C 代码 应用有限元法， 具有分段二次元， 到一个空间维度上的两点边界值问题 （BVP）.rar

03 March 2022 09:57:40 PM FEM1D_BVP_QUADRATIC_TEST 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