fem_check4th_la.c running 
Given x^3 u'''' + x^2 u''' + x u'' + u' + u = 
x^4 + 64 x^3 - x^2 - 4 x + 1 
0<x<1  Boundary u(0) = 1, u(1) = 1 
Analytic solution u(x) = x^4 - x^2 + 1 
Try 10 points 
x grid and analytic solution 
i=0, Ua( 0.000)= 1.000 
i=1, Ua( 0.111)= 0.988 
i=2, Ua( 0.222)= 0.953 
i=3, Ua( 0.333)= 0.901 
i=4, Ua( 0.444)= 0.841 
i=5, Ua( 0.556)= 0.787 
i=6, Ua( 0.667)= 0.753 
i=7, Ua( 0.778)= 0.761 
i=8, Ua( 0.889)= 0.834 
i=9, Ua( 1.000)= 1.000 

calling gauleg a=0, b=1, np=48 
compute stiffness matrix 
Legendre integration=125.116, at i=1, j=0 
adaptive integration=125.116, at i=1, j=0 
Legendre integration=-1245.05, at i=1, j=1 
adaptive integration=-1245.05, at i=1, j=1 
Legendre integration=5557.96, at i=1, j=2 
adaptive integration=5557.96, at i=1, j=2 
Legendre integration=-14699.2, at i=1, j=3 
adaptive integration=-14699.2, at i=1, j=3 
Legendre integration=25461.4, at i=1, j=4 
adaptive integration=25461.4, at i=1, j=4 
Legendre integration=-30062.3, at i=1, j=5 
adaptive integration=-30062.3, at i=1, j=5 
Legendre integration=24283.6, at i=1, j=6 
adaptive integration=24283.6, at i=1, j=6 
Legendre integration=-12970.2, at i=1, j=7 
adaptive integration=-12970.2, at i=1, j=7 
Legendre integration=4155.38, at i=1, j=8 
adaptive integration=4155.38, at i=1, j=8 
Legendre integration=-606.568, at i=1, j=9 
adaptive integration=-606.568, at i=1, j=9 
Legendre integration=-0.288071, f at i=1 
adaptive integration=-0.28807, f at i=1 
Legendre integration=-560.845, at i=2, j=0 
adaptive integration=-560.845, at i=2, j=0 
Legendre integration=5635.93, at i=2, j=1 
adaptive integration=5635.93, at i=2, j=1 
Legendre integration=-25345.3, at i=2, j=2 
adaptive integration=-25345.3, at i=2, j=2 
Legendre integration=67298.1, at i=2, j=3 
adaptive integration=67298.1, at i=2, j=3 
Legendre integration=-116796, at i=2, j=4 
adaptive integration=-116796, at i=2, j=4 
Legendre integration=138032, at i=2, j=5 
adaptive integration=138032, at i=2, j=5 
Legendre integration=-111554, at i=2, j=6 
adaptive integration=-111554, at i=2, j=6 
Legendre integration=59600.7, at i=2, j=7 
adaptive integration=59600.7, at i=2, j=7 
Legendre integration=-19098.5, at i=2, j=8 
adaptive integration=-19098.5, at i=2, j=8 
Legendre integration=2788.15, at i=2, j=9 
adaptive integration=2788.15, at i=2, j=9 
Legendre integration=1.91548, f at i=2 
adaptive integration=1.91548, f at i=2 
Legendre integration=1547.03, at i=3, j=0 
adaptive integration=1547.03, at i=3, j=0 
Legendre integration=-15441.6, at i=3, j=1 
adaptive integration=-15441.6, at i=3, j=1 
Legendre integration=69323.8, at i=3, j=2 
adaptive integration=69323.8, at i=3, j=2 
Legendre integration=-184135, at i=3, j=3 
adaptive integration=-184135, at i=3, j=3 
Legendre integration=319847, at i=3, j=4 
adaptive integration=319847, at i=3, j=4 
Legendre integration=-378286, at i=3, j=5 
adaptive integration=-378286, at i=3, j=5 
Legendre integration=305861, at i=3, j=6 
adaptive integration=305861, at i=3, j=6 
Legendre integration=-163451, at i=3, j=7 
adaptive integration=-163451, at i=3, j=7 
Legendre integration=52382.4, at i=3, j=8 
adaptive integration=52382.4, at i=3, j=8 
Legendre integration=-7647.64, at i=3, j=9 
adaptive integration=-7647.64, at i=3, j=9 
Legendre integration=-4.95146, f at i=3 
adaptive integration=-4.95146, f at i=3 
Legendre integration=-2820.37, at i=4, j=0 
adaptive integration=-2820.37, at i=4, j=0 
Legendre integration=28154.4, at i=4, j=1 
adaptive integration=28154.4, at i=4, j=1 
Legendre integration=-126249, at i=4, j=2 
adaptive integration=-126249, at i=4, j=2 
Legendre integration=334967, at i=4, j=3 
adaptive integration=334967, at i=4, j=3 
Legendre integration=-581557, at i=4, j=4 
adaptive integration=-581557, at i=4, j=4 
