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me180_hw01

# me180_hw01 - HOMEWORK 1 THE BASICS Solve the followng...

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HOMEWORK 1: THE BASICS Solve the followng boundary value problem, with domain Ω = (0 ,L ), analytically: d dx ± A 1 du dx ² = k 2 sin ( 2 πkx L ) A 1 = given constant = 0 . 1 k = given constant L = 1 u (0) = Δ 1 = given constant = 0 u ( L ) = Δ 2 = given constant = 1 (1) Now solve this with the ﬁnite element method using linear equal-sized elements. In order to achieve e N def = || u - u N || A 1 (Ω) || u || A 1 (Ω) TOL = 0 . 05 , || u || A 1 (Ω) def = s Z Ω du dx A 1 du dx dx (2) how many ﬁnite elements ( N ) are needed for k = 1 N =? k = 2 N =? k = 4 N =? k = 8 N =? k = 16 N =? k = 32 N =? (3) You should set up a general matrix equation and solve it using Gaus- sian elimination. Later we will use other types of more eﬃcient solvers. 1

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Plot the numerical solutions for N = 2 , 4 , 8 , 16 ,... , for each k , along with the exact solution. Also make a plot of the e N for each k . Remarks: You should write a general one-dimensional code where you specify the number of elements. Your code should partition the domain automatically. However, if you want to make the code more general (for future assignments), you should put in the following features:
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me180_hw01 - HOMEWORK 1 THE BASICS Solve the followng...

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