u(x, y) =f(x, y) Ω=(0, 1) (0, 1)

u(x, y) =f(x, y) Ω=(0, 1) (0, 1) u(x, y) =g(x, y) Γ=δΩ ={0, 1} {0, 1} Ω Ω Ω h Ω h h

ˆ Ω ˆ u v = fv Ω u = f in Ω v V H 1 (Ω) V V h V h ψ 1,ψ 2,...,ψ N, ˆ ˆ u v = Ω Ω fv v V ˆ ˆ u v = Ω ˆ ˆ u ψ i = Ω Ω Ω fψ i fv v V h i =1,...,N N N u = N i=1 α iψ i k h u h = f h α(ψ 1,ψ 2 ) α(ψ 1,ψ 2 )... K h = α(ψ n,ψ 1 )... α(ψ n,ψ n ) ˆ a(ψ i,ψ j )= ψ i ψ j Ω

α(ψ i,ψ j ) 4 1 0... 1 1 4 1 0 1 0 1 4 1 0 K h = 1 B 2 AC < 0 Au xx +2Bu xy + Cu yy + Du x + Eu y + F =0 K h und D 1 L T 1 L 1 D 2 L T 2 L n 1 L T n 1 D n K h

K h 2 1 0 0 K h = 1 2 1 0 0 1 2 1 0 0 1 2 (1, 2) 2 1 0 0 3 0 2 1 0 O(N 3 ) {k i,j : k i,j 0} = O(N)

Ku = b K n n u = K 1 b u = u K 1 (Ku b) K 1 M 1 M 1 u u M 1 (Ku b) u (i+1) = u (i) K 1 (Ku (i) b) M 1 M O(n) M 1 K 1 u M 1 = K 1 u (i+1) =(Id M 1 K)u (i) M 1 b Id M 1 K M = diag(m ii ) m ii = K ii M = L + D L K D = diag(k)

M =(L + D)D 1 (R + D) 0 0 k 1,1 k 2,1 0 L =,D= 0 k n,1... k n,n 1 0 0 k 1,2... k 1,n,R= kn 1,n 0 0 k n,n, O(n) (R + D)u = V }{{} O(n) Mu =(L + D)d 1 (R + D)u (L + D)D 1 V = u }{{} O(n) 4 1 0... 0 1 0... 0 1 4 1 0... 0 1 K = 4 0 M = D = 0 4 u (i+1) = u (i) D 1 (Ku (i) b) u (i+1) j = u (i) j 1 4 (4u(i) j u (i) j 1 u(i) j+1 u(i) j+n u(i) j n b j) = 1 4 (u(i) j 1 + u(i) j+1 + u(i) j n + u(i) j+n + b j)

4 0 1 4 M = L + D = 1 u (i+1) = u (i) (L + D) 1 (Ku (i) b) u (i+1) j = u (i) j = u (i) j 1 1 4 (L + D) 1 ((L + D + R)u (i) j b j ) u (i) j (L + D) 1 (Ru (i) j b j ) = (L + D) 1 (Ru (i) j b j ) u k+1 = u k M 1 (Ku k b) Ku = b M 1 u k+1 u = u k M 1 Ku k u M 1 b }{{} =Ku u k+1 u = Id M 1 K (u }{{} k u) G ρ(g) < 1 ρ(g) < 1 G := Id M 1 K K M v, Kv < 2 v, Mv,v R N ρ(g) = λ max (G) < 1

A M ρ(a) A G M (MG) T =(M(Id M 1 K)) T =(Id M 1 K) T M T ρ(g) { v 0 =(Id K T (M 1 ) T )M T =(Id KM 1 )M = M(Id M 1 K)=MG v, MGv v, Mv } = v 0 v, Kv = { 1 v 0 v, Mv } v,kv v,mv < 2 0 < v,kv v,mv K M v, Kv 0 < 1 v 0 v, Mv < 2 ρ(g) < 1 { v, M(Id M 1 K)v } v, Mv v, Kv < 2 v, Mv K = L + D + L T K Ku = b K K = L + D + L T D 1 (L + L T ) < 1 D 1 K < 2 D 1 L < 1

