> N[1](xi,eta):=(81/4)*xi*(1-xi)*(1-3*xi)*eta*(1-eta)*(1-3*eta);
> 

  N[1](xi, eta) :=

        81 xi (1 - xi) (1 - 3 xi) eta (1 - eta) (1 - 3 eta)
        ---------------------------------------------------
                                 4

> Di:=diff(N[1](xi,eta),xi):
> di:=diff(N[1](xi,eta),eta):
> f(xi,eta):=Di^2+di^2:
> K[11]:=Int(Int(f,xi=0..1),eta=0..1)=int(int(f(xi,eta),xi=0..1),eta=0..
> 1);
> 

                             1    1
                            /    /
                           |    |                1458
                 K[11] :=  |    |   f dxi deta = ----
                           |    |                175
                          /    /
                            0    0

> g(x,y):=subs({xi=(1+x)/2,eta=(1+y)/2},f(xi,eta)):
> k[11]:=(1/4)*Int(Int(g,x=-1..1),y=-1..1)=(1/4)*int(int(g(x,y),x=-1..1)
> ,y=-1..1);
> 

                                1    1
                               /    /
                              |    |             1458
                k[11] := 1/4  |    |   g dx dy = ----
                              |    |             175
                             /    /
                               -1   -1

# Werte des Integranden in den 3 x 3 GAUSS-Punkten:
> a:=sqrt(3/5);
> 

                                     1/2
                                   15
                              a := -----
                                     5

> G[1]:=subs({x=-a,y=-a},g(x,y)):
> G[1]:=expand(%);
> 

                                               1/2
                            2224179   452709 15
                    G[1] := ------- - ------------
                            200000       160000

> G[2]:=subs({x=a,y=-a},g(x,y)):
> G[2]:=expand(%);
> 

                                    452709
                            G[2] := ------
                                    200000

> G[3]:=subs({x=a,y=a},g(x,y)):
> G[3]:=expand(%);
> 

                                               1/2
                            2224179   452709 15
                    G[3] := ------- + ------------
                            200000       160000

> G[4]:=subs({x=-a,y=a},g(x,y)):
> G[4]:=expand(%);
> 

                                    452709
                            G[4] := ------
                                    200000

> G[5]:=subs({x=0,y=-a},g(x,y)):
> G[5]:=expand(%);
> 

                                               1/2
                            3562623   373977 15
                    G[5] := ------- - ------------
                            512000       256000

> G[6]:=subs({x=a,y=0},g(x,y)):
> G[6]:=expand(%);
> 

                                               1/2
                            3562623   373977 15
                    G[6] := ------- + ------------
                            512000       256000

> G[7]:=subs({x=0,y=a},g(x,y)):
> G[7]:=expand(%);
> 

                                               1/2
                            3562623   373977 15
                    G[7] := ------- + ------------
                            512000       256000

> G[8]:=subs({x=-a,y=0},g(x,y)):
> G[8]:=expand(%);
> 

                                               1/2
                            3562623   373977 15
                    G[8] := ------- - ------------
                            512000       256000

> G[9]:=subs({x=0,y=0},g(x,y)):
> G[9]:=expand(%);
> 

                                    59049
                            G[9] := -----
                                    8192

# Wichtungsfaktoren:
> w[1..4]:=25/81;  w[5..8]:=40/81;  w[9]:=64/81;
> 

                                        25
                           w[1 .. 4] := --
                                        81


                                        40
                           w[5 .. 8] := --
                                        81


                                      64
                              w[9] := --
                                      81

> GAUSS_Quadratur:=(1/4)*(w[1..4]*sum(G[i],i=1..4)+w[5..8]*sum(G[j],j=5.
> .8)+w[9]*G[9]);
> 

                                          13851
                       GAUSS_Quadratur := -----
                                          2000

> rel_Fehler:=(1458/175-GAUSS_Quadratur)/(1458/175);
> 

                                        27
                          rel_Fehler := ---
                                        160

