Numerical_Methods for Nonlinear Variable Problems Episode 10 potx

Solution of Dirichlet problems for second-order elliptic partial differential operators We shall now discuss the formulation and the solution via variational methods of Dirichlet problem

then (4.29) has a unique solution in H 1 (Q), which is also the unique solution of the variational problem:

Find u e H^Q) such that

I ( X V M ) • \ v dx + I aouv dx = f fv dx + j gv dT, V H E H \ Q )

J n Jn Jn J r

(4.95)PROOF It suffices to prove that the bilinear form occurring in (4.95) is H 1 (Q)-elliptic.

This follows directly from Lemma 4.1 and from the fact that if v = constant = C, then

Proposition 4.7 We consider the Neumann problem (4.29) with Q bounded and

A still obeying (4.47); if we suppose that a0 = 0, then (4.29) has a unique solution

u in H^nyU 20 if and only if

L ; (4.96)

(4.97)

f fdx+ LdT = 0;

u is also the unique solution in H 1 (Q)/U of the variational problem:

Find u e f / ' P such that

f ( S V u ) - \ v d x = f fv dx + f gv dT, V c e H\£l).

J n Jn Jr

PROOF For clarity we divide the proof into several steps.

Step 1 Suppose that a 0 = 0; if u is a solution of (4.29) and if C is a constant, it is clear

from

V(« + C) = \u that u + C is also a solution of (4.29).

If u is a solution of (4.29), we can show, as in Sec 4.2.2, that (4.97) holds; taking v = 1

the bilinear form a(-, •) is clearly continuous, and from Lemma 4.1, it is if'(fi)-elliptic (it suffices to observe that if v = constant = C, then

v 1/2

defines a norm equivalent to the H '(Sll-norm (4.49); henceforth we shall endow Vx with

the following scalar product:

7'

•In

• Vw dx.

From these properties of Vu and from (4.47), the variational problem:

Find ueV^ such that

f (KVu) • \v dx = [fvdx+ [gvdT, VueFj, (4.99)

has a unique solution.

Step 4 Returning to H l (Q) equipped with its usual product, let us introduce Vo a H 1 (Q)

Step 5 From (4.96), (4.99) it follows that we also have

f (AVu) • V(v + c)dx= f f(v + c)dx + \g(v + c)dT,

V v e Vu V c e R, u € Vx (4.101) From the results of Step 4, relation (4.101) implies that u is a solution of (4.97) (but the only one belonging to F J ; actually if we consider a second solution of (4.97), say u*,

we clearly have (from (4.47))

a f \V(u* - u)\ 2 dx < f (XV(«* - «)) • V(«* - u) dx = 0 (4.102)

From (4.102) it follows that u* — u = const; this completes the proof of the proposition.

•

Remark 4.10 In many cases where a 0 = 0 in (4.29), one is more interested

in Su than in u itself [this is the case, for example, in fluid mechanics (resp., electrostatics), where u would be a velocity potential (resp., an electrical potential) and Vu (resp., — Vu) the corresponding velocity (resp., electrical field)]; in such cases, the fact that u is determined only to within an arbitrary constant does not matter, since V(« + c) = Vu, V c e R

4.3 Solution of Dirichlet problems for second-order elliptic

partial differential operators

We shall now discuss the formulation and the solution via variational methods

of Dirichlet problems for linear second-order elliptic partial differentialoperators The finite element approximation of these problems will be dis-cussed in Sec 4.5

4.3.1 The classical formulation.

With Q, A, a 0 , f, g, and the notation as in Sec 4.2.1, we consider the following

Dirichlet problem:

- V • (XVu) + V • (PM) + a o u = / in Q, u = g on F, (4.103)

where P is a given vector function denned over Q and taking its values in U N

Remark 4.4 of Sec 4.2.1 still holds for the Dirichlet problem (4.103)

Remark 4.11 If X = I, a 0 = 0, p = 0, the Dirichlet problem (4.103) reduces to

-Au = / in Q, u = g on T, (4.104)

which is the classical Dirichlet problem for the Laplace operator A

4.3.2 A variational formulation of the Dirichlet problem (4.103)

Let v e 9{Q) (where ®(Q) is still denned by (4.38)); we then have

c = 0 o n r (4.105)

Multiplying the first relation (4.103) by v, we obtain (still using the

Green-Ostrogradsky formula, and taking (4.105) into account)

[ (SVu) • \v dx - tufl-Vvdx+ [aouvdx = [ fv dx, Vi>eS>(Q).

