procedure can be implemented, consider the model

problem,

Find u in V1{v G H (n); v=O on r1 } such that

I~u • 'i7vdx = J n fv dx gv dsfor all v in V (6)

This is the variational form of the model Poisson problem,

-tlu = f in n c. m2

u = o on r 1 c an

au g on r2 c= anan

(7)

5

2with 6 = ~,H1(n) the usual Sobolev space of functions with derivatives in

2 n --L ( ), and an = f1 LJ f2·

We define

(8)

p > 1, v~ interpolant v~

where Ql(K) is the usual set of bilinear functions defined on a quadrilateral

element K and P (K) is the space of polynomials of degree p defined on K.pThe residual r~ satisfies

L {f (-bu~ - f)v~ dxK K

1 1

+J 1 aUh aU*h- ( --- - ---- ) vP ds2 an an . haK-an

1aUh }+f (T - g) v~ ds

aKrian n

L < Rlh p > (9)K ,vhK

where u~ is the (coarse-grid-initial) finite element approximation of u deter-lh p

mined using the space Vh and RK is the functional on Vh defined as indicated,with aU;h/an an approximation of auh/an obtained by averaging this normal

derivative over adjacent elements to K.

It is not enough to simply calculate ~h as an indicator of the error in

element K. In general, we wish to have an indicator ~K of the error which

will bound the local error above and below and which will converges to zero

at the same rate as the ~ctual error; e.g.

= fK

6

(10)

such an error indicator is obtained as a solution of the auxiliary problem,

J 'VtflK. 'Vv~ dxK

for all v~ in v~O (K) (11)

We generally compute the solution of (11) using the concept of hierarchic

elements in which the stiffness matrics are only modified by the addition of

a row and column with the addition of each degree of freedom (see. e.g., Carey

and Oden [lO] for details). Using (11), (9), and (5), we have (to within

terms of O(hP))

(12 )

where C is a (hopefully) known constant. Though this estimate is global,

we use I I~KIIl,K as an estimate of the local error over each element K.In general, reducing' ItflKI'1,Kimplies a reduction in IIr~1,* which (par-ticularly for linear self-adjoint problems on Hilbert spaces) implies a

reduction in I lu-u~1 I·

Interpolation Error Estimates. It is well known (see, e.g. Oden and

Carey [11]) that for linear elliptic problems the approximation error

I lehl I~ = I lu-u~1Iv can be bounded above by the so called interpolationerror,

(13)

7

For the model problem (6), for example,

)

1/2

dx

(14 )

If u is smooth enough, a local interpolation error estimate can be derived

of th~ type (for Ql-elements)

1C hKlu/2,Klu-vhll,K ~

where

Z J W dxlulz,K =K

2 a2uW dx - (~ + 2) dx dy2aXl aX2

(15 )

(16)

The basic problem we face when attempting to make use of any of these

estimates is that we must calculate the higher order derivatives of the unknown

solution using only available information, i.e., through use of the currently

available finite element solution u~. There are numerous a priori techniques

for estimating the second derivatives u ,u or u ,but many are somewhatxx xy yyintuitive and not all are based on rigorous estimates. Exceptions are the

techniques based on so called "extraction formulas" introduced by Babuska

and Miller (see [12], [l3]). Following their idea one can prove that, if u

is regular enough, then the second derivatives at an arbitrary point (x ,y )o 0satisfy

J M u dxdyQ

a

- J (~ .~i) f dxdyn

-1 a (~ + ~)dsan u an+/ (~ + ~) ~ ds

(17)

an an

1 cos2Here ,= w 2 where (r, 8) are the polar coordinates centered at the point

r(xO' YO) under consideration and ~ is an arbitrary, regular function. By

the prope~ choice of ~ , one can eliminate the boundary terms in (l7). Of

course \1 on the right-hand side of (17) remains still unknown, but when

replaced by its element approximations u~ results in a formula for appro-

ximation of second-derivatives at (xO' YO) of the same order of accuracy as

the L2-error in the approximation of u by u~. For example, for the first

order approximation we can "extract" the difference of second order derivatives

with 0(h2) order of convergence! Formula (17), when combined with equation

(7), allows us to calculate each of the derivatives separately. Also, by

choosing =.!. si~2e in the same formula, we can "extract" the mixed deri-1f

2 r. a u ( )vat1ve axay xo' yo .

