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360

Theorem 3.1 There exist r > 0, p > 0, and a map $:Fn 5(0; r) -> W

n 5(0; p),

which is Lipschitzian as a map from

HQ(£1)

L°°(n) into

itself,

with $(0) = 0

;

and such

that:

(a) Vt)GFnS(0;r), Vw e W

d_B{0;p)

: f(v + w)> 0;

(b) VveVD 5(0; r),VmeIVnB(0;p)..iu^ $(v) : /(u + w) > /(v + $(u));

-0(0; r) and every w €W

5(0; p), we have:

(VzeWn L°°(Q) : f'{v + w){z) = 0) <=> w; = $(v).

Let ip(v) := f(v + ${v)), v eVH 5(0, r). Then, ip is of class C

onVH 5(0; r) with

Vv€Vn 5(0; r), VzeV:

f'(v)(z)

= /> + $(v))(z),

<p'

is strictly differentiable at 0 and tp"(0)(z, z) = Q(z), for all z £ V:

4 Conservation of critical groups in the HQ(CI)

r\L°°(Q,)

topology

In this section, we consider the functional / : Hg(fi)

—>

M defined by (10), assuming

that (a.l)-(a.5) and (g.l), (g.2) hold. We shall denote by f^ the restriction of / to

H%(n)nL°°(n).

Definition 4.1 Let Z be a metric space and g : Z

—>

R be a continuous function. For

u e Z, c := g(u), U a neighborhood of u, and q 6 Z+, set

(g; u) := H\g* n

\ {u}) n U),

where H

(A, 5) denotes the q-th relative Alexander-Spanier cohomology group of the pair

(A,B),

with coefficients in R (say; see [18]), and g

:= {v 6 Z : g(v) < c}. The vector

space C

(g; u) is called the q-th critical group of / at u. Due to the excision property of

the Alexander-Spanier

cohomology,

its definition does not depend on the particular choice

of the neighborhood U.

It is easy to see, as in [8, Proposition 3.4], that if / is the functional given by (10),

then

\d„^)f\{u) > 0 =>•

(f;u)

= {0} for all q.

Note that \d

i(fl)f\(u) = 0 if and only if u is a weak solution of the problem (*)

associated with /. Abusing a little the terminology adopted in [14, 11, 8] when dealing

with general continuous functionals defined on arbitrary metric spaces, we might say that

such a is a "critical point" of /, which thus accounts for the related notion of critical

group.

Theorem 4.2 Let f : H^(Q) -> R be defined by (10), such that (a.l) - (a.5) and

{g.l),

(g.2)

hold,

and let /«, denote the restriction of f to Hg(£l)

L°°(f2). Then,

{f;0)=C

(/«,; 0) for every qeZ+.

361

Proof.

Let Q be the quadratic form defined by (11). If m*(/;0) = 0, that is, if Q

is a nondegenerate, positive quadratic form, then the result follows from those of [10]

(where indeed singular homology was used instead of Alexander-Spanier cohomology, but

this clearly makes no difference here). More precisely, [10, Theorem 5.13] shows that

(f;0)

= (5,0-R for every q (where 6

j is the Kronecker symbol), and from [10, Lemma

5.9] we deduce that 0 is a

local

minimum of f (arguing in a similar way as in

[6,

Theorem

1.4-6]), hence a local minimum of f

, so that finally,

C^f^O)

= fi

o'R for every q.

Therefore, we now assume that -ffg(fi) = V © W, where Q is positive definite on W

and V C L°°(fi) with 0 < dim V = m*(f; 0) < oo, and we show that for every 9 £ Z

C,(/;0)

C,(

;0)

= C,(/

;0),

where tp is given by Theorem 3.1 (d).

Indeed,

the fact that

(f;0)

(tp;0)

was

already proved in [17, Theorem (6.4)], combining Theorem 3.1 with an abstract deforma-

tion theorem. In fact, Theorem 2.3 could be used instead of the latter for that purpose,

and our point here is to establish that, through the same argument but using specifically

Theorem 2.3 (d), we also have C

(/oo;0) = C,(</?;0) (q being fixed in the sequel).