M J = D K D 1 K 2 D 1 K < 2 v, Kv D 1 K 2 v, Dv < 2 v, Dv ρ(id D 1 A) < 1 K M = D + L u k+1 = u k + M 1 (b Ku k ) Mu k+1 = Mu k +(b Ku k ) (D + L)u k+1 =(D + L)u k (D + L)u k+1 = L T u k + b Ku }{{} k +b (L+D+L T )u k G V GS = Id (D + L) 1 K = (D + L) 1 L T M = D + L T (D + L T )uk +1= Lu k + b G R GS = Id (D + L) T K = Id (D + L) T (L + D + L T ) }{{}}{{} D=D T =D T +L T =(D+L) T = Id (D + L) T ((D + L) T + L) = (D + L) T L

G SGS = G R GSG V GS =(D + L) T L(D + L) 1 L T = Id M 1 SGS K G SGS = Id M 1 SGS K =(D + L) T L(D + L) 1 L T M SGS = K + LD 1 L T (ausrechnen) ρ(g SGS )= G SGS K = G GS 2 K ρ(g SGS )= G SGS K }{{} G SGS K = v 0 = AGV GS, AGV GS v = G GSv 2 K Kv,v v 2 K v, AGR GS GV GS v v, Kv = G GS 2 K G R GS GV GS GR GS = G GS AG GS =(AGV GS )T M SGS v, Kv < 2 v, Mv

M SGS = K + LD 1 L T v, M SGS v = v, K + LD 1 L T v = v, Kv + v, LD 1 L T v = v, Kv + L T v, D 1 L T v > v, Kv v, M SGS v > v, Kv v, Kv < 2 v, M SGS v G SGS K = ρ(g SGS ) < 1 Du k+1 = ω(l + L T )u k +(1 ω)du k + b G J (ω) =Id ωd 1 K 0 <ω<1 K 0 <ω<1 M J(ω) = 1 ω D ωd 1 K 2 ωd 1 K 2ω <2 ρ <1

Du k+1 = (L + L T )u k + b G J = Id D 1 K a i,i u i,k+1 = j i k i,j u j,k + b i i u k+1 (D + L)u k+1 = L T u k + b k i,i u i,k+1 + j<i k i,j u j,k+1 = j>i k i,j u j,k + b i i u k+1 u i,k+1 u j,k+1

Au = b A u { } 1 u = y R N 2 yt Ay b T y N N N N A R N N R N A :=,A. e k r k u k e k := u u k, r k := b Au k. r k = b Au k = Au Au k = Ae k r u

u 0 u k = u k 1 + α k d k A R N d 1,d 2,...,d N R N = {d 1,...,d N } (d k, Ad j )=δ k,j j, k {1,...,N}. e 0 := u u 0 = N λ i d i. i=1 k k {u k 1 + αd k,α R} f (y) = 1 2 yt Ay b T y 0! = d dα (f (u k 1 + αd k )) 0=(d k,a(u k 1 + αd k ) b) α = (d k,b Au k 1 ) (d k, Ad k ) u k = u k 1 + α k d k α k = (d k,r k 1 ) (d k, Ad k ). = (d k,r k 1 ) (d k, Ad k ). k α k λ k d T k A (d k, Ae 0 )= N λ i (d k, Ad i )=λ k (d k, Ad k ) i=1 λ k = (d k, Ae 0 ) (d k, Ad k ) = = (d k, Ae k 1 ) (d k, Ad k ) ( ( d k,a e 0 )) k 1 i=1 α id i = (d k,r k 1 ) (d k, Ad k ). (d k, Ad k )

N A d i k d k A r k 1 k 1 d k := r k 1 i=1 (r k 1, Ad i ) (d i, Ad i ) d i. (d k, Ad j )=0 j<k A r k 1 d j j<k d 1,...,d k 1 d k 0 e k A d 1,...,d k r k = Ae k (d j,r k )=(d j, Ae k )= =0 j k. ( d j,a N i=k+1 λ i d i ) r k 1 A d j j<k d k r i = b Au i = b A (u i 1 + α i d i )=r i 1 α i Ad i. Ad i d 0 = r 0