Jn J(i Jn Jn

(4.106)

Conversely it can be proved that if (4.106) holds, then u satisfies the

second-order partial differential equation in (4.103) (at least in a distribution sense).Let us now introduce the Sobolev space HQ(Q) defined by

H l0 (Q) = S>(Q)H'(n); (4.107)

if F = dQ is sufficiently smooth, we also have

Hj(fi) = {v\veHHQ), yo^ = 0}, (4.108)where y0 is the trace operator introduced in Sec 4.2.2 From (4.107), (4.108),

HQ(Q) is a closed subspace of HJ(Q)

An important property of HQ(Q) is the following:

Suppose that Q is bounded in at least one direction of U N ; then

\ l / 2

Ja ) defines a norm over Hj(Q) equivalent to the H 1 (Q)-norm.

Property (4.109) holds, for example, for

Q = ]a, /?[ x R^"1 with a, jS e U, a < )3,

but does not hold for

Returning to the Dirichlet problem (4.103), and to (4.106), we suppose that

the following hypotheses on A, a 0 , P, / , and g hold:

feL 2 (Q), lgeH\Q) such that g = y o g, (4.110)

a0 e L^fi), ao(x) > a0 > 0 a.e on Q, (4.111)

X satisfies (4.47), (4.112)

p e (LOO(Q))/V, V • p = 0 (in the distribution sense) (4.113)

We now define a: H\Q) x H\SX) -> U and L: H l (ii) -^Uby

is skew symmetric over HQ(Q,) X Hl(Q)).

PROOF Let v, w e 3){Q); we have

I up • Vw dx = I P • \(vw) dx- I wp • \v dx (4.117)

Jn Jn •'n

Since vw e @(£l) and V • p = 0, we also have

f p• \(vw) dx = <P, V(w)> = - <V • P, vw} = 0 (4.118) (where < •, • > denotes the duality between £)'(Q.) and

From (4.117), (4.118), we then have

PROOF From (4.111), (4.112), (4.114) we have

a(v, v) > Min(a, ao )|M|£i (n) - (" up • Vu rfx, VUG Hj(fl); (4.120)

since p satisfies (4.113), from Lemma 4.2 we have

yp • Vv dx = 0, V v e H&Q) (4.121)

[2

Combining (4.120) and (4.121), we have

a(v, v) > Min(a, ao )||y|||i ( jj), V » e Hj(fl),

i.e., the HJ(Q)-ellipticity of a{ •, •) •

Using the above results we n o w p r o v e :

Proposition 4.9 Suppose that (4.110)—(4.115) hold; then the variational

problem:

Find u e H 1 (Q) such that y o u — g and

a(u,v) = L(v), V c e r / J f Q ) (4.122) has a unique solution This solution is also the unique solution in H 1 (D.) of the

Find u e Hj(fi) such that

a(u, v) = L(v) - a(g, v), V v e H^(£i) (4.125) From Proposition 4.8, a{-, •) is bilinear, continuous over Hj(Q), and i/J(fl)-elliptic;

moreover, the linear functional

v -* L(v) - a(g, v) (4.126)

is clearly continuous over Ho(ii) From the properties of HQ(Q), a(-, •), and of the linear

functional (4.126), we can apply Theorem 2.1 of Sec 2.3 to prove that (4.125) has a unique

solution in HQ(Q); this, in turn, implies (taking part (1) into account) that (4.122) has a

unique solution as well.

(3) The Solution u of (4.122) Satisfies (4.103) and Conversely Taking v e &(Q) in (4.122),

we find that u satisfies (4.103) in the sense of distributions Conversely, if u e H1 (Q.), with

y o u = g, satisfies (4.103), we can easily prove that

a(u, v) = L(v), V D £ ®(Q), (4.127) and using the density of &(Q) in tfj(fi), we find that (4.127) also holds for all the v in

Hj(fl) •

Remark 4.12 Suppose that A is symmetric and that P = 0; this implies the

symmetry of the bilinear form a(-, •) There is then equivalence between

(4.122) and the minimization problem:

Find u e H g such that

4.3.3 Further remarks and comments

Remarks 4.7 and 4.8 of Sec 4.2.3.1 still hold for the Dirichlet problem (4.103)

in that we can replace L defined by (4.115) by more complicated linear tinuous functionals like the one in (4.63), or

con-L(v) = f fv dx + [hv

Jn Jy

dy

with h and y as in (4.68).