Withr, p> 0 as in Theorem 3.1, let X := [{VnB{0;r))

(Wr\B{0; p))]nf° and define

a : X

—>] —

oo,0] by a{u) := f(pv{u) + $(pv(u)))- Then, a(u) < f(u) for every u € X,

according to Theorem 3.1 (b), and it is clear that all the assumptions of Theorem 2.3

are satisfied, but for (4): the argument is given in the proof of

[17,

Theorem

6.20],

reproduce it for the sake of clarity. Let (u^) C X \ f

be such that e/, := \dwf\{u^)

—>

We thus have:

l/'K)

CO I

< eh||*Hi,2 for every z£WC\

{Q).

(12)

Let {</?!,...,

(p*

}

be an I?'-orthonormal basis in V, where m* := m*(/, 0). Then,

Pv(

) = 5Z ( / fk

zdx

Vk,

and writing (12) for z

—

Pv(z), we obtain

/ I ^

a,ij(x,

)DiU

DjZ + - V* D

g(x, u

)DiU

DjU

J a \^^

^tj=i

- / (g{x, u

) + Y^

f'(

h){>Pk)'Pkz)

<£h\\z\\i,2,

with (f'(u

)(ifi

))

bounded

for every k = l,...,m*, and

•= £h\\Id - Pv\\

—*

0. Hence,

f'(

€ iJ"

(fi) (more precisely,

f'{uh)

can be extended to a continuous linear form on

HQ(£1))

and, up to a subsequence,

f'(u

)

is strongly convergent in i?

(fi). It follows

from [5, Theorem 2.2.4] that, up to a further

subsequence,

(u^) is strongly convergent to

u € X, and passing to the limit in (12), we have

f'(u)(z)

= 0, for every z € W n L°°(Q).

According to Theorem 3.1 (c), it follows that u = v + $(?;) for some v G V D B(0; r), so

that f(u) = a(u), i.e., u € /". This establishes (4), upon arguing by contradiction. Now,

set:

M:={v + $(v) :veVr\B{0;r), f(v + ${v)) < 0} C

H£{Q.)

I~l

L°°(fi),

362

and observe that :

\ {0}) = {M,M\ {0}) = (/£,,/£, \ {0}). It follows from

Theorem 2.3 and the properties of Alexander-Spanier

cohomology

that the inclusions /£,

—>

Xao and /£, \ {0}

—>

Xoo \ {0} induce isomorphisms in Alexander-Spanier cohomology:

see [15, Lemma 2.6]; hence, from the Five lemma, so does the inclusion (/£,, /£, \ {0})

—>

(XoojXoo \ {0}), and iue have

(f«,;

0) := ^(Xc, ^ \ {0}) = #»(£,, /£ \ {0}) = H"{M, M \ {0}).

On the other

hand,

from the properties of $, the pair (M, M \ {0}) is clearly homeo-

morphic to the pair {ip°,<p°\{0}), so that:

(<p;0)

:= H"(ip

,tp

\{0}) = H

{M,M\{0}),

whence the conclusion. •

Remark J

.3 1.

(a) In [12], we improve Theorem 4-2 in two directions: firstly, by considering the re-

striction of f to Cg(n) instead of f^, secondly, by considering critical groups defined

through singular homology, which is possible using an extension of the so-called Sec-

ond deformation lemma of

[8,

9] in place of Theorem 2.3.

(b) As consequences of Theorem 3.1 mainly, it is shown in [17, Corollaries 6.5, 6.6]

that

(/;0)

= 0 if

£ [wi(/;0),m*(/;0)], and that ifO is an isolated weak solution

of(*),

thenC

(f;0) has finite dimension for every q. See [10,

17]

for an application

of nonsmooth Morse theory to the existence of weak solutions of (*) for functionals

f of the type (10).

References

[1] A. Ambrosetti, H. Brezis, and G. Cerami, Combined effects of concave and convex

nonlinearities in some elliptic problems. J. Punct. Anal. 122 (1994), 519-543.

