Differential Equations Solutions 130

Differential Equations Solutions 130 - 140 Chapter 23....

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Unformatted text preview: 140 Chapter 23. Solutions: Case Study: Finite Diﬀerences and Finite Elements Using coeﬃcient functions a(1) and c(3) with true solution u(1) Inﬁnity norm of the error at the grid points for various methods and numbers of interior grid points M M= 1st order ﬁnite diﬀerence 2nd order ﬁnite diﬀerence Linear ﬁnite elements Quadratic ﬁnite elements 9 1.9405e-03 1.9405e-03 4.3912e-04 2.8745e-05 99 1.9529e-05 1.9529e-05 4.3908e-06 3.5282e-09 999 1.9530e-07 1.9530e-07 4.3906e-08 3.6134e-13 PROBLEM 4 Using coeﬃcient functions a(2) and c(1) with true solution u(1) Inﬁnity norm of the error at the grid points for various methods and numbers of interior grid points M M= 1st order ﬁnite diﬀerence 2nd order ﬁnite diﬀerence Linear ﬁnite elements Quadratic ﬁnite elements 9 1.5788e-02 3.8465e-03 1.3904e-03 1.6287e-04 99 1.8705e-03 3.8751e-05 1.3930e-05 1.9539e-08 999 1.8979e-04 3.8752e-07 1.3930e-07 1.9897e-12 PROBLEM 5 Using coeﬃcient functions a(3) and c(1) with true solution u(1) Inﬁnity norm of the error at the grid points for various methods and numbers of interior grid points M M= 1st order ﬁnite diﬀerence 2nd order ﬁnite diﬀerence Linear ﬁnite elements Quadratic ﬁnite elements 9 1.1858e-02 3.6018e-03 8.3148e-04 1.0981e-04 99 1.4780e-03 3.6454e-05 8.2486e-06 1.6801e-06 999 1.5065e-04 3.6467e-07 1.2200e-06 2.5858e-06 PROBLEM 6 Using coeﬃcient functions a(1) and c(1) with true solution u(2) Inﬁnity norm of the error at the grid points for various methods and numbers of interior grid points M M= 1st order ﬁnite diﬀerence 2nd order ﬁnite diﬀerence Linear ﬁnite elements Quadratic ﬁnite elements 9 8.9200e-02 8.9200e-02 8.6564e-02 8.6570e-02 99 9.5538e-02 9.5538e-02 9.5219e-02 9.5224e-02 999 9.6120e-02 9.6120e-02 9.6086e-02 9.6088e-02 PROBLEM 7 Using coeﬃcient functions a(1) and c(1) with true solution u(3) ...
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