Concerning the case where a 0 = 0 in (4.103), (4.114), we have the following:

Proposition 4.10 Suppose that Q is bounded in at least one direction of U N ; also suppose that a 0 = 0, the hypotheses on A, P, / , g remaining the same Then

the variational problem (4.122) still has a unique solution which is also the unique solution in HJ(Q) of the Dirichlet problem (4.103).

PROOF It suffices to prove that the bilinear functional a: H^Sl) x H^il) -> R defined

by

a(v, w) = (A\v) -\wdx - \ vfl • \w dx

is Ho(£l)- elliptic This follows from (4.112) (and (4.47)) and from Lemma 4.2 which implies

that

(v, v) > a f

a(v, v)>a\ \Vv\ 2 dx, V v e tf£(Q), and from (4.109) which implies that v -> (Jn | Vu | 2 dx) 1/2 is a norm over HJ(Q) equivalent

With fi as in the above sections, we suppose that F ( = d£l) is the union of

Fo, Fj such that

Fo u Fi = F, ron r , = 0 ; (4.131)such a situation is described in Fig 4.1 of Chapter II, Sec 4.1

We now consider the mixed boundary-value problem

— V • (AVu) + a o u = / i n D., u = g 0 o n F0, (AVM) • n = g l on r1 (

(4.132)where A, ao, / a r e as in Sec 4.3.1 and where g 0 , g x are given functions definedover Fo, T u respectively

Taking a test function v "sufficiently smooth" and such that

These hypotheses on A,a o ,f,g 1 imply that a(-, •) is bilinear continuous over

H*(Q) x H ^ Q ) and H1(Q)-elliptic and that L is a linear continuous functional

over H^Q).

We now introduce (motivated by (4.133)) the following subspace V o of

H\ny.

V o = {v|v e H^Q), y o v = O a.e on Fo} ; (4.138)

actually V o is a closed subspace of H 1 (Q).

Using a variant of the proof of Proposition 4.9 in Sec 4.3.2, we should provethe following:

Proposition 4.11 Suppose that there exists g 0 e H 1 (Q) such that

For a proof see, e.g., Necas [1]

Most remarks of Sees 4.2.3 and 4.3.3 still hold for (4.132), (4.140); in

particular, if D is bounded and if |r o dT > 0, then we can suppose that a 0 = 0

[this follows from the fact that if Q, is bounded, then

\\v\ 2 dx V ' 2 defines a norm over V o which is equivalent to the H^Q^norm (Lemma 4.1can be used to prove this equivalence property)]

4.4 Solution of second-order elliptic problems with

Fourier boundary conditions

4.4.1 Synopsis

In this section we shall discuss the solution—via variational methods—of

the so-called Fourier problem for linear second-order elliptic partial differential

operators; our interest in this problem is twofold:

(i) The Fourier problem occurs in the modelling of several heat-transfer

phenomana

(ii) The Neumann and Dirichlet problems discussed in Sees 4.2 and 4.3,respectively, can be considered, in fact, to be particular cases21 of theFourier problem; this claim will be justified in the sequel.

4.4.2 Classical formulations of the Fourier problem

With Q, A, a 0 , f, g as in Sec 4.2, we consider the following boundary-value

problem:

Find u such that

- V • ( A V M ) + a 0 u= f in Q, (X\u) • n + ku = g onT, (4.141)

where k is a given function denned over F.

4.4.3 A variational formulation of the Fourier problem (4.141)

Let v be a smooth function defined over Q; multiplying the first relation (4.141) by v and integrating over Q, we obtain (using the Green-Ostrogradsky

From the above hypotheses and from the continuity of the trace operator

y 0 : H 1 (Q) -> L 2 {T) (see Sec 4.2.2), we find that a(-, •) is bilinear continuous

over H 1 (Q) x H 1 (Q) and /^(fij-elliptic, and that L(-) is linear continuous; we

can therefore apply Theorem 2.1 of Sec 2.3 to prove:

Proposition 4.12 / / the above hypotheses on A, a 0 , k, f, g hold, the linear variational problem:

Find UE H\Q) such that

a(u, v) = L(v), V v e H\Q) (4.150)

has a unique solution; this solution is also the unique solution in H *(Q) of the

Fourier problem (4.141).