[2] D. Aze, J.-N. Corvellec, and R. E. Lucchetti, Variational pairs and applications to

stability in nonsmooth analysis, Nonlinear Anal. Theory Methods Appl. 49 (2002),

643-670.

[3] H. Brezis and L. Nirenberg, H

versus C

local minimizers, C. R. Acad. Sci. Paris,

Ser. A 317 (1993), 465-472.

[4] A. Canino, Multiplicity of solutions for quasilinear elliptic equations, Topol. Methods

Nonlinear Anal. 6 (1995), 357-370.

[5] A. Canino and M. Degiovanni, Nonsmooth critical point theory and quasilinear el-

liptic equations. Topological Methods in Differential Equations and Inclusions (Mon-

treal, 1994), A. Granas and M. Prigon Eds., 1-50, NATO ASI Series, C 472, Kluwer,

Dordrecht, 1995.

[6] K. C. Chang, Infinite Dimensional Morse Theory and Multiple Solution Problems.

Progress in Nonlinear Differential Equations and Their Applications, 6, Birkhauser,

Boston, 1993.

363

[7] K. C. Chang, H

versus C

isolated critical points. C. R. Acad. Sci. Paris, Ser. A

319 (1994), 441-446.

[8] J.-N. Corvellec, Morse theory for continuous functionals. J. Math. Anal. Appl. 196

(1995),

1050-1072.

[9] J.-N. Corvellec, On the Second Deformation Lemma. Topol. Methods Nonlinear Anal.

17 (2001), 55-66.

[10] J.-N. Corvellec and M. Degiovanni, Nontrivial solutions of quasilinear equations via

nonsmooth Morse theory. J.

Diff.

Equations 136 (1997), 268-293.

[11] J.-N. Corvellec, M. Degiovanni, and M. Marzocchi, Deformation properties for contin-

uous functionals and critical point theory. Topol. Methods Nonlinear Anal. 1 (1993),

151-171.

[12] J.-N. Corvellec and H. Douik. In preparation.

[13] E. De Giorgi, A. Marino, and M. Tosques, Problemi di evoluzione in spazi metrici e

curve di massima pendenza. Atti Accad. Naz. Lincei Rend. CI. Sci. Fis. Mat. Natur.

68 (1980), 180-187.

[14] M. Degiovanni and M. Marzocchi, A critical point theory for nonsmooth functionals.

Ann. Mat. Pura Appl. (4) 167 (1994), 73-100.

[15] M. Degiovanni and L. Morbini, Closed geodesies with Lipschitz obstacle. J. Math.

Anal. Appl. 233 (1999), 767-789.

[16] I. Ekeland, Nonconvex minimization problems. Bull. Amer. Math. Soc. 1 (1979),

443-474.

[17] S. Lancelotti, Morse index estimates for continuous functionals associated with quasi-

linear elliptic equations. Adv. Differential Equations 7 (2002), 99-128.

[18] E. H. Spanier, Algebraic

Topology.

McGraw-Hill, New York, 1966.

Approximating exterior flows by flows on truncated

exterior domains: piecewise polygonal artificial

boundaries

Paul Deuring

Universite du Littoral,

Centre Universitaire de la Mi-Voix,

Laboratoire de Mathematiques Pures et Appliquees, B.P. 699,

62228 Calais Cedex, France

Abstract

The article deals with a mixed finite element method for discretizing the Stokes

system in truncated exterior domains. This method yields approximate solutions

of the Stokes equations on exterior domains, as is shown by error estimates. An

implementation of the method is discussed, and the results of test computations

are presented. In particular, it is indicated how a truncated exterior domain with

spherical outer boundary may be triangulated in a suitable way. Similar mixed

finite element methods work for other exterior problems which may be written as

mixed variational problems.

Introduction

In this article, we discuss how exterior flows may be approximated by solutions of mixed

variational problems in truncated exterior domains. In particular we will be concerned

with truncated exterior domains which are polyhedral and thus may easily be decomposed

into grids suitable for finite element methods.