One method we have used successfully in applying the estimate (15) is

to construct the function ~ using a bivariate blending function of Gordon and

Hall [14,l5] type. For example, for an element K centered in a patch Q ofK

nine Ql - elements as shown in Fig. 1, we use the cut-off function,

~ (x,y) + x~ (y)r

+ (1 - y) tJ.b(x)

+ y ~ (x)u

(1 - x)(l - y) ~b(O)

- x(l - y) ~ (0)r xy ~ (l)u

(18)

where ~t' 'r' 'b' 'u are the restrictions of , on the left, right, bottom,

9

K

Figure l. Patch QK surrounding element K over which blending

functions are introduced.

10

and upper sides of nK and ~b(O), $r(O), ... , etc. are the corner values ofthese functions.

Note that we still have a global estimate although we "apply it" to

K

c 2lul2,K (19)

MESH REFINEMENT STRATEGIES BASED ON THE A POSTERIORI ERROR ESTIMATES

While many issues remain open in the area of reliable a-posteriori

error estimation, still further complications exist in designing efficient

adaptive algorithms based on these estimates. The basic problem can assume

the form of an optimal control problem in which one has to attain a discrete

approximation which is optimal in some sense determined by the error measures

and the strategy used to reduce error. The entire problem is further compli-

cated by the fact that our a-posteriori estimates are global in nature

(particularly the residual-type estimates discussed earlier) even though

they are used locally as a basis for local enrighments of the solution.

In this section, we describe three methods developed in [l-4).

An h- method (See [4])

Consider a quadratic mesh presented in Fig. 2 and the associated error

estimate (19). If we define a function h(x, y) specifying a "density" of

the mesh by:

h(x, y) if (x, y) G K (20)

the error estimate (19) may be ~ritten in the form of a functional

J(h) h2W2dxdy, (2l)

where W is given by (16).

11

x

<

l

\

12

This leads to a natural minimization problem:

Find h

constraint:

h minimizing the functional (21) and subjected to theopt

J !dxdyn h N (22)where N is simply the number of elements. This minimization problem leads

to the following optimality condition:

2 1(hw - a-) c5hh2

o for any c5h

where a is a respective Lagrange muliplier. Thus, we have

const.

or when rewritten elementwise, we have the redistribution law

h2 J W dxdyK K

const. for every element K (23)

which results in a very simple iterative - mesh refinement scheme, provided

we can estimate the function ~ For additional details we refer to [4].

A p-method [1,3]

It is difficult to predict the rate of convergence of local interpo-

lation error in the case of the p-version since it depends only on the un-

known regularity of the solution [2], and it is almost impossible to say

anything about a local order of convergence of the error or the residual-type

estimate. One way that we have used estimates of the type is (12) effectively

is to employ the following steps:

1. Solve local problem (11) and determine local contribution I~Kll,Kto the global error estimate.

2. Normalize the local indicators I~KI1,K by subdividing by the largestone.

13

3. If

o < I41KI < °l1,K

°1 < I~KI < °21,K

°2 < I~KI < 11,K

the first order approximation is retained.

a second-order approximation is used.

a third-order approximation is used.

Here the numbers 01 and 02 are chosen rather arbitrarily and u~ corres-

pondes to first-order approximation on a uniform mesh.

III. 2 Moving Mesh Strategy [2]

A popular moving finite element, method due to Miller, is based on an

L2-residual estimate in its basic form finds a justification in only the one-

dimensional case. In [2], we present a moving mesh strategy based on the

interpolation error estimate that can be considered as a generalization of the

Brackbill and Saltzman strategy [161.A

We consider a fixed mesh of elements K shown in Fig. 2 that is mapped

onto a distorted mesh in such a way as to minimize the interpolation error.

We may penalize the interpolation error functional further by appending

to it additional terms to smooth the mesh am prevent the Jacobian of the map

from vanishing.

In the method discussed in [2], the optimal mesh results from minimizing

the functional

J = IO + aI + 8J

where

= r. fA (x2 2 2 210 + Yn + x~ + y~)d~dnnA KK

= I: fA 1 2 2 2I J [ (xF; + y~)uK nn

K (i 2 2+ + Yn)u~~ ]df;dnn

(24)

(25)

(26)

J ~fK K

+

14

(27)

Here a and B are constant parameters, IO is the Dirichlet integral whichtreats as a constraint the invertibility of the transformation defining

the mesh. In the absence of other terms, its minimization yields a con-

formal mesh and then the method is merely a generalization of the well-known

conformal-map mesh generators. The functional K forces the mesh lines to be

orthogonal and this orthogonality allows us to simplify the form of the error

functional I.