D i D i = { r 0, Ar 0,A 2 r 0,...,A i 1} r 0,...,r i 1 d 1,...,d i r k 1 D k 1 D k 2 A r k 1 (r k 1,r i )=(r k 1,r i 1 α i Ad i ) (r k 1, Ad i )= (r k 1,r i 1 ) (r k 1,r i ) α i. (r k 1,r j )=0 j k 2 r j {,..., + } d j+1 r k 1 d 1,...,d k 1 d k (r k 1,r k 1 ) α (r k 1, Ad i )= k 1, i = k 1. 0,, (r k 1,r k 1 ) d k = r k 1 + α k 1 (d k 1, Ad k 1 ) d k 1 = r k 1 + (r k 1,r k 1 ) (d k 1,r k 2 ) d k 1. (d k,r k 1 )= ( r k 1 + (r ) k 1,r k 1 ) (d k 1,r k 2 ) d k 1,r k 1 =(r k 1,r k 1 )

κ 2 κ 2 := λ max (A) λ min (A). u u k A 2 ( κ2 1 κ2 +1) k u u 0 A ε u u k A ε u u 0 A, k = ( ) κ2 2 2 +1 ε

λ min 1 Āū = b Ā = AP 1 ū = Pu P Ā N P P = A 1 P P 1 P Ā P P = KK T K 1 K } 1 AK {{ T } K }{{ T u } = K }{{ 1 } b. =:à =:ũ =: b à à P d r ũ ˆd = K d ˆr = K r û = K T ũ s K T ũ 0 = K T K T u 0 = u 0

Ãũ = b Au = b r = b ) ( b Ãũ ˆr = b Aû = K Ãũ = K r d = r ˆd = P 1ˆr = K T d s = P 1ˆr = K T r ( r, r) ε 2 ( r 0, r 0 ) (ˆr, s) ε 2 (ˆr 0,s 0 ) ( r, r) ε 2 ( r 0, r 0 ) ( r, r) ( d,ã d) (ˆr,s) (K T r,k r) = α T d) α = ˆα = = ( ˆd,A ˆd) (K T d,kãkt K ũ =ũ + α d û =û +ˆα ˆd = K (ũ T + α d ) r = r αã d ˆr =ˆr ˆαA ˆd = K r αkãkt K T d = K ( r αã d ) s = P 1ˆr = K T r β = ( r, r) ( r, r ) ˆβ = (ˆr,s) (ˆr,s ) = (K r,k T r) (K r,k T r = β d = r + β d ( ) ˆd = s + ˆβ ˆd = K T r + β d

u =0 Ω=[0, 1] [0, 1], u = g Ω. Ω h h = 1/N K ψ ν,µ (K)(x, y) = (νπx) (µπy), (x, y) Ω h, λ ν,µ (K) = 4 ( ( ) ( )) νπh µπh h 2 2 + 2 2 2, 1 υ, µ N 1. G J = Id ωd 1 K G J K λ ν,µ (G J )=1 ω h2 4 λ υ,µ (K) =1 ω h2 4 =1 ω 4 h 2 =1 2ω + ω ( ( νπh 2 2 ) ( ( νπh 2 2 2 ( 2 ( νπh 2 ) ( )) µπh + 2 2 ( )) µπh 2 2 ) ( )) µπh + 2 2

ω = 1 2 λ ν,µ (G J( 1 2) ) = 1 ( ( νπh 2 2 2 ( ) = ρ G J( 1 < 1. 2) ) ( )) µπh + 2 2 h h u u h u 0 u = k N 1 ν,µ=1 u k u = G k J(ω) (u 0 u) = α ν,µ ψ ν,µ. N 1 ν,µ=1 α ν,µ λ k ν,µψ ν,µ. e k := u k u ν µ e k ω =1 ω = 1 2 ω =1 ω =0.5