The model problem we consider is the Stokes system in the exterior domain fl :=

\fi, with a Dirichlet boundary condition on dQ and a decay condition near infinity:

-AU

VTT

= /, div« = 0 in If, u\dQ = b, (1.1)

/ Hz)|

= 0(BT

) for R^oo,

JdBi

where Q. C M

is either the empty set, or a bounded open polyhedron with Lipschitz

boundary and with 0 G Q. We further suppose b G H

1/2

(dQf,f G L

)

, and we

assume there is a e (4, oo), 5,7 G (0, oo) with IT C B

, \}{x) \ < 7-|z|~

for x G W?\B

The quantities a, S, 7 and the functions b and / will be kept fixed throughout. Here and

in the following, for e G (0, 00), we denote by B

the open ball in M

with radius e > 0

and center in the origin.

364

365

Under the preceding assumptions, problem (1.1) admits a unique solution (u,7r) with

the properties u 6 #,

)

6 ^(if) n L

(ff), Vu e L^jf)

("exterior Stokes

flow"); see [6, Theorem V.2.1,_V.1.1].

Now take R e

[S,

oo) with f2 C B

, and denote by

that part of the exterior domain

Q which lies inside the ball B

, that is, Q

:= B

\Q. Then we consider the boundary

value problem

-Av + VQ = f\n

, divu = 0

iuU

v\dQ = b, (1.2)

)(v,

Q){X)

= 0 for x £ dB

where CR is defined as follows:

jCflMCiD,*),•(*) := 3

•

-(a;)/(2

•

R) + £ (£>*«;,• - 1^/2 - <S#

•

a){x)

•

\x)

= i

for j e {1, 2, 3}, R e (0, oo), a; £ dA, A C E

a Lipschitz domain, n'-

' its outward unit

normal, w a vector-valued function and a a scalar function with the property that w, Vw

and a are defined on dA. The symbols

and Dj denote partial derivatives.

This choice of C

is based on two considerations: first, it allows to write problem (1.2)

in mixed variational form, and second, the difference between the exterior Stokes flow and

the solution of (1.2) ("truncation error"), measured in some suitable norm, decays with

an optimal rate. More precisely, setting for v, w £

^(QR)

, Q

(QR),

(v,w):= 3/(2

•

Ri)

•

[ v

•

w do

+ [ £

(DjV

-DjW

(1.3)

Jen

- (1/2)

•

DjV

(x)

•

Wj(x)\ dx, p

{y,Q):=-l Ayvv-gdx,

JUR

we could show (see [1, Theorem 4.2]) there is a uniquely determined pair of functions

(wH,te)e#

(n*)

xi/

(nfl) with

,w) + fi

{w, g

) = I

f-wdx

for we

H^QR)

with w\

8Q.

= 0, (1.4)

JSIR

div v

= 0, VR

dQ = b.

Note that problem (1.4) is a weak form of (1.2); see [1, Lemma 4.2]. The equation

divVR = 0 may be replaced by the relation fi

,<j) = 0 for a £

so problem

(1.4) is a mixed variational problem. It is further shown in [1] that

|| (u - v

)

+ || V(u

- v

) h + R

1/2

•

(« - VR)

(1.5)

+ \\x\n

-QR\\

< C

(|| /

||a + 7 + ||6||i/2,2)foranyi?e[5,oo),

with a constant C > 0 which only depends on Q, S and a ([1, Theorem 5.1, Lemma 5.1]).

Actually, only the case b = 0 is considered in [1], but a technique for extending this result

to the case 6^0 may be found in [5]. The important point of this extension is that

the constant C in (1.5) remains independent of R. We further remark that the results

in [1] were inspired by a similar theory in [8], [9] pertaining to the Poisson equation.

366

An artificial boundary condition for the Stokes system different from the one in (1.2) is

presented in [10]. Similar estimates as in (1.5) are proved in [5] for Oseen flows.