GENERALIZATIONS

All of the methods and results described for the model Poisson problem

can be generalized to significant problems in solid and fluid mechanics. We

list examples of these without proofs; for details, see [1,2].

A-Posteriori and Interpolation Estimates. We list the forms of the local

error estimates for some representative problems.

Plane elasticitI' The local problem is

Find the vector field ~K G ~~O (K) such that

f 0ij (!K)Eij (~~) dxK

f 1 (G + >') div ':~ vP dx(- G6~h + - f) .K

_h

+ J 1 1 1*'2 (: (~h) - : (uh )) dselK-elQ

+ J 1 vP ds( ~(~h) - g ) .- _helK1"\ rt

for all ~~ in ~~O (K)

15

Here

°i'(c/l)J -

stress tensor at displacement ~

GE .. (~) +~J -

E •. (

16

In the above t(u) is a stress vector associated with displacement u, ~- - -is an arbitrary, regular function and {u, v} is one of four linear com-

binations of second order derivatives of u and v at (xO

' YO) corresponding

to the four extraction functions ~ in the following way:

1. {u,v}

2. {u, v}

a2

u a2u a2vn{(2G + A) --z - G --z + (3G + A) axa-}

ax ay y

(cos 26 sin 26)2 ' 2r r

if ~ (_ sin 26

r2cos 26)

2r

3. {u, v} nG 2G + A

= (COS 46 cos 26

2 + a 2r r

sin 462r

- a sin 26)2r

where a =3G + A2(G + A)

4. {u,v}222

n 2G + A {~ _ ~ _ 2 au}G G + A ay2 ax2 oxay

=,(_.sin462r

- asin 26

2r

cos 462r

- a cos 262r

with a as before. As previously, (r, 6) denote polar coordinates centered at

(xO' yO)'

Stokes problem A similar a-posteriori error estimate for the Stokes

problem of incompressible vicious flow is possible, the main difference

being the necessity of introducing a pair of error indicators, (~K' ~ )_ K

corresponding to errors in the velocity field and the hydrostatic pressure

field, respectively. This means that a pair of local equations must be

solved.

17

For the interpolation estimates, we have

+ (33)

wnere ~, ~h' ~~ are velocity rielas and p, p~, gh are pressures. the localvelocity estimate is treated in the same manner as the displacement field

in the preceding elasticcity problem and one can show that the pressures

satisfy,

K(34)

where ~,n are master element coordinates. Thus, an extraction formula for

first derivative of pressures can be derived and used to estimate this bound.

Parabolic problems; heat conduction, The above procedures can also be

used to derive bases for adaptive schemes for time dependent problems. As

an example, consider the heat conduction problem,

au f in D- - 6u =atu = uo on ru (35)

au = g on rTnu = Uo for t = 0

In the above D = U nt' where n C R2 is a (possibly) variable domaint G (O,T) t

in R2, r = U an , rT = u arT with aQ and ar two disjointu tG (O,T) u tG(O,T) u t

parts of the boundary ant where the corresponding "kinematic boundary con-

ditions" Uo and "tractions" g are prescribed respectively. Finally, uospecifies initial conditions for t = O.

18

A variational formulation of (35) can be formulated as follows:

Find u G Uo + V such that:

J dV{-u - + lJu·lJv}D dt

dx dt + J u v dxnT

J fv dxdtD

+ J gv ds dtfT

for all v 6 V (36 )

Here the space of admissible displacements is definded by V = {v G H1(D)1

v = 0 on r }, and Uo is an arbitrary HI - function coinciding with u on r .u uInstead of applying the principle (35) over the entire space-time domain

D, we partition the time interval [O,T] into a series

to < t1< 1... 1 < tN = T and apply the principle over

of intervals [t I' t ] 0n- na domain D = U n

n t.t l

19

{J 1 2 i J (ul _ u)2'}! ~~(uh - u) + nD n

n

~ cl

{L I~ 12 }! +jc2 112 (38)- zllun - ull L (nn_l)K K 1,KThe first term in (38) corresponds to the residual estimated in an appro-

priate norm and the second one to initial conditions.