> rel_Fehler:=evalf(%);
> 

                      rel_Fehler := 0.1687500000

# Der relative Fehler betrgt 16,9 % aufgrund der zu geringen Anzahl von
# 3 x 3 GAUSS-Punkten. Die GAUSS-LEGENDRE-Quadratur mit 4 x 4
# Sttzstellen liefert den exakten Integralwert, wie der folgende
# MAPLE-Output zeigt. 
> 
> N[1](xi,eta):=(81/4)*xi*(1-xi)*(1-3*xi)*eta*(1-eta)*(1-3*eta);
> 

  N[1](xi, eta) :=

        81 xi (1 - xi) (1 - 3 xi) eta (1 - eta) (1 - 3 eta)
        ---------------------------------------------------
                                 4

> Di:=diff(N[1](xi,eta),xi):
> Di:=simplify(%):
> di:=diff(N[1](xi,eta),eta):
> di:=simplify(%):
> f(xi,eta):=Di^2+di^2:
> f(xi,eta):=expand(%):
> K[11]:=Int(Int(f,xi=0..1),eta=0..1)=int(int(f(xi,eta),xi=0..1),eta=0..
> 1);
> 

                             1    1
                            /    /
                           |    |                1458
                 K[11] :=  |    |   f dxi deta = ----
                           |    |                175
                          /    /
                            0    0

> g(x,y):=subs({xi=(1+x)/2,eta=(1+y)/2},f(xi,eta)):
> g(x,y):=expand(%):
> g(x,y):=simplify(%):
> k[11]:=(1/4)*Int(Int(g,x=-1..1),y=-1..1)=(1/4)*int(int(g(x,y),x=-1..1)
> ,y=-1..1);
> 

                                1    1
                               /    /
                              |    |             1458
                k[11] := 1/4  |    |   g dx dy = ----
                              |    |             175
                             /    /
                               -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]:=subs({x=-a,y=-a},g(x,y)):
> G[1]:=simplify(%):
> G[2]:=subs({x=a,y=-a},g(x,y)):
> G[2]:=simplify(%):
> G[3]:=subs({x=a,y=a},g(x,y)):
> G[3]:=simplify(%):
> G[4]:=subs({x=-a,y=a},g(x,y)):
> G[4]:=simplify(%):
> G[5]:=subs({x=-b,y=-b},g(x,y)):
> G[5]:=simplify(%):
> G[6]:=subs({x=b,y=-b},g(x,y)):
> G[6]:=simplify(%):
> G[7]:=subs({x=b,y=b},g(x,y)):
> G[7]:=simplify(%):
> G[8]:=subs({x=-b,y=b},g(x,y)):
> G[8]:=simplify(%):
> G[9]:=subs({x=-a,y=-b},g(x,y)):
> G[9]:=simplify(%):
> G[10]:=subs({x=a,y=-b},g(x,y)):
> G[10]:=simplify(%):
> G[11]:=subs({x=a,y=b},g(x,y)):
> G[11]:=simplify(%):
> G[12]:=subs({x=-a,y=b},g(x,y)):
> G[12]:=simplify(%):
> G[13]:=subs({x=-b,y=-a},g(x,y)):
> G[13]:=simplify(%):
> G[14]:=subs({x=b,y=-a},g(x,y)):
> G[14]:=simplify(%):
> G[15]:=subs({x=b,y=a},g(x,y)):
> G[15]:=simplify(%):
> G[16]:=subs({x=-b,y=a},g(x,y)):
> G[16]:=simplify(%):
# 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:=(w[1..4]/4)*sum(G[i],i=1..4)+(w[5..8]/4)*sum(G[j],j=5
> ..8)+(w[9..16]/4)*sum(G[k],k=9..16):
> GAUSS_Quadratur:=expand(%);
> 

                                          1458
                       GAUSS_Quadratur := ----
                                          175

# Dieser Wert stimmt mit dem exakten Integralwert (7) aus  7.1.11
# berein.
# 
> 