Remark 4.13 Suppose that A is symmetric; this, in turn implies the symmetry

of the bilinear form a( •, • )• From Proposition 2.1 of Sec 2.4 it then follows that

(4.150) is equivalent to the minimization problem:

Find u e H^Q) such that

J(u)<J(v), VveH^n), (4.151)

where

J(v) = i f l(K\v) • \v + a 0 v 2 ^ dx + \ f kv 2 dY - f fv dx - [gv dT.

(4.152)

Remark 4.14 Using Lemma 4.1, we should prove that Proposition 4.6 of

Sec 4.2.3.2 still holds for problem (4.141), (4.150) in that if Q is bounded and if

a 0 e LCO(Q), a o (x) > 0 a.e on Q, a o (x) dx > 0,

Jn

then problem (4.141) has a unique solution in H ^ Q ) which is also the uniquesolution of the variational problem (4.150)

Concerning the case a 0 = 0 on Q, we shall prove the following:

Proposition 4.13 We suppose that Q is bounded and that a 0 = 0 on Q; we

also suppose that k satisfies

fcGL°°(r), k(x) > 0 a.e on T, f k(x) dY > 0 (4.153)

Then, the hypotheses onf, g, A being (4.146), (4.149), respectively, the variational problem:

Find u e H l (Q) such that

f (SVu) • \v dx + [kuvdY= [ fv dx + f gv dY, V v e H\n),

Jn Jr Jn Jr

(4.154)

has a unique solution which is also the unique solution in H l (Q) of the value problem

boundary V • (XVu) = / in Q, (X VM) • n + ku = g on T (4.155)

PROOF It suffices to prove that the bilinear form occurring in (4.154) is H^^-elliptic

This follows directly from Lemma 4.1 (see Sec 4.2.3.2) and from the fact that if v =

trivial since it suffices to take k = 0 on F in (4.141) and (4.144); obtaining the

Dirichlet problem from the Fourier problem is less simple

First consider the variational formulation of the standard nonhomogeneousDirichlet problem:

Find u e H 1 (Q) such that y o u = g, and

1'[(AVu) -\v + a 0 uif] dx = L(v), V v e HJ(Q), (4.156) Jn

where g = y o g, g e H^Q.), and where L is linear continuous from H^Q) to U;

taking L linear continuous from H l {Q) to U in (4.156) is not restrictive since

any functional linear continuous over Hl(Q) can be considered as the restriction

over Ho(fi) °f a functional linear continuous over H^fi)

With g and L as in (4.156) and e > 0, we now consider the following

par-ticular Fourier problem:

Find u t e H:(Q) such that

I [(SVu£) • Vu + a o u e v] dx + - u e v dT

= L(v) + - f gv dT, VveH\O) (4.157)

e Jr

Remark 4.15 If A" is symmetric, there is equivalence between (4.157) and the

following minimization problem:

Find n e eif'ffl) such that

J£u t ) < J£v), V v e H\Q), (4.158)

where

Jlv) = \ f [_(X\v) • \v + a 0 v 2 1 dx - L(v) + i - f (v - gf dT (4.159)

Thus (4.157), (4.158) is obtained from (4.156) by using a penalty procedure

to handle the boundary condition y o u = g (see Chapter I, Sec 7, for more

details on penalty procedures)

The relationship between (4.156) and (4.157) follows from:

Proposition 4.14 We suppose that X satisfies (4.47) and that a 0 satisfies

a 0 £ L°°(Q), a o (x) > oc0 > 0 a.e on Q, (4.160)

(or possibly—if Q is bounded—

a 0 e L°°(Q), a o (x) > 0 a.e on Q) (4.161)

We then have

lira \\u e - u\\ HHa) = 0, (4.162)

where u (resp u e ) is the solution of the Dirichlet problem (4,156) (resp., of the Fourier problem (4.157)).

