# Ermittlung der Fehlernorm in  7.1.6 mit Hilfe der
# GAUSS-LEGENDRE-Quadratur:
> F(xi,eta):=(10-6*xi-5*eta)*eta/63-((1-eta/3)^2-xi^2)*eta/6;
> 

                                            //    eta\2     2\
                                            ||1 - ---|  - xi | eta
                  (10 - 6 xi - 5 eta) eta   \\     3 /       /
    F(xi, eta) := ----------------------- - ----------------------
                            63                        6

> L_2:=sqrt(Int(Int(F^2,eta=0..1-xi),xi=0..1))=sqrt(int(int(F(xi,eta)^2,
> eta=0..1-xi),xi=0..1));
> 

                   /   1    1 - xi            \1/2
                   |  /    /                  |        1/2
                   | |    |         2         |      15
            L_2 := | |    |        F  deta dxi|    = -----
                   | |    |                   |      1890
                   |/    /                    |
                   \  0    0                  /

> G(x,y):=(1/16)*(2-x-y)*subs({xi=(1+x)*(3-y)/8,eta=(1+y)*(3-x)/8},F(xi,
> eta)^2):
> G(x,y):=simplify(%);
> 

                                   2         2
  G(x, y) := - (-2 + x + y) (1 + y)  (-3 + x)  (9 y + 27 x - 60 x y

               2       2           2      2  2      2
         + 70 x  - 49 x  y + 21 x y  + 7 x  y  - 33) /9364045824

> l_2:=sqrt(Int(Int(G,y=-1..1),x=-1..1))=sqrt(int(int(G(x,y),y=-1..1),x=
> -1..1));
> 

                       /   1    1        \1/2
                       |  /    /         |        1/2
                       | |    |          |      15
                l_2 := | |    |   G dy dx|    = -----
                       | |    |          |      1890
                       |/    /           |
                       \  -1   -1        /

# Werte des Integranden in den 4 x 4 GAUSS-Punkten:
> a:=sqrt((3+2*sqrt(6/5))/7);   b:=sqrt((3-2*sqrt(6/5))/7);
> 

                                        1/2 1/2
                            (525 + 70 30   )
                       a := -------------------
                                    35


                                        1/2 1/2
                            (525 - 70 30   )
                       b := -------------------
                                    35

> g[1]:=simplify(subs({x=-a,y=-a},G(x,y))):
> g[2]:=simplify(subs({x=a,y=-a},G(x,y))):
> g[3]:=simplify(subs({x=a,y=a},G(x,y))):
> g[4]:=simplify(subs({x=-a,y=a},G(x,y))):
> g[5]:=simplify(subs({x=-b,y=-b},G(x,y))):
> g[6]:=simplify(subs({x=b,y=-b},G(x,y))):
> g[7]:=simplify(subs({x=b,y=b},G(x,y))):
> g[8]:=simplify(subs({x=-b,y=b},G(x,y))):
> g[9]:=simplify(subs({x=-a,y=-b},G(x,y))):
> g[10]:=simplify(subs({x=a,y=-b},G(x,y))):
> g[11]:=simplify(subs({x=a,y=b},G(x,y))):
> g[12]:=simplify(subs({x=-a,y=b},G(x,y))):
> g[13]:=simplify(subs({x=-b,y=-a},G(x,y))):
> g[14]:=simplify(subs({x=b,y=-a},G(x,y))):
> g[15]:=simplify(subs({x=b,y=a},G(x,y))):
> g[16]:=simplify(subs({x=-b,y=a},G(x,y))):
# Wichtungsfaktoren:
> w[1..4]:=59/216-(1/6)*sqrt(5/6);
> 

                                            1/2
                                    59    30
                       w[1 .. 4] := --- - -----
                                    216    36

> w[5..8]:=59/216+(1/6)*sqrt(5/6);
> 

                                            1/2
                                    59    30
                       w[5 .. 8] := --- + -----
                                    216    36

> w[9..16]:=49/216;
> 

                                        49
                          w[9 .. 16] := ---
                                        216

> GAUSS_Quadratur:=sqrt(w[1..4]*sum(g[i],i=1..4)+w[5..8]*sum(g[j],j=5..8
> )+w[9..16]*sum(g[k],k=9..16)):
> GAUSS_Quadratur:=expand(%);
> 

                                                     1/2
                                717233481216000000000
             GAUSS_Quadratur := ------------------------
                                     13069123200000

> GAUSS_Quadratur:=simplify(%);
> 

                                            1/2
                                          15
                       GAUSS_Quadratur := -----
                                          1890

# Dieses Wert stimmt mit dem exakten Wert berein.
> 
