17 September 2021 9:31:11.849 AM FEM1D_BVP_LINEAR_TEST FORTRAN90 version Test the FEM1D_BVP_LINEAR 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 1 0.00 0.00000 0.00000 0.00000 2 0.10 0.147773E-01 0.147663E-01 0.110125E-04 3 0.20 0.287010E-01 0.286795E-01 0.214230E-04 4 0.30 0.409088E-01 0.408782E-01 0.306160E-04 5 0.40 0.505213E-01 0.504834E-01 0.379499E-04 6 0.50 0.566333E-01 0.565906E-01 0.427426E-04 7 0.60 0.583042E-01 0.582599E-01 0.442572E-04 8 0.70 0.545491E-01 0.545074E-01 0.416870E-04 9 0.80 0.443287E-01 0.442945E-01 0.341391E-04 10 0.90 0.265389E-01 0.265183E-01 0.206168E-04 11 1.00 0.00000 0.00000 0.00000 l1 norm of error = 0.258586E-04 L2 norm of error = 0.426196E-03 Seminorm of error = 0.156388E-01 Max norm of error = 0.115940E-02 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 1 0.00 0.00000 0.00000 0.00000 2 0.10 0.994655E-01 0.994654E-01 0.133423E-06 3 0.20 0.195425 0.195424 0.247563E-06 4 0.30 0.283471 0.283470 0.339433E-06 5 0.40 0.358038 0.358038 0.405613E-06 6 0.50 0.412181 0.412180 0.442187E-06 7 0.60 0.437309 0.437309 0.444681E-06 8 0.70 0.422888 0.422888 0.407976E-06 9 0.80 0.356087 0.356087 0.326231E-06 10 0.90 0.221364 0.221364 0.192775E-06 11 1.00 0.00000 0.00000 0.00000 l1 norm of error = 0.267262E-06 L2 norm of error = 0.400665E-02 Seminorm of error = 0.138667 Max norm of error = 0.121390E-01 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 1 0.00 0.00000 0.00000 0.00000 2 0.10 0.995976E-01 0.994654E-01 0.132179E-03 3 0.20 0.195686 0.195424 0.261061E-03 4 0.30 0.283852 0.283470 0.381845E-03 5 0.40 0.358526 0.358038 0.487632E-03 6 0.50 0.412749 0.412180 0.568904E-03 7 0.60 0.437921 0.437309 0.612904E-03 8 0.70 0.423491 0.422888 0.602870E-03 9 0.80 0.356604 0.356087 0.517106E-03 10 0.90 0.221692 0.221364 0.327866E-03 11 1.00 0.00000 0.00000 0.00000 l1 norm of error = 0.353851E-03 L2 norm of error = 0.369835E-02 Seminorm of error = 0.138675 Max norm of error = 0.119751E-01 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 1 0.00 0.00000 0.00000 0.00000 2 0.10 0.995489E-01 0.994654E-01 0.835035E-04 3 0.20 0.195591 0.195424 0.166483E-03 4 0.30 0.283718 0.283470 0.247341E-03 5 0.40 0.358361 0.358038 0.322738E-03 6 0.50 0.412567 0.412180 0.386818E-03 7 0.60 0.437739 0.437309 0.430206E-03 8 0.70 0.423327 0.422888 0.438689E-03 9 0.80 0.356478 0.356087 0.391498E-03 10 0.90 0.221623 0.221364 0.259052E-03 11 1.00 0.00000 0.00000 0.00000 l1 norm of error = 0.247848E-03 L2 norm of error = 0.377892E-02 Seminorm of error = 0.138671 Max norm of error = 0.120095E-01 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 1 0.00 0.00000 0.00000 0.00000 2 0.10 0.998202E-01 0.994654E-01 0.354837E-03 3 0.20 0.196115 0.195424 0.690400E-03 4 0.30 0.284455 0.283470 0.985074E-03 5 0.40 0.359254 0.358038 0.121595E-02 6 0.50 0.413540 0.412180 0.135997E-02 7 0.60 0.438703 0.437309 0.139455E-02 8 0.70 0.424186 0.422888 0.129771E-02 9 0.80 0.357134 0.356087 0.104777E-02 10 0.90 0.221987 0.221364 0.622818E-03 11 1.00 0.00000 0.00000 0.00000 l1 norm of error = 0.815371E-03 L2 norm of error = 0.338872E-02 Seminorm of error = 0.138705 Max norm of error = 0.118277E-01 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 1 0.00 