>> fem1d_nonlinear 29-Apr-2007 10:08:16 FEM1D_NONLINEAR MATLAB 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 equation is to be solved for X greater than XL = and less than XR = The boundary conditions are: At X = XL, U = 0.000000 At X = XR, U' = 1.000000 This is test problem #2: P(X) = 1, Q(X) = 0, F(X) = -0.5*pi*cos(0.5*pi*X) + 2*sin(0.5*pi*X)*(1-cos(0.5*pi*X)/pi. Boundary conditions: U(0) = 0, U'(1) = 1. The exact solution is U(X) = 2*(1-cos(pi*x/2))/pi Number of quadrature points per element is 1 Number of iterations is 10 Node Location 0 0.000000 1 0.100000 2 0.200000 3 0.300000 4 0.400000 5 0.500000 6 0.600000 7 0.700000 8 0.800000 9 0.900000 10 1.000000 Subint Length 1 0.100000 2 0.100000 3 0.100000 4 0.100000 5 0.100000 6 0.100000 7 0.100000 8 0.100000 9 0.100000 10 0.100000 Subint Quadrature point 1 0.050000 2 0.150000 3 0.250000 4 0.350000 5 0.450000 6 0.550000 7 0.650000 8 0.750000 9 0.850000 10 0.950000 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 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 20.000000 -10.000000 -0.154454 2 -10.000000 20.000000 -10.000000 -0.147799 3 -10.000000 20.000000 -10.000000 -0.136149 4 -10.000000 20.000000 -10.000000 -0.119284 5 -10.000000 20.000000 -10.000000 -0.097292 6 -10.000000 20.000000 -10.000000 -0.070600 7 -10.000000 20.000000 -10.000000 -0.039979 8 -10.000000 20.000000 -10.000000 -0.006511 9 -10.000000 20.000000 -10.000000 0.028472 10 -10.000000 10.000000 1.023081 Computed solution: Node X(I) U(X(I)) 0 0.000000 0.000000 1 0.100000 0.027948 2 0.200000 0.071342 3 0.300000 0.129516 4 0.400000 0.201305 5 0.500000 0.285022 6 0.600000 0.378468 7 0.700000 0.478975 8 0.800000 0.583479 9 0.900000 0.688634 10 1.000000 0.790942 Computed solution: Node X(I) U(X(I)) 0 0.000000 0.000000 1 0.100000 0.001484 2 0.200000 0.018456 3 0.300000 0.050411 4 0.400000 0.096523 5 0.500000 0.155664 6 0.600000 0.226426 7 0.700000 0.307153 8 0.800000 0.395967 9 0.900000 0.490808 10 1.000000 0.589466 Computed solution: Node X(I) U(X(I)) 0 0.000000 0.000000 1 0.100000 0.009782 2 0.200000 0.035023 3 0.300000 0.075125 4 0.400000 0.129110 5 0.500000 0.195640 6 0.600000 0.273059 7 0.700000 0.359430 8 0.800000 0.452588 9 0.900000 0.550198 10 1.000000 0.649816 Computed solution: Node X(I) U(X(I)) 0 0.000000 0.000000 1 0.100000 0.007772 2 0.200000 0.031014 3 0.300000 0.069153 4 0.400000 0.121249 5 0.500000 0.186017 6 0.600000 0.261860 7 0.700000 0.346904 8 0.800000 0.439050 9 0.900000 0.536021 10 1.000000 0.635418 Computed solution: Node X(I) U(X(I)) 0 0.000000 0.000000 1 0.100000 0.007765 2 0.200000 0.030999 3 0.300000 0.069131 4 0.400000 0.121221 5 0.500000 0.185985 6 0.600000 0.261824 7 0.700000 0.346867 8 0.800000 0.439012 9 0.900000 0.535982 10 1.000000 0.635380 Printout of tridiagonal linear system: Equation ALEFT ADIAG ARITE RHS 1 20.000000 -9.984501 -0.154214 2 -10.003882 20.000000 -9.965435 -0.146619 3 -10.015499 20.000000 -9.939390 -0.132716 4 -10.034565 20.000000 -9.907008 -0.111832 5 -10.060610 20.000000 -9.869088 -0.083827 6 -10.092992 20.000000 -9.826567 -0.049168 7 -10.130912 20.000000 -9.780494 -0.008934 8 -10.173433 20.000000 -9.732009 0.035229 9 -10.219506 20.000000 -9.682310 0.081216 10 -10.267991 10.317690 1.052189 Computed solution: Node X(I) U(X(I)) 0 0.000000 0.000000 1 0.100000 0.007765 2 0.200000 0.030999 3 0.300000 0.069131 4 0.400000 0.121221 5 0.500000 0.185985 6 0.600000 0.261824 7 0.700000 0.346867 8 0.800000 0.439012 9 0.900000 0.535982 10 1.000000 0.635380 Computed solution: Node X(I) U(X(I)) 0 0.000000 0.000000 1 0.100000 0.007765 2 0.200000 0.030999 3 0.300000 0.069131 4 0.400000 0.121221 5 0.500000 0.185985 6 0.600000 0.261824 7 0.700000 0.346867 8 0.800000 0.439012 9 0.900000 0.535982 10 1.000000 0.635380 Computed solution: Node X(I) U(X(I)) 0 0.000000 0.000000 1 0.100000 0.007765 2 0.200000 0.030999 3 0.300000 0.069131 4 0.400000 0.121221 5 0.500000 0.185985 6 0.600000 0.261824 7 0.700000 0.346867 8 0.800000 0.439012 9 0.900000 0.535982 10 1.000000 0.635380 Computed solution: Node X(I) U(X(I)) 0 0.000000 0.000000 1 0.100000 0.007765 2 0.200000 0.030999 3 0.300000 0.069131 4 0.400000 0.121221 5 0.500000 0.185985 6 0.600000 0.261824 7 0.700000 0.346867 8 0.800000 0.439012 9 0.900000 0.535982 10 1.000000 0.635380 Computed solution: Node X(I) U(X(I)) 0 0.000000 0.000000 1 0.100000 0.007765 2 0.200000 0.030999 3 0.300000 0.069131 4 0.400000 0.121221 5 0.500000 0.185985 6 0.600000 0.261824 7 0.700000 0.346867 8 0.800000 0.439012 9 0.900000 0.535982 10 1.000000 0.635380 Compare computed and exact solutions: X U(X) U(exact) 0.000000 0.000000 0.000000 0.125000 0.013573 0.012232 0.250000 0.050065 0.048460 0.375000 0.108198 0.107290 0.500000 0.185985 0.186462 0.625000 0.283085 0.282933 0.750000 0.392940 0.392996 0.875000 0.511740 0.512421 1.000000 0.635380 0.636620 FEM1D_NONLINEAR: Normal end of execution. 29-Apr-2007 10:08:19 >>