Vector Curl and Gradient, Lecture Notes - Mathematics - 9, Study notes of Mathematics

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Approximation Properties Recall the^ L

(Ω)^ inner product and associated norm 2

considered in the first lecture:^ 〈u, v

∫ 〉 :=^ u v,^ Ω

√ ‖u‖ :=

√ 〈u, u〉 = ∫^2 u^ Ω

where unambiguously for a vector function v^ = (v, v^12

T, v) (^3) ∫ (^2) ‖v‖=^ v^ ·^ v Ω ∫^2 =^ v+^ v^1 Ω 22 +^ v=^ ‖ 2 3

(^2) v‖+^ ‖v 12 (^2 2) ‖+^ ‖v‖^3

(^2) u(x)≤^

∫^ x (x − a) a ′^2 (u(s))ds ≤^ (x^ −^ a)

∫^ b ′^2 (u(s))a

ds

∫^ b ⇒^ u(x) a

∫^2 dx ≤

∫^ b (x − a) a b ′^2 (u(s))ds a

dx (^1) = (b^ −^ a 2 ∫^ b 2 ′)^ (u(s))a 2 ds

Thus

(^1) ‖u‖ ≤ √^2

′(b − a)‖u

which is the Poincaré inequality for a 1-dimensionaldomain. A similar result holds if

u(b) = 0.

If^ Ω = [a, b

]^ ×^ [c, d]^ ⊂

2 Ris a rectangle and

u^ = 0^ on^ ∂

then the arguement is similar to the above:

(^1) H=^ {u^ ∈ H E^

1 (Ω)^ |^ u^ =^

gon^ ∂ΩD^

},D

(^1) H=^ {v^ E^0

1 ∈ H(Ω)^ |^

v^ = 0 on^ ∂

Ω},D^

S= span^0 {φ, φ,... , φ^12

(^1) } ⊂ Hn E^0

S=^ {r^ +E^

(^1) w| r ∈ H E , w^ ∈^ S} ⊂ H^0

1 E

(can take^ r

= 0^ if^ g= 0D^

in which case 11 H= H, S=E^ E E^0

S^0

If for^ u, v^ ∈ H

1 we define E^ a(u, v) :=

∫^ ∇u^ · ∇v^ Ω

and the linear form

∫ `(v) :=^ Ω ∫ vf +^ ∂ΩN vgN

then we can restate:Weak form: Find

1 u ∈ Hsuch that E^ a(u, v) = `(v)^

for all^ v^ ∈ H

1. E^0

Finite element approximation: Find

u∈^ S⊂ Hh^ E^

1 such E^

that^ a(

u, v) =^ `(h

v)^ for all

v^ ∈^ S⊂ H^0

1 E^0 FEM – p.8/

Given this inner product a norm (the

energy norm

) is

naturally defined by

√ ‖u‖= (^) E a(u, u)

We will often prefer to use the norm

‖ · ‖^ associated with

the^ L(Ω)^ inner product^2 〈u, v

∫ 〉 =^ u v,^ Ω

√ ‖u‖ =

√ 〈u, u〉 = ∫^2 u^ Ω

and therefore write (as above)^ ‖

u‖=^ ‖∇E^

u‖^ or^ a(u, u

(^2) ) = ‖∇u‖

Immediately we have uniqueness of the weak solution:

Theorem:‖∇

(u^ −^ u)‖ ≤ ‖∇h

(u^ −^ v)‖^ h

for all^ v∈h^

SE

Proof: Let^ v

∈^ S. Noteh E^

u−^ v∈h^ h^

S,^ u^ −^ u^0 h

1 ∈ Hso E^0

‖∇(u^ −^

(^2) u)‖h = a(u −^ u, uh −^ u)h =^ a(u^ −^ u

, u^ −^ v+hh^ v−^ u)h^ h =^ a(u^ −^ u

, u^ −^ v) +hh

a(u^ −^ u, vh

−^ u)h h

=^ a(u^ −^ u

, u^ −^ v)^ hh

(Galerkin orthogonality)

≤^ ‖∇(u^ −

u)‖‖∇(uh

−^ v)‖^ (Cauchy-Schwarz)h

Hence for either

‖∇(u^ −^ u

)‖^ = 0^ or^ h

‖∇(u^ −^ uh

)‖ 6^ = 0

‖∇(u^ −^ uh

)‖ ≤ ‖∇(u

−^ v)‖^ for allh

v∈^ Sh^ E

Notice that the minimum is achieved since

u∈^ S.h^ E^

An^ a priori^

error bound is now obtained by describing how

well any function in

1 Hcan be approximated by a function E^

in^ Sin the energy norm.E^