0.00000 0.00000 0.00000 2 0.10 0.999805E-01 0.994654E-01 0.515151E-03 3 0.20 0.196432 0.195424 0.100789E-02 4 0.30 0.284924 0.283470 0.145384E-02 5 0.40 0.359566 0.358038 0.152843E-02 6 0.50 0.413603 0.412180 0.142291E-02 7 0.60 0.438574 0.437309 0.126559E-02 8 0.70 0.423939 0.422888 0.105136E-02 9 0.80 0.356861 0.356087 0.774081E-03 10 0.90 0.221791 0.221364 0.426454E-03 11 1.00 0.00000 0.00000 0.00000 l1 norm of error = 0.858701E-03 L2 norm of error = 0.349352E-02 Seminorm of error = 0.138709 Max norm of error = 0.119258E-01 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 L2 norm and seminorm of error for various N. N L1 error L2 error Seminorm error Maxnorm error 11 0.000004 0.005798 0.201186 0.012153 21 0.000000 0.001453 0.100697 0.003073 41 0.000000 0.000363 0.050361 0.000770 81 0.000000 0.000091 0.025182 0.000193 161 0.000000 0.000023 0.012591 0.000048 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 L2 norm and seminorm of error for various N. N L1 error L2 error Seminorm error Maxnorm error 11 0.010523 0.054894 2.119623 0.272576 21 0.004689 0.015170 1.069906 0.066475 41 0.001210 0.004950 0.685573 0.025421 81 0.000303 0.001267 0.350963 0.007090 161 0.000075 0.000317 0.176055 0.001801 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 1 0.00 0.00000 0.00000 0.00000 2 0.10 0.845329E-01 0.994654E-01 0.149325E-01 3 0.20 0.165559 0.195424 0.298650E-01 4 0.30 0.238673 0.283470 0.447975E-01 5 0.40 0.298308 0.358038 0.597300E-01 6 0.50 0.337518 0.412180 0.746626E-01 7 0.60 0.347713 0.437309 0.895952E-01 8 0.70 0.319447 0.409024 0.895770E-01 9 0.80 0.251919 0.311637 0.597180E-01 10 0.90 0.145437 0.175296 0.298590E-01 11 1.00 0.00000 0.00000 0.00000 l1 norm of error = 0.447942E-01 L2 norm of error = 0.595979E-01 Seminorm of error = 0.240692 Max norm of error = 0.103643 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 1 0.00 0.00000 0.00000 0.00000 2 0.10 0.729598E-01 0.994654E-01 0.265056E-01 3 0.20 0.142413 0.195424 0.530111E-01 4 0.30 0.203954 0.283470 0.795167E-01 5 0.40 0.252016 0.358038 0.106022 6 0.50 0.279652 0.412180 0.132528 7 0.60 0.278275 0.437309 0.159034 8 0.70 0.240438 0.210000 0.304383E-01 9 0.80 0.180292 0.160000 0.202922E-01 10 0.90 0.100146 0.900000E-01 0.101461E-01 11 1.00 0.00000 0.00000 0.00000 l1 norm of error = 0.561358E-01 L2 norm of error = 0.822364E-01 Seminorm of error = 0.233968 Max norm of error = 0.179063 FEM1D_BVP_LINEAR_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 1 0.387837E-01 0.129787 0.578696E-01 1 2 0.104315E-01 0.750012E-01 0.214296E-01 2 4 0.265160E-02 0.387482E-01 0.647518E-02 3 8 0.665607E-03 0.195299E-01 0.177789E-02 4 16 0.166571E-03 0.978440E-02 0.465830E-03 5 32 0.416532E-04 0.489464E-02 0.119228E-03 6 64 0.104140E-04 0.244762E-02 0.301600E-04 log(E1) E1 / E2 L2rate H1rate Maxrate 0 1/ 2 1.89451 0.791158 1.43320 1 2/ 4 1.97601 0.952785 1.72662 2 4/ 8 1.99412 0.988446 1.86475 3 8/ 16 1.99854 0.997126 1.93229 4 16/ 32 1.99963 0.999282 1.96608 5 32/ 64 1.99991 0.999821 1.98302 Created graphics data file "data.txt". Created plot file "l2.png". Created plot file "h1.png". Created plot file "mx.png". FEM1D_BVP_LINEAR_TEST Normal end of execution. 17 September 2021 9:31:11.856 AM