S 0 S 1 S j H 1 0 (Ω) A k u k = f k, k =0,...,j. A k S k = N k j +1 S 0,...,S j

G h u u h S k ū k v k 1 u k ū k S k 1 v k 1 Av k 1,ϕ = A(u k ū k ),ϕ = f k Aū }{{ k,ϕ } Residuum A(u k ū k )=f k Aū k r k = f k Aū k }{{} Residuum, d k = Aū k f k }{{} Defekt ū k r p Ω 1 Ω 1 Ω 2 Ω 1 r p

u (0) u (i+1) u (i) ( ) A 2 u 2 = f 2 A 1 u 1 = f 1 ū 2 = G ν 1 2 u(i) 2 ν 1 d 2 = A 2 ū 2 f 2 d 1 = rd 2 Ω 2 d 2 Ω 1 û 1 = A 1 1 d 1 û 1 = A 1 1 r(a 2ū 2 f 2 ) û 2 =ū 2 pû 1 ū i+1 2 = G ν 2 2 û2 ν 2 u (i+1) = G ν 2 2 (Gν 1 1 u(i) pa 1 1 r(a 2G ν 1 1 u(i) f 2 )) = G ν 2 2 (Id pa 1 1 ra 2)G ν 1 1 u (i) + pa 1 1 }{{} rf 2 Zweigitteriterationsmatrix A k u k = f k k =0,...,j Ω 0 Ω 1 Ω j u (i+1) = u (i) ωd 1 (Au (i) f) ω<1

4 1 0... 1 1 4 1 0 1 0 1 4 1 0 K = 1

4 1 0 1... 4 1 K = = 1 4 0 1... 1 4 ( D L L T D ) 0 ν 1 ν 2 ν 1 + ν 2 4

##BH/mM; 9Xj, o@wvfhmb, γ = 2 ##BH/mM; 9X9, o@wvfhmb, γ = 2 qb` b2?2m,.2` m7r M/ b+?2bmi KBi xmm2?k2m/2m γ xm bi2b;2mx m7r M/b M Hvb2, "2i` +?i2 Ω0 Ω1 Ωj `2;2HK ĽB; p2`72bm2`i- /X?X nk = 2dk n0 - KBi nk ;H2B+? /2` Mx?H /2` EMQi2M pqm Ωk mm/ _ mk/bk2mbbqm dx.2` m7r M/ m7 D2/2K :Bii2` Bbi HBM2 `- /X?X m7r M/ = G(ν1, ν2, G, p) = Cnk jk

Level Auf wand Ω 0 Cn 0 Ω 1 Cn 1 + γcn 0 Ω 2 Cn 2 + γ(cn 1 + γcn 0 ) k 1 Ω k Cn k + Cγ k l n l l=0 n k =2 dk n 0 k 1 k Cn k + Cγ k l n l = C( γ k l n l ) l=0 l=0 n k 1 = 1 2 d n k n k 2 = 1 (2 d ) 2 n k n l = 1 2 d(k l) n k k k C( Cγ k l n l )=C γ k l 1 2 d(k l) n k l=0 l=0 k ( γ ) k l = Cn k 2 d l=0 γ 2 d < 1 k l=0 ( γ 2 d ) k l C k ( γ ) k l 1 = 2 d l=0 k ( γ ) k l 2 d l=0 }{{} A(γ,d) 1 γ 2 d = C 1 γ 2 d

γ =1: A =2C γ =2: A = kc γ =1: A = 4 3 C γ =2: A =2C γ =3: A =4C p

u O = 1 2 (u L O + u RO ), u L = 1 2 (u L O + u LU ) u R = 1 2 (u R O + u RU ), u U = 1 2 (u L U + u RU ) u Z = 1 4 (u L O + u RO + u LU + u RU ) 1 1 1 1 [ ] 2 4 2 4 p = 1 1 2 1 2 1 = 1 1 2 1 2 1 2 = 1 1 2 1 2 4 2 4 1 2 1 1 4 1 2 1 4 Ω k 1 Ω k r = 1 1 2 1 2 4 2 4 1 2 1

p = 1 1 2 1 2 4 2 16 1 2 1 u i+1 = G ν 2 2 (Id pa 1 1 ra 2)G ν 1 2 u(i) + pa 1 1 rf 2 T 2,l (ν 1,ν 2 )=G ν 2 2 (Id pa 1 l 1 ra l)g ν 1 2 e i+1 T 2,l (ν 1,ν 2 ) e (i) T 2,l (ν 1,ν 2 ) ζ<1 ν 1 + ν 2 =: ν T 2,l (ν, 0) = (Id pa 1 l 1 ra l)g ν 2 T 2,l (ν, 0) = (Id pa 1 l 1 ra l)g ν 2 = (A 1 l = A 1 l pa 1 l 1 r)a lg ν 2 pa 1 l 1 r A lg ν 2