PROOF (1) A priori estimates for {u£ } E Let us define a{-, •) bilinear continuous from H\a) x H\Q) into R by

a(v, w)= f l(KVv)-Vw + a o vw']dx, Vi),weff'(Q); (4.163)

•Jn

taking v = us — u in (4.157), and using the notation of (4.163), we obtain

1 f

a(u e , u s -u) + - \(u e - gf dT = L(« E - u),

which, in turn, implies

a(u E -u,u c -u) + -\\u c -g\\i2 ir) = L(u E -u)-a(u,u s -u), V e > 0 (4.164)

Using (4.52) (which still holds), from (4.164) we obtain

a{u E -u,u c -u) + - ||uE - g\\lHT) < C^u, - u\\ HHa) , (4.165)

where

If (4.160) holds, from (4.165) we obtain

lim uz = u* weakly in H^fi); (4.169)

ToshowthatM* = w,itsufficestoshow(from(4.156),(4.171))thaty 0 «* = g;the

bounded-ness of {uc } e in H 1 (fi) and (4.164) imply

Since a(v, v) > 0, V v e H^O), it follows from (4.164), (4.172) that

lim a{ut -u,u,-u) = 0, (4.173)

lim - ||u £ - g\\ii<x) = 0 (4.174)

If (4.160) (resp., (4.161)) holds, the strong convergence property (4.162) follows from (4.173) and from

a(u e - u,u B - u)> M i n ( a , ao ) | | u t - u\\j,Htl)

(resp., from (4.173), (4.174), and from the fact that

\\v\ 2 dx + \v\ 2 dT)

a h i

defines a norm equivalent to the //'(Q^norm) •

Remark 4.16 It is possible to prove Proposition 4.14 using Theorem 7.1

of Chapter I, Sec 7.3; we should take V = H\Q), a(-, •), and L(-) as above, and j and K defined by

K={v\veH 1 (Q),y o v = g},

respectively

Remark 4.17 Let us consider the mixed boundary-value problem (4.132)

(of Neumann-Dirichlet type), whose variational formulation is given by(4.140) In fact, the solution of this problem can also be obtained (if the usual

hypotheses on A, a 0 hold) as the limit, as e —> 0, of the solution u E of thefollowing problem of Fourier type:

Find u e e H 1 (Q) such that

The proof of the convergence of the solution u £ of (4.175) to the solution u of

(4.132), (4.140) is left to the reader as an exercise

The above approximation properties are, in fact, of practical interest sincemany modern finite element codes for solving elliptic partial differentialequation problems compute the solution of the (discrete) Dirichlet, Neumann,Neumann-Dirichlet, etc problems via discrete Fourier formulations (derivedfrom (4.157), (4.175)) with e > 0 "very small." In the finite element code,MODULEF, for example, one uses e = 10~3 0 (see also Perronnet [1] formore details about MODULEF-and further references)

From the practical interest of these approximations by penalization of the

Dirichlet boundary condition y 0 u = g on F (or Fo), a quite natural problem

which arises is to estimate the approximation error as a function of s; the

following two propositions give partial answers to this problem:

Proposition 4.15 We suppose that the hypotheses ona 0 ,A are those of tion 4.14 and that

Proposi-L(v) = f fvdx, V c £ H\n), (4.176)

•Jn where fe L2(Q) We also suppose that the solution u of (4.156) satisfies

(K\u) • n G L2(F) (4.177)

We then have

I k - «l|H.(n) = 0{yfe\ (4.178)

where u (resp., u j is the solution of (4.156) (resp., (4.157)).

PROOF Let us define X e L 2 {Y) by

Since a(v, v) > 0, V I I E H ' P , from (4.182) and from the Schwarz inequality in L2 (T)

we obtain

Ik - dWm-n < eUlvHn- (4 - 183 )

Combining (4.182) and (4.183), we obtain, V e, 0 < e < 1,

a(u e -u,u.-u)+ J |ti, - s i2 rfF < e\\X\\ 2Hr)

which clearly implies (4.178) (if (4.160) holds, we have

where u (resp., u £ ) is the solution o/(4.156) (resp., (4.157)).

PROOF With I = (5Vu) • n, we define u1 e H^Q) as the (unique) solution of the Dirichlet

23 We have y0 H\Q) = {n\n e L2(r), 3 p e H^O) such that n = yo p); actually y 0 H\Sl) =

H 1I2 (T) (for the definition and properties of the Sobolev spaces H s with s e R, see, e.g., Adams [1],

Lions and Magenes [1] and Necas [1]).

