29 April 2007 06:43:45 AM

FEM1D_NONLINEAR
  C++ version

  Solve a nonlinear boundary value problem:

    -d/dx (p(x) du/dx) + q(x)*u + u*u' = f(x)

  on an interval [xl,xr], with the values of
  u or u' specified at xl and xr.

  The interval [XL,XR] is broken into N = 10 subintervals
  Number of basis functions per element is NL = 2

  The equation is to be solved for
  X greater than XL = 0
  and less than XR = 1

  The boundary conditions are:

  At X = XL, U = 0
  At X = XR, U' = 1

  This is test problem #1:

  P(X) = 1, Q(X) = 0, F(X) = X.
  Boundary conditions: U(0) = 0, U''(1) = 1.

  The exact solution is U(X) = X

  Number of quadrature points per element is 1
  Number of iterations is 10

  Node      Location

         0               0
         1             0.1
         2             0.2
         3             0.3
         4             0.4
         5             0.5
         6             0.6
         7             0.7
         8             0.8
         9             0.9
        10               1

Subint    Length

         1             0.1
         2             0.1
         3             0.1
         4             0.1
         5             0.1
         6             0.1
         7             0.1
         8             0.1
         9             0.1
        10             0.1

Subint    Quadrature point

         1            0.05
         2            0.15
         3            0.25
         4            0.35
         5            0.45
         6            0.55
         7            0.65
         8            0.75
         9            0.85
        10            0.95

Subint  Left Node  Right Node

         1         0         1
         2         1         2
         3         2         3
         4         3         4
         5         4         5
         6         5         6
         7         6         7
         8         7         8
         9         8         9
        10         9        10

  Number of unknowns NU = 10

  Node  Unknown

         0        -1
         1         1
         2         2
         3         3
         4         4
         5         5
         6         6
         7         7
         8         8
         9         9
        10        10

Printout of tridiagonal linear system:

Equation  ALEFT  ADIAG  ARITE  RHS

         1               0              20             -10            0.01
         2             -10              20             -10            0.02
         3             -10              20             -10            0.03
         4             -10              20             -10            0.04
         5             -10              20             -10            0.05
         6             -10              20             -10            0.06
         7             -10              20             -10            0.07
         8             -10              20             -10            0.08
         9             -10              20             -10            0.09
        10             -10              10               0          1.0475

Computed solution:

Node    X(I)        U(X(I))

       0             0             0
       1           0.1       0.14975
       2           0.2        0.2985
       3           0.3       0.44525
       4           0.4         0.589
       5           0.5       0.72875
       6           0.6        0.8635
       7           0.7       0.99225
       8           0.8         1.114
       9           0.9       1.22775
      10             1        1.3325

Computed solution:

Node    X(I)        U(X(I))

       0             0             0
       1           0.1     0.0849454
       2           0.2      0.170164
       3           0.3      0.255932
       4           0.4      0.342534
       5           0.5      0.430266
       6           0.6      0.519437
       7           0.7      0.610377
       8           0.8      0.703436
       9           0.9      0.798989
      10             1      0.897438

Computed solution:

Node    X(I)        U(X(I))

       0             0             0
       1           0.1      0.105202
       2           0.2      0.210298
       3           0.3      0.315181
       4           0.4      0.419746
       5           0.5      0.523887
       6           0.6      0.627501
       7           0.7      0.730483
       8           0.8      0.832732
       9           0.9      0.934149
      10             1       1.03464

Computed solution:

Node    X(I)        U(X(I))

       0             0             0
       1           0.1      0.100039
       2           0.2      0.200077
       3           0.3      0.300111
       4           0.4       0.40014
       5           0.5      0.500163
       6           0.6      0.600179
       7           0.7      0.700189
       8           0.8      0.800194
       9           0.9      0.900195
      10             1        1.0002

Computed solution:

Node    X(I)        U(X(I))

       0             0             0
       1           0.1           0.1
       2           0.2           0.2
       3           0.3           0.3
       4           0.4           0.4
       5           0.5           0.5
       6           0.6           0.6
       7           0.7           0.7
       8           0.8           0.8
       9           0.9           0.9
      10             1             1

Computed solution:

Node    X(I)        U(X(I))

       0             0             0
       1           0.1           0.1
       2           0.2           0.2
       3           0.3           0.3
       4           0.4           0.4
       5           0.5           0.5
       6           0.6           0.6
       7           0.7           0.7
       8           0.8           0.8
       9           0.9           0.9
      10             1             1

Computed solution:

Node    X(I)        U(X(I))

       0             0             0
       1           0.1           0.1
       2           0.2           0.2
       3           0.3           0.3
       4           0.4           0.4
       5           0.5           0.5
       6           0.6           0.6
       7           0.7           0.7
       8           0.8           0.8
       9           0.9           0.9
      10             1             1

Computed solution:

Node    X(I)        U(X(I))

       0             0             0
       1           0.1           0.1
       2           0.2           0.2
       3           0.3           0.3
       4           0.4           0.4
       5           0.5           0.5
       6           0.6           0.6
       7           0.7           0.7
       8           0.8           0.8
       9           0.9           0.9
      10             1             1

Computed solution:

Node    X(I)        U(X(I))

       0             0             0
       1           0.1           0.1
       2           0.2           0.2
       3           0.3           0.3
       4           0.4           0.4
       5           0.5           0.5
       6           0.6           0.6
       7           0.7           0.7
       8           0.8           0.8
       9           0.9           0.9
      10             1             1

Computed solution:

Node    X(I)        U(X(I))

       0             0             0
       1           0.1           0.1
       2           0.2           0.2
       3           0.3           0.3
       4           0.4           0.4
       5           0.5           0.5
       6           0.6           0.6
       7           0.7           0.7
       8           0.8           0.8
       9           0.9           0.9
      10             1             1

Compare computed and exact solutions:

      X      Computed U      Exact U

             0             0             0
         0.125         0.125         0.125
          0.25          0.25          0.25
         0.375         0.375         0.375
           0.5           0.5           0.5
         0.625         0.625         0.625
          0.75          0.75          0.75
         0.875         0.875         0.875
             1             1             1

FEM1D_NONLINEAR:
  Normal end of execution.

29 April 2007 06:43:45 AM