A 1 l pa 1 l 1 r C Ah 2m A l G ν l C Sη(ν)h 2m η(ν) 0 ν 2m h T 2,l (ν, 0) < 1 ν ν 0 T 2,l (ν, 0) A 1 l pa 1 l 1 r A lg ν l C A h 2m C S η(ν)h 2m C A C S η(ν) T 2,l (ν, 0) G = Id M 1 A A = M N G = M 1 N A l G ν 2 C S η(ν)h 2m

AG ν =(M N)(M 1 N) ν AG ν X (M 1 N) ν (M 1 N) ν = M 1 N M 1 N M 1 N M 1 N }{{} ν mal = M 1 2 M 1 2 NM 1 2 M 1 2 N M 1 2 M 1 2 }{{ N } M 1 1 2 M 2 ν mal = M 1 2 X ν M 1 2 X := M 1 2 NM 1 2 X G X = M 1 2 GM 1 2 X = M 1 2 NM 1 1 2 = M 2 M 1 }{{ N } G M 1 2 AG ν =(M N)(M 1 N) ν =(M N)(M 1 2 X ν M 1 2 ) =(M 1 2 NM 1 2 )(X ν M 1 2 ) = M 1 2 (Id M 1 2 NM 1 2 }{{} )X ν M 1 2 X = M 1 2 (Id X)X ν M 1 2 AG ν 2 M 1 2 2 (Id X)X ν 2 M 1 2 2 M (1 ξ)ξ ν ξ

g(ξ) :=(1 ξ)ξ ν ν g (ξ) = ξ ν + ν(1 ξ)ξ ν 1 = ξ ν + νξ ν 1 νξ ν = νξ ν 1 (ν + 1)ξ ν! =0 ξ 0 ν (ν + 1)ξ =0 ξ = ν ν +1 ν g( ν +1 )=(1 ν ν +1 )( ν ν +1 )ν = ν ν (ν + 1) ν+1 ν ν ν (ν + 1) ν+1 ν ν (ν) ν+1 = 1 ν 0 ν g(ξ) =(1 ξ)ξ ν η(ν) := (1 ξ)ξ ν ξ AG ν 2 M 2 η(ν) M 2 C S h 2m A 2 {}}{ 8 h 2 }{{} A h 2m C S A 2 C S h 2m A l G ν 2 C S η(ν)h 2m

A 1 l pa 1 l 1 r C Ah 2m A 1 l pa 1 l 1 r = A 1 l pra 1 l + pra 1 l } {{ } =(Id pr)a 1 l =(Id pr)a 1 l =(Id pr)a 1 l =(Id pr)a 1 l + p(ra 1 l pa 1 l 1 r A 1 l 1 r) + pa 1 l 1 (A l 1r ra l )A 1 l + pa 1 l 1 (A l 1r ra l pr + ra l pr }{{} neu ra l )A 1 l + pa 1 l 1 ((A l 1 ra l p)r + r(a l pr A l ))A 1 l A 1 l C 1 ( A 1 l 1 C 3) A l C 5 r C 4 p C 2 A l 1 ra l p C K h 2m A 1 l Id pr C I h 2m pa 1 l 1r Id pr A 1 l + p A 1 l 1 ( A l 1 ra l p r + r Id pr A l ) A 1 l C I h 2m C 1 + C 2 C 3 (C K h 2m C 4 + C 4 C I h 2m C 5 )C 1 =(C I C 1 + C 1 C 2 C 3 C 4 C K + C 1 C 4 C 5 )h 2m = C 2m A

ϕ u E p ϕ + (ϕu) =0 t (ϕu) + (u (ϕu)) + p =0 t E + (u(e + p)) = 0 t u = v = u v = n α i e i i=1 n β i d i i=1 n α i β j (e i d j ) i,j=1 f(u),f: D X R f X f (u) =0 F (u) :=f (u)