Legendre integration=687866, at i=4, j=5 
adaptive integration=687866, at i=4, j=5 
Legendre integration=-556356, at i=4, j=6 
adaptive integration=-556356, at i=4, j=6 
Legendre integration=297395, at i=4, j=7 
adaptive integration=297395, at i=4, j=7 
Legendre integration=-95316.5, at i=4, j=8 
adaptive integration=-95316.5, at i=4, j=8 
Legendre integration=13915.9, at i=4, j=9 
adaptive integration=13915.9, at i=4, j=9 
Legendre integration=10.038, f at i=4 
adaptive integration=10.038, f at i=4 
Legendre integration=3588.84, at i=5, j=0 
adaptive integration=3588.84, at i=5, j=0 
Legendre integration=-35823.2, at i=5, j=1 
adaptive integration=-35823.2, at i=5, j=1 
Legendre integration=160683, at i=5, j=2 
adaptive integration=160683, at i=5, j=2 
Legendre integration=-426203, at i=5, j=3 
adaptive integration=-426203, at i=5, j=3 
Legendre integration=739386, at i=5, j=4 
adaptive integration=739386, at i=5, j=4 
Legendre integration=-873874, at i=5, j=5 
adaptive integration=-873874, at i=5, j=5 
Legendre integration=706549, at i=5, j=6 
adaptive integration=706549, at i=5, j=6 
Legendre integration=-377734, at i=5, j=7 
adaptive integration=-377734, at i=5, j=7 
Legendre integration=121110, at i=5, j=8 
adaptive integration=121110, at i=5, j=8 
Legendre integration=-17682.7, at i=5, j=9 
adaptive integration=-17682.7, at i=5, j=9 
Legendre integration=-11.1016, f at i=5 
adaptive integration=-11.1016, f at i=5 
Legendre integration=-3291.06, at i=6, j=0 
adaptive integration=-3291.06, at i=6, j=0 
Legendre integration=32850.2, at i=6, j=1 
adaptive integration=32850.2, at i=6, j=1 
Legendre integration=-147372, at i=6, j=2 
adaptive integration=-147372, at i=6, j=2 
Legendre integration=391036, at i=6, j=3 
adaptive integration=391036, at i=6, j=3 
Legendre integration=-678350, at i=6, j=4 
adaptive integration=-678350, at i=6, j=4 
Legendre integration=800983, at i=6, j=5 
adaptive integration=800983, at i=6, j=5 
Legendre integration=-646453, at i=6, j=6 
adaptive integration=-646453, at i=6, j=6 
Legendre integration=344930, at i=6, j=7 
adaptive integration=344930, at i=6, j=7 
Legendre integration=-110479, at i=6, j=8 
adaptive integration=-110479, at i=6, j=8 
Legendre integration=16146, at i=6, j=9 
adaptive integration=16146, at i=6, j=9 
Legendre integration=13.0132, f at i=6 
adaptive integration=13.0132, f at i=6 
Legendre integration=2281.68, at i=7, j=0 
adaptive integration=2281.68, at i=7, j=0 
Legendre integration=-22782.9, at i=7, j=1 
adaptive integration=-22782.9, at i=7, j=1 
Legendre integration=102267, at i=7, j=2 
adaptive integration=102267, at i=7, j=2 
Legendre integration=-271579, at i=7, j=3 
adaptive integration=-271579, at i=7, j=3 
Legendre integration=471744, at i=7, j=4 
adaptive integration=471744, at i=7, j=4 
Legendre integration=-557696, at i=7, j=5 
adaptive integration=-557696, at i=7, j=5 
Legendre integration=449819, at i=7, j=6 
adaptive integration=449819, at i=7, j=6 
Legendre integration=-238861, at i=7, j=7 
adaptive integration=-238861, at i=7, j=7 
Legendre integration=75667.1, at i=7, j=8 
adaptive integration=75667.1, at i=7, j=8 
Legendre integration=-10858.8, at i=7, j=9 
adaptive integration=-10858.8, at i=7, j=9 
Legendre integration=-4.40335, f at i=7 
adaptive integration=-4.40335, f at i=7 
Legendre integration=-1553.47, at i=8, j=0 
adaptive integration=-1553.47, at i=8, j=0 
Legendre integration=15554.4, at i=8, j=1 
adaptive integration=15554.4, at i=8, j=1 
Legendre integration=-70079, at i=8, j=2 
adaptive integration=-70079, at i=8, j=2 
Legendre integration=187053, at i=8, j=3 
adaptive integration=187053, at i=8, j=3 
Legendre integration=-327355, at i=8, j=4 
adaptive integration=-327355, at i=8, j=4 
Legendre integration=391745, at i=8, j=5 