Due to its mixed variational form, problem (1.4) may be discretized by finite element

methods. A finite element solution to (1.4) may then be considered an approximation

of the exterior Stokes flow

(U,TT).

The corresponding error consists of the sum of the

truncation and the discretization error, the latter one arising due to the transition from

(1.4) to a finite-dimensional problem. In [1, Theorem 6.3, Corollary 6.2], we estimated

this sum, under the assumption that the Babushka-Brezzi condition [1, (6.9)] is valid

with a constant independent of R. This assumption is satisfied by the Mini element ([3,

Theorem 4.1]).

When we performed test computations ([2], [4]), we did not actually decompose QR.

Instead we used tetrahedral grids of such a type that the union of the tetrahedrons of each

grid yields a polyhedron P, depending on the respective grid, with P C £l

and 0

\P a

slender set near the sphere dB

. The artificial boundary condition was then imposed on

the outer part dP\dQ. of the boundary dP of P. It is the purpose of this paper to give

a precise description of the transition from Q

to these polyhedra, and to quantify the

error related to this transition.

To this end, we fix some index set I. For igl, let n; e No with n,- > 3, and put

Ri := 2

• S. (The parameter S was introduced at the beginning of this section and

refers to the size of Q and the decay behaviour of /.) Assume that T, = (Kf

)

1<i<T

with Ti € N, is a family of open tetrahedrons in B^\Q. We will write K\ instead of K\

since the dependence of Ki on i should be obvious. Denote by P

the interior of the set

U{ Ki : 1 < I < ^ }. We suppose that Pi C

Q,R

and

\Pi

is a remainder set near

dB^.

Let hi denote an upper bound of diamKi for tetrahedrons Ki near Q. Following

Goldstein [8], [9], we suppose the mesh size of T; to augment from this value h

near

hi- Ri/S near dB^. These and other properties of Ti will be made precise in Section 2.

For each iel, the polyhedron Pi is used for setting up a mixed variational problem.

In fact, define the bilinear forms a,- : H

(Pif x H

f i->- K, ft : H

x L

(Pi) M-1 in

the same way as respectively a

and fi

in (1.3), but with R = Rt, and with the domains

of integration QR and dB

replaced by Pi and dPi\dQ, respectively. Then we consider

the boundary value problem

-Avi + Vft = /

Pi, divvi = 0 in Pi, Vi\dQ = b,

CR,

{Pi

Q) (vi,

Qi)

(x) = 0 for x e dPi\dtt,

which admits a weak solution for any i el with hi small enough:

Theorem 1.1 There is S

G (0, S) such that for i £ I with h

{

6 (0, S

], a uniquely

determined pair of functions (vi,gi) exists in H

x L

(Pi) with

ai(vi,w) + PAw,

Qi)

= /

f-wdx

forweWi, divvi = 0, v

<9f2

6,(1.6)

J Pi

where Wi :=

{

w G

(Pi)

: w

d£l = 0

Moreover, for i as before, we have

|| (u

|| v(«

+ R;

1/2

•

dPi\dQ

(1.7)

+ Iklfi-ftll2

367

< C -I$

.((Vi- V^

+ (V

+ V^

•

(

•

hi)

+ Vi +

(V~

+ Vi)

•

T>i),

with a constant C > 0 independent ofi, and with the terms V

T>i

and

T>i

defined by

A:=sup{|£

)(w,7r)(s)| : xedB^},

Vi := sup{ | Fi(t

•

x) | : x e dB^, t e [(1 -

hj/S

)

1] },

Vt := sup{ | Fi(x) -Fi(t-x)\ : x e OB^, t € [(1 -

1] },

where

Fi(x) := (R^

•

u(x),

VM(I),

TT(X)

) for x e fi .

