17 September 2021 9:31:12.547 AM FEM1D_NONLINEAR FORTRAN90 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 = 0.00000 and less than XR = 1.00000 The boundary conditions are: At X = XL, U= 0.00000 At X = XR, U'= 1.00000 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 0.00000 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 1.00000 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 0.500000E-01 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 20.0000 -10.0000 0.100000E-01 2 -10.0000 20.0000 -10.0000 0.200000E-01 3 -10.0000 20.0000 -10.0000 0.300000E-01 4 -10.0000 20.0000 -10.0000 0.400000E-01 5 -10.0000 20.0000 -10.0000 0.500000E-01 6 -10.0000 20.0000 -10.0000 0.600000E-01 7 -10.0000 20.0000 -10.0000 0.700000E-01 8 -10.0000 20.0000 -10.0000 0.800000E-01 9 -10.0000 20.0000 -10.0000 0.900000E-01 10 -10.0000 10.0000 1.04750 Computed solution: Node X(I) U(X(I)) 0 0.00000 0.00000 1 0.100000 0.149750 2 0.200000 0.298500 3 0.300000 0.445250 4 0.400000 0.589000 5 0.500000 0.728750 6 0.600000 0.863500 7 0.700000 0.992250 8 0.800000 1.11400 9 0.900000 1.22775 10 1.00000 1.33250 Computed solution: Node X(I) U(X(I)) 0 0.00000 0.00000 1 0.100000 0.849454E-01 2 0.200000 0.170164 3 0.300000 0.255932 4 0.400000 0.342534 5 0.500000 0.430266 6 0.600000 0.519437 7 0.700000 0.610377 8 0.800000 0.703436 9 0.900000 0.798989 10 1.00000 0.897438 Computed solution: Node X(I) U(X(I)) 0 0.00000 0.00000 1 0.100000 0.105202 2 0.200000 0.210298 3 0.300000 0.315181 4 0.400000 0.419746 5 0.500000 0.523887 6 0.600000 0.627501 7 0.700000 0.730483 8 0.800000 0.832732 9 0.900000 0.934149 10 1.00000 1.03464 Computed solution: Node X(I) U(X(I)) 0 0.00000 0.00000 1 0.100000 0.100039 2 0.200000 0.200077 3 0.300000 0.300111 4 0.400000 0.400140 5 0.500000 0.500163 6 0.600000 0.600179 7 0.700000 0.700189 8 0.800000 0.800194 9 0.900000 0.900195 10 1.00000 1.00020 Computed solution: Node X(I) U(X(I)) 0 0.00000 0.00000 1 0.100000 0.100000 2 0.200000 0.200000 3 0.300000 0.300000 4 0.400000 0.400000 5 0.500000 0.500000 6 0.600000 0.600000 7 0.700000 0.700000 8 0.800000 0.800000 9 0.900000 0.900000 10 1.00000 1.00000 Computed solution: Node X(I) U(X(I)) 0 0.00000 0.00000 1 0.100000 0.100000 2 0.200000 0.200000 3 0.300000 0.300000 4 0.400000 0.400000 5 0.500000 0.500000 6 0.600000 0.600000 7 0.700000 0.700000 8 0.800000 0.800000 9 0.900000 0.900000 10 1.00000 1.00000 Computed solution: Node X(I) U(X(I)) 0 0.00000 0.00000 1 0.100000 0.100000 2 0.200000 0.200000 3 0.300000 0.300000 4 0.400000 0.400000 5 0.500000 0.500000 6 0.600000 0.600000 7 0.700000 0.700000 8 0.800000 0.800000 9 0.900000 0.900000 10 1.00000 1.00000 Computed solution: Node X(I) U(X(I)) 0 0.00000 0.00000 1 0.100000 0.100000 2 0.200000 0.200000 3 0.300000 0.300000 4 0.400000 0.400000 5 0.500000 0.500000 6 0.600000 0.600000 7 0.700000 0.700000 8 0.800000 0.800000 9 0.900000 0.900000 10 1.00000 1.00000 Computed solution: Node X(I) U(X(I)) 0 0.00000 0.00000 1 0.100000 0.100000 2 0.200000 0.200000 3 0.300000 0.300000 4 0.400000 0.400000 5 0.500000 0.500000 6 0.600000 0.600000 7 0.700000 0.700000 8 0.800000 0.800000 9 0.900000 0.900000 10 1.00000 1.00000 Computed solution: Node X(I) U(X(I)) 0 0.00000 0.00000 1 0.100000 0.100000 2 0.200000 0.200000 3 0.300000 0.300000 4 0.400000 0.400000 5 0.500000 0.500000 6 0.600000 0.600000 7 0.700000 0.700000 8 0.800000 0.800000 9 0.900000 0.900000 10 