F (u) =0 F (u) =0 Ω Ω h Fh (u h )=0 F h (u h ) F h (u)+δu h F h (u h) F h (u h)δu h = F h (u h ) F (u) =0 F (u)δu = F (u) Ω Ω h F h (u h)δu h = F h (u h ) f(u) = f : D R N R f F (u) :=f (u) T =0 F : D R N

F (u) F (u) F F (u) =f (u) F G = C F C g(ũ) =f(cũ) =min G(ũ) =C T F (Cũ) =0 G (ũ) =C T F (u)c u = Cũ F (u) w, z F (u) = w T F (u)z u,w,z D w, z G (ũ) = w T C T F (u)c z = w, z F (u) u = Cũ, w = C w, z = C z w F (u) = ( ) 1 w, w F 2 (u) = ( w T F (u)w ) 1 2 F (u) =0 F (u) =f (u) T =0 F (u)δu = F (u)

F (u k )δu k = F (u) u k+1 = u k + λ k δu k,λ k ]0, 1] δu k = (F (u k )) 1 F (u k ) u k+1 = u k λ k (F (u k )) 1 F (u k ) u k u λ k =1 λ<1 δu k δu k δu k δu k = (F (u k )) 1 F (u k )+ F (u k ) δu k = F (u k )+r k F (u k )( δu k δu k )=r k u k+1 = u k + δu k Residuum {}}{ r k

r k i i =1,...,n δu k i δu k i δuk i F (u k ) δu k i, δu k i δu k i F (u k ) = δu k i,r k i =0 e k i = δu k i δuk i F (u k ) δu k i F (u k )

c i,j i j u i,j c i,j F ( ) := i,j (u i,j ( ) c i,j ) 2 F D u i,j (D)

u(x, t) = div(d u(x, t)) + t u(x, t 0 )=u 0 (x) u(x, t) =f(t) δω {}}{ S(t) Ω ˆ Ω ˆ u t = = Ω ˆ div(d u) = m ˆ i=1 b i D u n Ω u t u(t k+1,x) u(t k,x) t D u n ˆ ˆ ˆ u(t k+1,x i )dx b i u(t k,x)dx = t b i δb i j u(t k+1,x j )D ξ j (γ) n i dγ b i (u(t k+1,x i ) u(t k,x i )) = t j,l b i b l u(t k+1,x j )D ξ(x i,l ) n i,l b i : b i b j : b i b j F (D) = (u i,j (D) c i,j ) 2 D i,j

F (D) = i,j ( u i,j (D) c i,j + u ) 2 i,j D D D ( u i,j (D (q) ) c i,j + u i,j D(q) D (q) D (q) i,j D F (q) (D (q) )=0 2 ( i,j i,j i,j ( D (q) = u i,j (D (q) ) c i,j + u i,j D(q) D (q) ) u i,j (D (q) ) c i,j + u i,j D(q) D (q) ( ) c i,j u i,j (D (q) ) u i,j(d (q) ) i,j D (q) ) 2 ) ( ) u i,j (D (q) ) D (q) + u i,j(d (q) ) D (q) =0 u i,j(d (q) ) D (q) =0 = i,j ( ui,j D (q) ) D (q) ( (ci,j u i,j (D (q) ) ) u i,j(d (q) ) ( ) ui,j 2 i,j D (q) ) 2 D (q) D (q) u i,j D (q) b i D (q) (u(t k+1,x i ) u(t k,x i )) = t ( ) b i b j D (q) u(t k+1,x j )D (q) + u(t k+1,x j ) ξ j (x i,l ) n i j,l ( b i D (q) u(t k+1,x i ) ) D (q) u(t k,x i ) = t ( ) b i b j D (q) u(t k+1,x j )D (q) + u(t k+1,x j ) ξ j (x i,l ) n i j,l g(t k,x j,d (q) ):= D (q) u(t k,x i )