This theorem, which will be proved in Section 2, yields an analogue of inequality (1.5)

once we are able to find upper bounds for Vi, Vi and Vi in terms of ||/ | fis||

, 7 and

||6||

/2,2-

In other words, we still have to establish pointwise estimates of the exterior

Stokes flow

(w,

7r)

in the annular domain Bn

..(

2y/2. Such estimates follow from

our assumptions on /. For lack of space, we can only give the result here, which says there

is a constant C > 0 only depending on Q, S and a such that

Wi<C-Rf-T,

Vi<C-Ri

Vi <C-(hi/Ri)

-{l + \n(S/hi))-r (1.8)

for i el, where T := ||/ | Bs||2 + 7+||6||i/2,2- A proof of (1.8) may be based on an integral

representation of u and 7r similar to the one in [1, (3.7), 3.8)]. In fact, the term X>, is

evaluated in this way in [1, p. 259], under the assumption 6 = 0. Combining (1.7) and

(1.8) finally yields an estimate of the truncation error which corresponds to one given in

(1.5) for the case of spherical truncating surfaces, and which states that the left-hand side

of (1.7) is bounded by a constant times a sum of terms of the form

/i"

•

(1+ln(S//ij))

with a + b > 3/2, c > 0.

Also for lack of

space,

we cannot establish error estimates associated with finite element

discretisations of the mixed variational problem (1.6). We only mention that the error

estimates in

[1,

Theorem 6.3, Corollary 6.2], which pertain to finite element discretisations

of (1.4), must be modified by adding to their right-hand side some additional terms which

may be traced back to (1.8) and (1.7). The proof of this result does not differ much

from that of [1, loc.

cit.].

The main changes concern [3, Theorem 4.1], which yields the

Babushka-Brezzi condition [1, (6.9)] for the Mini element, with a constant independent

of Ri. In fact, [3, Theorem 4.1] pertains to domains Q^; if we consider Pi instead of

fij^, we have to replace [3, Corollary 3.1] by Theorem 3.1 below, and [3, Theorem 2.2] by

Theorem 3.5 below. But otherwise, the transition from Q^ to Pi simplifies the proof of

[3,

Theorem 4.1]. In particular, we may drop the smallness condition on hi required in [3,

loc.cit].

Partitionings of truncated exterior domains

We begin by describing our grids Ti in a more precise way. First we characterize the set

(Al) Let i £ I. Then Ki C B^Q for I e {1, ...,

TS};

the set P

is a connected

polyhedron; any vertex of

is either a vertex of

or it is located on dB^.

368

We do not admit hanging nodes, and we require that our grids do not degenerate:

(A2) Ifiel, k,l G {1, ..., Ti} with k^l, and if F :=

n~K~

/ 0, then F is either a

common face or side or vertex of K^ and Ki.

(A3) Put B

(x) :=

{

y G ffi

: \y - x\ < r } for x G K

, r > 0; B

:= B

(0) for r > 0;

:= B

\Ti; U

:= Bv.s^&pTs for j

Then there is a constant ay > 0 and a family {hi)

z in (0, S/4) such that

diamKi < V

•

sup{ r G (0, oo) : B

(x) C K

for some x G K

} >

o~\

•

fori&X,

G {1, ..., T^, j G {0, ..., nj with Uj n #, ^ 0.

Note that although the mesh size of Ti increases with growing distance from fl, as already

indicated in Section 1, our assumptions in (A3) imply that the family (Tj)

i(S

j is non-

degenerate in the sense that the ratio of the smallest circumscribing sphere to the largest

inscribing sphere of any element Kf' is bounded independently of i

I and / G {1, ...,

T;}.

In the three-dimensional situation we are considering, it might happen that all four

vertices of a tetrahedron are situated on the sphere dB^. We want to exclude this case,

so we require

(A4) For i

X, I G {1, ..., Ti}, at least one vertex of Ki is located in the open set Pi.

Finally we prescribe a condition which allows us to project the points of 8BR onto dPi\dU

and vice versa.

(A5) For i

X, the ray { r

•

(cos g

•

cos $, sin

g • cos

i9,

sin 0) : re(0, oo) } has precisely

one point of intersection with 3Pi\dQ, for any g, d G R.

Let i G I. By (A4) and (Al), we may assume there is some K, G {1, ..., T

—

1} with

the properties to follow: For / G {1, ..., «,}, at most two vertices of K

are located on

dB^.