1.00000 1.00000 Compare computed and exact solutions: X Computed U Exact U 0.00000 0.00000 0.00000 0.125000 0.125000 0.125000 0.250000 0.250000 0.250000 0.375000 0.375000 0.375000 0.500000 0.500000 0.500000 0.625000 0.625000 0.625000 0.750000 0.750000 0.750000 0.875000 0.875000 0.875000 1.00000 1.00000 1.00000 The equation is to be solved for X greater than XL = 0.00000 and less than XR = 1.00000 The boundary conditions are: At X = XL, U= 0.00000 At X = XR, U'= 1.00000 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 0.00000 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 1.00000 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 0.500000E-01 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 20.0000 -10.0000 -0.154454 2 -10.0000 20.0000 -10.0000 -0.147799 3 -10.0000 20.0000 -10.0000 -0.136149 4 -10.0000 20.0000 -10.0000 -0.119284 5 -10.0000 20.0000 -10.0000 -0.972917E-01 6 -10.0000 20.0000 -10.0000 -0.706000E-01 7 -10.0000 20.0000 -10.0000 -0.399792E-01 8 -10.0000 20.0000 -10.0000 -0.651058E-02 9 -10.0000 20.0000 -10.0000 0.284722E-01 10 -10.0000 10.0000 1.02308 Computed solution: Node X(I) U(X(I)) 0 0.00000 0.00000 1 0.100000 0.279485E-01 2 0.200000 0.713424E-01 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 1.00000 0.790942 Computed solution: Node X(I) U(X(I)) 0 0.00000 0.00000 1 0.100000 0.148389E-02 2 0.200000 0.184564E-01 3 0.300000 0.504114E-01 4 0.400000 0.965231E-01 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 1.00000 0.589466 Computed solution: Node X(I) U(X(I)) 0 0.00000 0.00000 1 0.100000 0.978230E-02 2 0.200000 0.350230E-01 3 0.300000 0.751252E-01 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 1.00000 0.649816 Computed solution: Node X(I) U(X(I)) 0 0.00000 0.00000 1 0.100000 0.777244E-02 2 0.200000 0.310140E-01 3 0.300000 0.691526E-01 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 1.00000 0.635418 Computed solution: Node X(I) U(X(I)) 0 0.00000 0.00000 1 0.100000 0.776462E-02 2 0.200000 0.309987E-01 3 0.300000 0.691306E-01 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 1.00000 0.635380 Computed solution: Node X(I) U(X(I)) 0 0.00000 0.00000 1 0.100000 0.776462E-02 2 0.200000 0.309987E-01 3 0.300000 0.691306E-01 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 1.00000 0.635380 Computed solution: Node X(I) U(X(I)) 0 0.00000 0.00000 1 0.100000 0.776462E-02 2 0.200000 0.309987E-01 3 0.300000 0.691306E-01 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 1.00000 0.635380 Computed solution: Node X(I) U(X(I)) 0 0.00000 0.00000 1 0.100000 0.776462E-02 2 0.200000 0.309987E-01 3 0.300000 0.691306E-01 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 1.00000 0.635380 Computed solution: Node X(I) U(X(I)) 0 0.00000 0.00000 1 0.100000 0.776462E-02 2 0.200000 0.309987E-01 3 0.300000 0.691306E-01 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 1.00000 0.635380 Computed solution: Node X(I) U(X(I)) 0 0.00000 0.00000 1 0.100000 0.776462E-02 2 0.200000 0.309987E-01 3 0.300000 0.691306E-01 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 1.00000 0.635380 Compare computed and exact solutions: X Computed U Exact U 0.00000 0.00000 0.00000 0.125000 0.135731E-01 0.122325E-01 0.250000 0.500647E-01 0.484598E-01 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 1.00000 0.635380 0.636620 FEM1D_NONLINEAR: Normal end of execution. 17 September 2021 9:31:12.548 AM