b i (g(t k+1,x i ) g(t k,x i )) = t ( ) b i b j g(t k+1,x j )D (q) + u(t k+1,x j ) ξ j (x i,l ) n i j,l ( ) ( ) = t j,l ( b i b j g(t k+1,x j )D (q)) ξ j (x i,l ) n i ( ) + t j,l b i b j (u(t k+1,x j )) ξ j (x i,l ) n i ( ) div(d (q) g(x, t)) div( u(x, t)) u i,j D (q) D (q) F (D (q) ) u i,j D (q) t g i,j(d (q) )=div(d (q) g(x, t)) + div( u(x, t)) u(x, t) t g(x, t) t = div(d u(x, t)) Ω = div(d g(x, t)) + div( u(x, t)) Ω D (q) D (q+1) = D (q) D (q) D (q+1) D (q) <TOL

( ) 2 ϵ u = x 2 ϵ 2 y 2 ϵ =1 ϵ 0 u = f ϵ 0 h h G ν 2 (Id pa 1 l 1 ra l)g ν 1 2 ξ<1 ϵ ϵ ϵ ϵ u = f Ω=(0, 1) 2, ϵ = 2 x 2 + ϵ 2 y 2 u = g Γ= Ω ϵ = 1 ϵ h 2 1 2+2ϵ 1 = 1 ] [ 1 h 2 2 1 ϵ + ϵ h 2 1 2 1

ϵ 0 2 1 ϵ=0 = 1 0 h 2 1 2 1 = 1 1 2 1 h 2 0 1 2 ϵ 0 K(ϵ) K 0 < 1 κ(ϵ) 1 ϵ 0 1 h 2 ϵ 1 2+2 ϵ 1 1 h 2 ϵ 0 1 2 1 0 y K 1 l (ϵ) pk 1 l 1 (ϵ)r C h 2 A ϵ K l (ϵ) ϵ

ϵ ϵ = 1 ] [ 1 h 2 2 1 + 1 h 2 ϵ 2ϵ h ϵ 0 ϵ ϵ = 1 h 2 x h y << h x [ ] 1 2 1 + 1 h 2 y ϵ 2ϵ ϵ

ϵ K 1 l (ϵ) pk 1 l 1 (ϵ)r C h 2 A ϵ K l G ν l C S η(ν) ϵ h 2 ϵ h K K l (ϵ) M(ϵ) N(ϵ) K l (ϵ)+αn(ϵ) 0 α G = M 1 N N(ϵ) ϵ h 2 C N K l (ϵ)g ν l C S η(ν) ϵ h 2 N N(ϵ) C N ϵ h 2

2(1 + ϵ) 1 ( ) ϵ 0 ( ) 1 2(1 + ϵ) 1 ϵ 0 K h (ϵ) = 1 h 2 ϵ ϵ 1 ϵ 1 2(1 + ϵ) K h (ϵ) N ( ) I. 2(1 + ϵ) 1 : 2(1 + ϵ) II. +1 2(1 + ϵ) I. 2(1 + ϵ) 1 II. 0 4(1+ϵ) 2 1 2(1+ϵ) ( ) I. ϵ 0 : 2(1 + ϵ) II. 0 ϵ I. ϵ 0 II. 0 ϵ 2(1+ϵ) N N N = 1 h 2 ϵ d ij 0 0 0 0 0 0 0 ϵ d ij 2(1 + ϵ) i =1,j =1 2(1 + ϵ) 1 d d ij = i,j 1 i =1,j >1 2(1 + ϵ) ϵ2 d i 1,j i>1,j =1 2(1 + ϵ) 1 d i,j 1 ϵ2 d i 1,j i>1,j >1

d ij δ = 2(1 + ϵ) 1 ϵ2 δ δ 2 (2 + 2ϵ)δ +(1+ϵ 2 )=0 = 2(1 + ϵ)δ (1 + ϵ2 ) δ δ 1,2 = 2+2ϵ ± (2 + 2ϵ) 2 4(1 + ϵ 2 ) 2 δ 1,2 = 2+2ϵ ± 4+8ϵ +4ϵ 2 4 4ϵ 2 2 =1+ϵ ± 2ϵ N 1 h 2 ( ϵ δ + ϵ δ )=2ϵ δ C N =2 1 h 2 < 2ϵ 1 δ<1 h 2 u t = ϵ u }{{} + v }{{} u Konv. ϵ v Dif.

