15 January 2017 08:42:36 PM

FEM1D_BVP_QUADRATIC_PRB
  C version
  Test the FEM1D_BVP_QUADRATIC library.

TEST00
  Solve -( A(x) U'(x) )' + C(x) U(x) = F(x)
  for 0 < x < 1, with U(0) = U(1) = 0.
  A(X)  = 1.0
  C(X)  = 1.0
  F(X)  = X
  U(X)  = X - SINH(X) / SINH(1)

  Number of nodes = 11

     I    X         U         Uexact    Error

     0  0.000000        0.000000        0.000000   2.914335e-16
     1  0.100000        0.014766        0.014766   4.253521e-08
     2  0.200000        0.028679        0.028680   5.717636e-08
     3  0.300000        0.040878        0.040878   1.369556e-07
     4  0.400000        0.050483        0.050483   1.012851e-07
     5  0.500000        0.056591        0.056591   2.601080e-07
     6  0.600000        0.058260        0.058260   1.181175e-07
     7  0.700000        0.054508        0.054507   4.334600e-07
     8  0.800000        0.044294        0.044295   9.111253e-08
     9  0.900000        0.026519        0.026518   6.820897e-07
    10  1.000000        0.000000        0.000000   0.000000e+00

  l1 norm of error  = 1.74804e-07
  L2 norm of error  = 3.87933e-05
  Seminorm of error = 0.001502
  Max norm of error = 7.64223e-05

TEST01
  Solve -( A(x) U'(x) )' + C(x) U(x) = F(x)
  for 0 < x < 1, with U(0) = U(1) = 0.
  A1(X)  = 1.0
  C1(X)  = 0.0
  F1(X)  = X * ( X + 3 ) * exp ( X )
  U1(X)  = X * ( 1 - X ) * exp ( X )

  Number of nodes = 11

     I    X         U         Uexact    Error

     0  0.000000       -0.000000        0.000000   5.551115e-16
     1  0.100000        0.099473        0.099465   8.053079e-06
     2  0.200000        0.195424        0.195424   1.063711e-09
     3  0.300000        0.283482        0.283470   1.150505e-05
     4  0.400000        0.358038        0.358038   1.730682e-09
     5  0.500000        0.412197        0.412180   1.620006e-05
     6  0.600000        0.437309        0.437309   1.883466e-09
     7  0.700000        0.422911        0.422888   2.254381e-05
     8  0.800000        0.356087        0.356087   1.371135e-09
     9  0.900000        0.221395        0.221364   3.106609e-05
    10  1.000000        0.000000        0.000000   0.000000e+00

  l1 norm of error  = 8.12492e-06
  L2 norm of error  = 0.000475788
  Seminorm of error = 0.0183976
  Max norm of error = 0.00128552

TEST02
  Solve -( A(x) U'(x) )' + C(x) U(x) = F(x)
  for 0 < x < 1, with U(0) = U(1) = 0.
  A2(X)  = 1.0
  C2(X)  = 2.0
  F2(X)  = X * ( 5 - X ) * exp ( X )
  U2(X)  = X * ( 1 - X ) * exp ( X )

  Number of nodes = 11

     I    X         U         Uexact    Error

     0  0.000000        0.000000        0.000000   1.249001e-15
     1  0.100000        0.099471        0.099465   5.501657e-06
     2  0.200000        0.195419        0.195424   5.088203e-06
     3  0.300000        0.283475        0.283470   4.733163e-06
     4  0.400000        0.358029        0.358038   8.496040e-06
     5  0.500000        0.412187        0.412180   7.162985e-06
     6  0.600000        0.437299        0.437309   9.625453e-06
     7  0.700000        0.422902        0.422888   1.403186e-05
     8  0.800000        0.356079        0.356087   7.384697e-06
     9  0.900000        0.221392        0.221364   2.728618e-05
    10  1.000000        0.000000        0.000000   0.000000e+00

  l1 norm of error  = 8.11911e-06
  L2 norm of error  = 0.000475222
  Seminorm of error = 0.0183976
  Max norm of error = 0.00128743

TEST03
  Solve -( A(x) U'(x) )' + C(x) U(x) = F(x)
  for 0 < x < 1, with U(0) = U(1) = 0.
  A3(X)  = 1.0
  C3(X)  = 2.0 * X
  F3(X)  = - X * ( 2 * X * X - 3 * X - 3 ) * exp ( X )
  U3(X)  = X * ( 1 - X ) * exp ( X )