If I G {^ + 1, ..., Ti}, three vertices of Ki are situated on dB^.

For I G {1, ..., Ti}, let a'

' = a^(i,l), ..., a'

' = a'

'(i,I) denote the four vertices of

Ki. If I > Ki, then, by the choice of K

, we may suppose that aW, a'

), a'

' G dB^ and

Pi.

We note a well known consequence of assumption (A3):

Lemma 2.1 There is a constant 02 G (0,1) such that

a« - a'

' I"

• I a<

- a<

> |~

•

(a™ - a<

•

(a™ - a<

>) < a

fori el, l£ {1, ..., T

{

}, where

a.W

:= a<

>(i,/) for k G {1, 2, 3}.

If i G I, Z €

{KJ

+ 1, ..., Tj}, we write T, , or more briefly Xj, for the plane open

triangle in B

with vertices a

(1)

(i,I), a

(2)

(i,Z), a

(3)

(i,

Z).

Due to (Al), (A2) and (A4), we

have

\dQ = U{T, :

+ 1 < I < n} for i

X. (2.1)

369

We introduce a two- and a three-dimensional standard tetrahedron K^ and A^

', respec-

tively, by setting

{2)

:={x€ (0,1)

: x

< 1 - x

(3)

:= {x € (0, l)

: x

< 1 -Ti, a

-xi - x

For iel, /

{1, ..., Ti}, we put

A, := Af» := (a« - oW, a

(2)

- a

(4)

, a'

' - a<

>) e

3x3

where the vectors a^

—

' = aW

(i,

—

(i,

I) are to be understood as columns

(v G {1, 2, 3}). We further set

Ai(x) := A?\x) := Af

•

x + o

(4)

(i,0 for i

X, 1 <

< 7y, X G M

Denote by n^

\ n'

flK

;' the outward unit normal to P

and B^, respectively. Define the

functions tp

:= yf : K™ ^ QP^dQ, Vi := #

(2)

«i> ft

= &

W :

(2)

E, i/( := v,

: if'

) Hlby

<p,(r,)

:= -4,(?7, 1 - iji - %) = a

(3)

+ (a

(1)

- a

(3)

)

•

ift + (a

(2)

- a

(3)

)

•

(„)

( |

(D -

(3)|2 . |

(2) _

(3)|2 _

( (a

(l) _

(3)

}

(ffl

(2) _

(

3)) )3

y/^

vi{rj) := {det{DjMv)

•

DkipiCn))^^^)

for

G K

\ where a^ := a^(i, I), j G {1, 2, 3}, m + 1 < I <

TV,

X. Put

= i?W —

{ipf\ri)

•

v e

{2)

} for i

{

+ 1 < I < T

{

We note that ipi is a local parameter of dPi, with range

(ipi)

= XJ, and gi is the corre-

sponding element of surface area. The function Ai

K^ is the usual affine mapping from

K^ onto Ki, and Fi is defined by a radial projection of the triangle T

(

on dB^. Due to

(A5),

ipi is one-to-one. We shall show below (Corollary 2.3) that if hi is small enough,

ipi

is regular and hence a local parameter of dB^. The related element of surface area is

given by the function

V[.

As a consequence of (A5) and (2.1), we note

dBfl, = V{¥i :

+ 1 < I < n}, (2.2)

/ fdoy = £ / fdo

= •£ L

(f°<Pi)(l)-9i(v)dri (2.3)

for i

X, / G L

(dPi\dQ). In order to show that

ipi

is a regular mapping, and to estimate

the difference between integrals over dPi\Q and dBn, we begin by recalling a result from

[3]:

Lemma 2.2 ([3, Lemma 4.2]) We have B ,

u/^I^cPj fori el.

We note that the quantity ft/4 in [3] corresponds to hi in the present context. This

explains why a factor h

•

arises here instead of h

•

S)~

as in [3, Lemma 4.2]. By

Lemma 2.2 and (A3) we obtain