ϵ u [ ϵ u] h = ϵ h 2 1 1 4 1 1 v u [ v u] h = v [ ] 1 1 0 1 + v 1 2 0 2h 2h 1 [ ϵ u + v u] h = ϵ h 2 1 1 4 1 + v 1 2h 1 = 1 ϵ + hv 2 2 h 2 ϵ hv 1 2 4ϵ ϵ + hv 1 2 ϵ hv 2 2 [ ] 1 0 1 + v 2 2h 1 0 i j K i,j = ϵ + hv 2 2 + ϵ + hv 1 2 + ϵ hv 1 2 + ϵ hv 2 2 K i,j i j K i,j v 1,v 2 > 0 1 ϵ hv 1 2 ϵ hv 2 2 4ϵ! ϵ hv 2 2 + ϵ hv 1 2 + ϵ + hv 1 2 + ϵ + hv 2 2 0 0 K ϵ hv 1 2 ϵ hv 2 2 4ϵ ϵ! + hv 2 2 ϵ + hv 1 2 + ϵ + hv 1 2 + ϵ + hv 2 2 4ϵ hv 1 + hv 2 h( 2ϵ h )+h(2ϵ h )=4ϵ K

ϵ< hv 1 2 ϵ hv 2 2 4ϵ! ϵ hv 2 2 ϵ + hv 1 2 + ϵ + hv 1 2 + ϵ + hv 2 2 =2ϵ + hv 1 2ϵ hv 1! 0 2ϵ! hv 1 ϵ! hv 1 2 K ϵ h 2 v 1 ϵ h 2 v 2 M h M v u 1 [ ] 1 1 v 1 1 0 1 + v 2 0 2h 2h 1 v 1,v 2 > 0 v 1,v 2 < 0 [ v u] h 1 [ ] 0 1 = v 1 1 1 0 + v 2 1 2h 2h 1 [ v u] v h < 0 = v 1 1 2h [ ] 1 0 1 1 + v 2 2h 1 1 0

v > 0 [ ϵ u + v u] h = ϵ 1 h 2 1 4 1 + v 1 h 1 = 1 ϵ h 2 ϵ hv 1 4ϵ + hv 1 + hv 2 ϵ ϵ hv 2 [ ] 1 1 0 + v 2 h 0 1 1 v + p = f v =0 v p ( )( ) ( ) v f T = 0 p 0 ( ) ( ) ( ) v K = T,u=, f f = 0 p 0 K 1,1 K 1,m K = K m,1 u 1 u m K m,m f 1 f m u =,f = K i,j n n ( ) A B B T 0

2 2 L = ( ( 0 ) Id 1 0 B T A 1 Id 2 ) ( ) ( ) ( Id 1 0 A B A L K = B T A 1 Id 2 B T = 0 B T A 1 A + B T ( ) A B = 0 B T A 1 B B B T A 1 B ) L ( )( ) ( ) v f 0 T 1 = p 0 A B T A 1 B ( ) A B 0 B T A 1 B a(u, v) b(v, p) = f,v v U b(u, q) = g, q q V a : U U R b : V V R a b

a U 0 a(x, x C E x 2 V ) U 0 := {v U : b(v, q) =0 q V } q V {0} v U {0} b(v, q) v U q V ζ>0 dim(u h )=n dim(v )=m (dim) (n ) (m ) R h = (n ) A h B h (m ) Bh T 0 m n R h (n+m) (n+m) K h Rang(K h ) Rang(A h )+Rang(B h ) = n + Rang(B h ) B h n m m>n Rang(B h ) n<m Rang(K h ) n + n<n+ m Rang(K h ) <n+ m K h m n

U V V x v g p x y dim(u h )=2 dim(v h )=9 1=8 Ω p m>n U h V h dim(u h )=2 dim(v h )=9 1 1=7 Ω p = c m>n dim(u h ) dim(v h ) 2h U h := {v C 2 ( Ω) : v Γ =0,v T P 2 } V h := {p C( Ω) : p T P 1 }

b k R 2 dim(u h )=8 2 = 16 dim(v h )=8