  Number of nodes = 11

     I    X         U         Uexact    Error

     0  0.000000        0.000000        0.000000   1.387779e-16
     1  0.100000        0.099472        0.099465   6.783610e-06
     2  0.200000        0.195422        0.195424   2.638322e-06
     3  0.300000        0.283478        0.283470   7.811195e-06
     4  0.400000        0.358033        0.358038   4.907359e-06
     5  0.500000        0.412191        0.412180   1.078859e-05
     6  0.600000        0.437302        0.437309   6.155136e-06
     7  0.700000        0.422905        0.422888   1.702168e-05
     8  0.800000        0.356081        0.356087   5.213245e-06
     9  0.900000        0.221393        0.221364   2.864151e-05
    10  1.000000        0.000000        0.000000   0.000000e+00

  l1 norm of error  = 8.17824e-06
  L2 norm of error  = 0.000475415
  Seminorm of error = 0.0183976
  Max norm of error = 0.00128668

TEST04
  Solve -( A(x) U'(x) )' + C(x) U(x) = F(x)
  for 0 < x < 1, with U(0) = U(1) = 0.
  A4(X)  = 1.0 + X * X
  C4(X)  = 0.0
  F4(X)  = ( X + 3 X^2 + 5 X^3 + X^4 ) * exp ( X )
  U4(X)  = X * ( 1 - X ) * exp ( X )

  Number of nodes = 11

     I    X         U         Uexact    Error

     0  0.000000        0.000000        0.000000   1.637579e-15
     1  0.100000        0.099477        0.099465   1.137923e-05
     2  0.200000        0.195421        0.195424   3.926512e-06
     3  0.300000        0.283499        0.283470   2.850301e-05
     4  0.400000        0.358030        0.358038   7.912516e-06
     5  0.500000        0.412238        0.412180   5.815353e-05
     6  0.600000        0.437299        0.437309   9.790475e-06
     7  0.700000        0.422990        0.422888   1.024294e-04
     8  0.800000        0.356079        0.356087   7.582612e-06
     9  0.900000        0.221528        0.221364   1.634191e-04
    10  1.000000        0.000000        0.000000   0.000000e+00

  l1 norm of error  = 3.5736e-05
  L2 norm of error  = 0.00047883
  Seminorm of error = 0.018419
  Max norm of error = 0.00137041

TEST05
  Solve -( A(x) U'(x) )' + C(x) U(x) = F(x)
  for 0 < x < 1, with U(0) = U(1) = 0.
  A5(X)  = 1.0 + X * X for X <= 1/3
         = 7/9 + X     for      1/3 < X
  C5(X)  = 0.0
  F5(X)  = ( X + 3 X^2 + 5 X^3 + X^4 ) * exp ( X )
                       for X <= 1/3
         = ( - 1 + 10/3 X + 43/9 X^2 + X^3 ) .* exp ( X )
                       for      1/3 <= X
  U5(X)  = X * ( 1 - X ) * exp ( X )

  Number of nodes = 11

     I    X         U         Uexact    Error

     0  0.000000        0.000000        0.000000   1.942890e-16
     1  0.100000        0.099690        0.099465   2.241951e-04
     2  0.200000        0.195842        0.195424   4.175568e-04
     3  0.300000        0.284132        0.283470   6.611607e-04
     4  0.400000        0.358565        0.358038   5.268467e-04
     5  0.500000        0.412668        0.412180   4.876947e-04
     6  0.600000        0.437633        0.437309   3.247078e-04
     7  0.700000        0.423209        0.422888   3.213542e-04
     8  0.800000        0.356238        0.356087   1.512860e-04
     9  0.900000        0.221550        0.221364   1.859622e-04
    10  1.000000        0.000000        0.000000   0.000000e+00

  l1 norm of error  = 0.000300069
  L2 norm of error  = 0.000628343
  Seminorm of error = 0.0184672
  Max norm of error = 0.0014469

TEST06
  Solve -( A(x) U'(x) )' + C(x) U(x) = F(x)
  for 0 < x < 1, with U(0) = U(1) = 0.
  A6(X)  = 1.0
  C6(X)  = 0.0
  F6(X)  = pi*pi*sin(pi*X)
  U6(X)  = sin(pi*x)

  Compute l1norm, L2norm and seminorm of error for various N.

     N        l1 error       L2 error      Seminorm error  Maxnorm error

    11      2.3654e-05     0.000838808       0.0325225      0.00183654
    21     1.54072e-06     0.000105326       0.0081608     0.000239035
    41     9.85135e-08     1.31807e-05      0.00204209     3.01793e-05
    81     6.23112e-09     1.64806e-06      0.00051064     3.78181e-06
   161     3.91905e-10     2.06021e-07     0.000127667      4.7302e-07

TEST07
  Solve -( A(x) U'(x) )' + C(x) U(x) = F(x)
  for 0 < x < 1, with U(0) = U(1) = 0.
  Becker/Carey/Oden Example

  Compute l1 norm, L2 norm and seminorm of error for various N.