Similarly, interpolation estimates can also be derived for this problem.

Moreover, the same techniques can ce used for the Navier-Stokes equations

and highly nonlinear problems. We omit details.

NUMERICAL EXPERIMENTS

In this section, we present some representative examples.

A Heat Conduction Problem with a Moving Domain

We first consider a linear transient heat conduction problem defined on

(0, 4 + O.lt) X (0; 3) with purely homogeneous Dirichlet=a moving domain Qt

boundary and initial data. We choose the data on the right-hand side of the

(4 + O.lt - x) Y (3 - y) . c. where:u

equation to correspond with a prescribed exact solution:

210 e-5 (x - 1 - 0.2t) x

C [t for 0 < t < 0.5

1 for t > 0.5

This problem has been solved for a mesh with 24 elements (see Fig. 3). The

time step has been chosen 6t = O.l and the solutions was computed for 20 timesteps. The constants °1 and °2 described in the previous chapter has beenchosen as follows:

1/20, 0'2 1/2.

Figure 3 shows the mesh enrichment for terms t = 0.5, 1.0, 1.S and 2.0

while the Figs. 4, 5, 6 and 7 present the computed first-order and enriched

solutions on the section AA (see Fig. 3) compared to the exact solution.

For additional details we refer to [2].

A A A. .A

A

t =0.5

t = 1.5

"A

t = 1.0

t = 2.0

.A

No

~ 10 order A 20 order ~ 30 orderFigure 3 Mesh enrichment for the heat conduction problem

21

TIME = 0.5--0--0- 1st Order~ Enriched

'Ex-act

oo.oC")

oo•o

N

ooio...

o. 1 ~OO 3.00 4.00

Figure 4 Heat conduction problem. Computed solutions on section

AA for p = 1 and adaptive correction for time t = 0.5.

22

TIME = 1.0---- 1st Order-0-0- Enriched

Exact

ooi

oC')

oo

•oC'.I

oo.o,..

o. 1~00 2:00 3:00 4:00

Figure 5 Heat conduction problem. Computed solutions on section

AA for p = 1 and adaptive correction for time t = l.O

oo·oM

oo·o~

oo·o.,..

o. 1.00

23

2.00

TIME 1.51st OrderEnrichedExact

4.00

Figure 6 Heat conduction problem. Computed solutions on

section AA for p = 1 and adaptive correction fortime t = 1.5

24

TIME = 2.0-0-0- 1 stOrder-0-0- Enriched

Exact

0.00 ./ 1.00 2.00 3.00 4~00

oo·oM

oo·o('I.l

oo·o...

Figure 7 Heat conduction problem. Computed solutions on section

AA for p = 1 and adaptive correction for time t = 2.0

25

A Stokes Problem

The second problem we consider deals with the application of the moving

mesh strategy described earlier rather than the penalty-formulation. In

this example, is the rectangle (0, 2) x (0, 1) and we define on this domain,

an initial uniform 16 x 8 mesh of rectangular elements. Dirichlet boundary

conditions are used with the velocity ~ prescribed as ~o = (1, 0) along thetop edge and u = (0, 0) along the remaining sides (driven cavity problem).

The optimal mesh results from a minimization of the functional

J I +o

NL a.I. + BJ + yK1. 1.

i=l

Where ai' i = 1, 2, a and y stand for positive (given) real numbers, Iiis defined by (26) with u replaced by the i-th component of the velocity

field, J is a similar term for the hydrostatic pressure given by (34).oFigure 8 presents a computed mesh for al = a2 = 4, B = 4, y = 0.1.

A Model Elliptic Problem

As an example of the application of the h-method combined with the ~nter-

polation error estimate we have solved the model elliptic problem (21) with

homogeneous Dirichlet boundary conditions. The right-hand side of the equations

corresponds to the following (exact) solution defined in Q = (0, 1) 2

With Xo = 0.55, Yo = 0.5,0.05, after 19-mesh refinements we have obtained a mesh

u(x, y) = Hx)$(y), where

IHx)

-e: +Ax+B= e 2e(x-xO) + e:

with A and B chosen in that way that (0) = (1) = o.e: = 0.02, and e:x ypresent in Fig. 9. Figure 10 shows a behavior of the error (measured in

Hl_ seminorm) compared with the error estimate - exact and the one based on

the extraction formula. As one can see, the extraction formula yields a

very precise approximation of the error estimate. In Fig. 10 N stands for

number of freedom degrees. The vertical distance between the eror and error

estimates is because of the absence of values of constant C in the estimate,

which we have not determined.