adaptive integration=391745, at i=8, j=5 
Legendre integration=-322383, at i=8, j=6 
adaptive integration=-322383, at i=8, j=6 
Legendre integration=176521, at i=8, j=7 
adaptive integration=176521, at i=8, j=7 
Legendre integration=-58312.9, at i=8, j=8 
adaptive integration=-58312.9, at i=8, j=8 
Legendre integration=8811.06, at i=8, j=9 
adaptive integration=8811.06, at i=8, j=9 
Legendre integration=8.80945, f at i=8 
adaptive integration=8.80945, f at i=8 
k computed stiffness matrix 
 1.000  0.000  0.000  0.000  0.000  0.000  0.000  0.000  0.000  0.000 
125.116 -1245.055 5557.964 -14699.180 25461.434 -30062.332 24283.616 -12970.202 4155.383 -606.568 
-560.845 5635.933 -25345.293 67298.073 -116795.964 138031.980 -111554.260 59600.704 -19098.470 2788.155 
1547.027 -15441.583 69323.770 -184135.293 319846.619 -378285.595 305861.078 -163450.535 52382.371 -7647.643 
-2820.368 28154.373 -126248.880 334966.748 -581556.610 687866.284 -556355.732 297394.834 -95316.472 13915.887 
3588.836 -35823.198 160682.787 -426202.728 739386.116 -873873.783 706548.897 -377733.700 121109.511 -17682.675 
-3291.061 32850.216 -147372.214 391036.462 -678350.219 800982.733 -646452.770 344929.606 -110478.583 16146.047 
2281.683 -22782.904 102267.225 -271579.443 471743.556 -557696.212 449818.973 -238861.176 75667.061 -10858.750 
-1553.468 15554.352 -70079.036 187052.761 -327355.452 391744.677 -322383.319 176521.469 -58312.872 8811.064 
 0.000  0.000  0.000  0.000  0.000  0.000  0.000  0.000  0.000  1.000 
f computed forcing function 
 1.000 -0.288  1.915 -4.951 10.038 -11.102 13.013 -4.403  8.809  1.000 
u computed Galerkin, Ua analytic, error 
u[0]= 1.000, Ua= 1.000, err=-5.55112e-16 
u[1]= 0.988, Ua= 0.988, err=3.61802e-11 
u[2]= 0.953, Ua= 0.953, err=-1.09901e-11 
u[3]= 0.901, Ua= 0.901, err=-1.18383e-10 
u[4]= 0.841, Ua= 0.841, err=-2.45258e-10 
u[5]= 0.787, Ua= 0.787, err=-3.51625e-10 
u[6]= 0.753, Ua= 0.753, err=-4.03897e-10 
u[7]= 0.761, Ua= 0.761, err=-3.75513e-10 
u[8]= 0.834, Ua= 0.834, err=-2.46054e-10 
u[9]= 1.000, Ua= 1.000, err=0 
                                        maxerr=4.03897e-10, avgerr=1.7879e-10 


kd computed adaptive stiffness matrix 
 1.000  0.000  0.000  0.000  0.000  0.000  0.000  0.000  0.000  0.000 
125.116 -1245.055 5557.964 -14699.180 25461.434 -30062.332 24283.616 -12970.202 4155.383 -606.568 
-560.845 5635.933 -25345.293 67298.073 -116795.964 138031.980 -111554.260 59600.704 -19098.470 2788.155 
1547.027 -15441.583 69323.770 -184135.293 319846.619 -378285.595 305861.078 -163450.535 52382.371 -7647.643 
-2820.368 28154.373 -126248.880 334966.748 -581556.610 687866.284 -556355.732 297394.834 -95316.472 13915.887 
3588.836 -35823.198 160682.787 -426202.728 739386.116 -873873.783 706548.897 -377733.700 121109.511 -17682.675 
-3291.061 32850.216 -147372.214 391036.462 -678350.219 800982.733 -646452.770 344929.606 -110478.583 16146.047 
2281.683 -22782.904 102267.225 -271579.443 471743.556 -557696.212 449818.973 -238861.176 75667.061 -10858.750 
-1553.468 15554.352 -70079.036 187052.761 -327355.452 391744.677 -322383.319 176521.469 -58312.872 8811.064 
 0.000  0.000  0.000  0.000  0.000  0.000  0.000  0.000  0.000  1.000 
fd computed forcing function 
 1.000 -0.288  1.915 -4.951 10.038 -11.102 13.013 -4.403  8.809  1.000 
ud computed Galerkin, Ua analytic, error 
ud[0]= 1.000, Ua= 1.000, err=2.73115e-14 
ud[1]= 0.988, Ua= 0.988, err=4.2918e-08 
ud[2]= 0.953, Ua= 0.953, err=1.2536e-06 
ud[3]= 0.901, Ua= 0.901, err=3.09843e-06 
ud[4]= 0.841, Ua= 0.841, err=4.97347e-06 
ud[5]= 0.787, Ua= 0.787, err=6.35529e-06 
ud[6]= 0.753, Ua= 0.753, err=6.83005e-06 
ud[7]= 0.761, Ua= 0.761, err=6.07976e-06 
ud[8]= 0.834, Ua= 0.834, err=3.86296e-06 
ud[9]= 1.000, Ua= 1.000, err=0 
                                        maxerr=6.83005e-06, avgerr=3.24965e-06