     N        l1 error      L2 error      Seminorm error  Maxnorm error

    11       0.0236359       0.0698852         1.72248        0.278261
    21      0.00526296       0.0175705        0.975957       0.0869379
    41     0.000771555      0.00366719        0.502186       0.0260734
    81     5.37619e-05     0.000407677        0.119887      0.00399866
   161     3.21083e-06     4.76744e-05       0.0291324     0.000543592

TEST08
  Solve -( A(x) U'(x) )' + C(x) U(x) = F(x)
  for 0 < x < 1, with U(0) = U(1) = 0.
  A8(X) = 1.0
  C8(X) = 0.0
  F8(X) = X * ( X + 3 ) * exp ( X ),   X <= 2/3
        = 2 * exp ( 2/3),                   2/3 < X
  U8(X) = X * ( 1 - X ) * exp ( X ),   X <= 2/3
        = X * ( 1 - X ) * exp ( 2/3 ),      2/3 < X

  Number of nodes = 11

     I    X         U         Uexact    Error

     0  0.000000        0.000000        0.000000   3.053113e-16
     1  0.100000        0.084636        0.099465   1.482981e-02
     2  0.200000        0.165749        0.195424   2.967573e-02
     3  0.300000        0.238968        0.283470   4.450209e-02
     4  0.400000        0.298686        0.358038   5.935145e-02
     5  0.500000        0.338007        0.412180   7.417312e-02
     6  0.600000        0.348281        0.437309   8.902718e-02
     7  0.700000        0.319995        0.409024   8.902872e-02
     8  0.800000        0.252243        0.311637   5.939495e-02
     9  0.900000        0.145599        0.175296   2.969747e-02
    10  1.000000        0.000000        0.000000   0.000000e+00

  l1 norm of error  = 0.0445164
  L2 norm of error  = 0.0569727
  Seminorm of error = 0.212209
  Max norm of error = 0.0967138

TEST09
  Solve -( A(x) U'(x) )' + C(x) U(x) = F(x)
  for 0 < x < 1, with U(0) = U(1) = 0.
  A9(X) = 1.0
  C9(X) = 0.0
  F9(X) = X * ( X + 3 ) * exp ( X ),   X <= 2/3
        = 2 * exp ( 2/3),                   2/3 < X
  U9(X) = X * ( 1 - X ) * exp ( X ),   X <= 2/3
        = X * ( 1 - X ),                    2/3 < X

  Number of nodes = 11

     I    X         U         Uexact    Error

     0  0.000000       -0.000000        0.000000   5.828671e-16
     1  0.100000        0.073447        0.099465   2.601876e-02
     2  0.200000        0.143371        0.195424   5.205363e-02
     3  0.300000        0.205401        0.283470   7.806894e-02
     4  0.400000        0.253931        0.358038   1.041073e-01
     5  0.500000        0.282062        0.412180   1.301179e-01
     6  0.600000        0.281148        0.437309   1.561609e-01
     7  0.700000        0.243386        0.210000   3.338608e-02
     8  0.800000        0.181953        0.160000   2.195308e-02
     9  0.900000        0.100977        0.090000   1.097654e-02
    10  1.000000        0.000000        0.000000   0.000000e+00

  l1 norm of error  = 0.055713
  L2 norm of error  = 0.080793
  Seminorm of error = 0.222691
  Max norm of error = 0.17056

TEST10
  Solve -( A(x) U'(x) )' + C(x) U(x) = F(x)
  for 0 < x < 1, with U(0) = U(1) = 0.
  A(X)  = 1.0
  C(X)  = 1.0
  F(X)  = X
  U(X)  = X - SINH(X) / SINH(1)

 log(E)    E         L2error         H1error        Maxerror

     0     2      0.00471405       0.0362083      0.00766215
     1     4     0.000602037      0.00930851      0.00108553
     2     8     7.57116e-05      0.00234472     0.000146767
     3    16     9.47875e-06     0.000587307     1.91493e-05
     4    32     1.18531e-06     0.000146898     2.44769e-06
     5    64     1.48178e-07     3.67288e-05     3.09464e-07
     6   128     1.85228e-08     9.18249e-06     3.89059e-08

 log(E1)  E1 / E2          L2rate          H1rate         Maxrate

     0     2 /    4         2.96904          1.9597         2.81935
     1     4 /    8         2.99127         1.98913          2.8868
     2     8 /   16         2.99775         1.99723         2.93816
     3    16 /   32         2.99943          1.9993          2.9678
     4    32 /   64         2.99986         1.99983         2.98358
     5    64 /  128         2.99996         1.99996         2.99171

  Created graphics data file "data.txt".
  Created graphics command file "commands_l2.txt".
  Created graphics command file "commands_h1.txt".
  Created graphics command file "commands_mx.txt".

FEM1D_BVP_QUADRATIC_PRB
  Normal end of execution.

15 January 2017 08:42:36 PM