< eMax(||X|| l0 o (n), \\ao \\ L ^\W\\ mm \\w E - u c \\ ma> (4.191)

By a now standard reasoning, (4.191) implies that

I K - « J H i ( n ) = O(e),

which completes the proof of the proposition •

Remark 4.18 The hypotheses (4.177), (4.184) concerning the regularity of

(AVM) • n on F are often encountered in practice If the ay, / , g are sufficiently

smooth functions of their respective arguments, and if F is a sufficiently smoothcurve or surface, properties (4.177), (4.184) will follow from regularity properties

of the solutions of the Dirichlet problem (4.156) discussed in, e.g., Necas [1]

4.5 Finite element approximations of the Neumann, Dirichlet,

and Fourier problems

4.5.1 Synopsis

We now consider the finite element approximation of the second-order ellipticproblems discussed in Sees 4.2, 4.3, and 4.4; actually the finite element tech-niques to be described in this section will also be used to approximate thesecond-order elliptic problem discussed in Sec 4.6 (and have been used allalong in this book to solve nonlinear problems much more complicated thanthose linear problems discussed in this appendix)

The following subsections cover only a small part of this very importantsubject; for more details (theoretical and practical) concerning finite elementapproximations, see Aubin [2], Ciarlet [l]-[3], Aziz and Babuska [1],Raviart and Thomas [1], Oden and Reddy [1] ,Mercier [1], Strang and Fix[1], Zienkiewicz [1], etc

4.5.2 Basic hypotheses: triangulations ofQ and fundamental discrete spaces For simplicity we suppose that Q is a bounded polygonal domain of U 2 ;

we then define a family {&~ h } h of triangulations of Q such that:

(i) V h, 9~ h is a finite collection of closed24 triangles contained in Q;

(ii) u Terh T = Q;

(iii) h is the maximal length of the edges of the T e ^ j ;

(iv) if T, T e T h , T # T, we have

fnf' = 0

and either T n V = 0 or T and V have in common a whole edge or

only one vertex

Various triangulations for which conditions (i)-(iv) hold are shown inseveral chapters of this book (see also Appendix II)

From 2T h we now define the two following finite-dimensional spaces:

Hl = {v h \v h eC°(Q), v h \ T eP u V T e ^ } , (4.192)

Hh h ={v h \v h eH}:, wfc = 0 o n r } , (4.193)

where P 1 is the space of the polynomials in two variables of degree less or

equal to 1 (i.e., if q e P u then q{x r , x 2 ) = axt + fSx 2 + y, a, fi, y s U).

Let Sv h be the gradient of v h (in the sense of distributions); it can be shown

that if v h e Hi, then

Let Efc (resp., H Oh ) be the (finite) set of the vertices of ^ (resp., of the vertices

of 2T h which do not belong to F); we denote by N h (resp., N Oh ) the number of

elements of 2, h (resp., T, Oh ) and we suppose that

\, £» = Z Oh u {Qi}^ Noh+1 (4.195)

We recall that any polynomial of P 1 is entirely defined by the values it

takes at the three vertices of a triangle; furthermore, if p, qe P l and satisfy

p(x) = q(x), p(x') = q(x'), where x / x', then p and q coincide on the line xx'.

From these properties it follows that T h (resp., ZOft) is Hj, (resp., Hg h )

uni-solvent (in the sense of Ciarlet [1], [2], [3]), since

V a = {a,}; e U Nh (resp., R*0*), there exists a unique v h e Hi

(resp, H^) such that v h (Qi) = ct h V i = 1, , N h (resp., V i = 1 , , N Oh ).

Remark 4.19 In this appendix we consider piecewise linear approximations

on triangles only; for more sophisticated methods making use of degree polynomials, quadrilateral elements, curved elements, and also forfinite element methods for three-dimensional problems, see the referencesgiven in Sec 4.5.1

higher-4.5.3 Some fundamental results on finite element approximations

4.5.3.1 Generalites: synopsis In this section (Sec 4.5.3, which closely follows

Ciarlet [1], [2], [3] and Mercier [1]), we shall give some results about the