..

N0\

A'~\ --~~~-1~:~1~

~I - ,,-""' -"

1-~ --.".r --.,...-..- _-J- -,,'7' ,,--/ . ./flI _ '.... "" I----~\ ------ .....--~.(.//.... .I II I- ,~,,- .I .... , I-----y ,,-""'./"' I ~-'t--t. --- -,'f-""-- ...../~;i /' '/.~11l .....t- ----I /" ',., I.,,--- , -"'" ,.I I .J' I I

- I -- ~ ,'''. I..,.,.--- -.l I.; / I~-- __-r / A i II ---,/ / // i II I--- / . - / .I' -- .;~-- ./ / I I--- --I-- ~ / " I--t'"' _, /- .I / . I (I ---I' l I l i Ir I LrT7~'-t--

I I Ir t I II / l II I l

~-;' " I I.;F::.-:r---r-_~

~

K-r:-:::.~__ I,I 'i'~'--~~t- ......,'

• - ,--....... ---...... I-.....- ---...... /"-"'P, ,,-........... __~\ \ \ ........._~ --_.....__ t _I J. \ '. ~__ "-_.~\'r-\\~\ .\ \--~

\ "'\ '\ ~---.....\ ''1 '.... _........I \ '\... '\ .~ _____\ \ \-......_- \ ----- ........I \ .........._ .\--\ . -

''\ -~\ \ \ -~\ \, \, .\ \ ----,-I \ \ ,-A/T\\- \- \ \

1\\\ \ \I \ I

A

Figure 8 Driven cavity problem. Optimal mesh after 8 FE recalculations

a=4.,B=4., y=O.l.

\ 27

II

~~

,

I

~::~

Ln II error II\\\

':'.:. ,......\

....'".\

....\.....\.....\

....\....:\

". '\.'..:,

....:

1.0

o

-1.0

-2.0

fo 2.0 "'. Ln(N112)".\\\

"

H1 •- semmorm error

error estimate

computational errorestimate based onthe extraction formula

"

Nco

29

Acknowledgement. This work was supported by the u.s. Office of Naval Researchunder Contract N00014-84-K-D409. The work reported here was the result of

joint work with Dr. L. Demkowicz and Messrs. T. Strouboulis and Ph. Devloo.

REFERENCES

1. Oden, J T, Demkowicz, L, Strouboulis, T, and Devloo, P, Adaptive Methods

for Problems in Solid and Fluid Mechanics, Adaptive Methods and Error Refinement

in Finite Element Computation, Edited by I. Babuska and O.C. Zienkiewicz,

John Wiley and Sons. Ltd., London (to appear).

2. Demkowicz, Land Oden, J T, On a Mesh Optimization Method Based on a

Minimization of Interpolation Error", Int'l. Journal of Engineering Science

(to appear).

3. Demkowicz, L, Oden, J T, and Strouboulis, T, Adaptive Methods for Flow

Problems with Moving Boundaries. I. Variational Principles and A-Posteriori

Estimates, Computer Methods in Applied Mechanics and Engineering (to appear).

4. Demkowicz, L, Oden, J T, and Ph. Devloo, On an H-type mesh Refinement

Strategy Based on Minimization of Interpolation Errors, (in review).

s. Babuska, I and Rheinboldt, W C, Error Estimates for Adaptive Finite ElementComputations, SIAM J. Numer. Anal., Vol. 15, No.4, August, 1978.

6. Babuska, I and Rheinboldt, W C, A-Posteriori Error Estimates for the

Finite Element Method, International Journal for Numerical Methods in Eng.,

Vol. 12, l597-l6l5, 1978.

7. Babuska, I, and Dorr, M R, Error Estimates for the Combined hand p Versions

of the Finite Element Method, Numer. Math. 37, 257-277 (1981).

8. Babuska, I, Szabo, B A, and Katz, I N, The p-Version of the Finite Element

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30

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