1. Write down the equation for the linear piecewise shape functions , 2, and 3 shown...

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1. Write down the equation for the linear piecewise shape functions ₁, 2, and 3 shown in the figure below that are used to construct trial functions on two finite element subdomains of the and their derivatives for the following finite element subdomains (1 point for each one, and 1 point for checking that ₂ € H¹). φ(x) $1(x) 1 -1 $2(x) 3(x) (u,xx-3= 0 on (-1,1) u(-1) = 1 u(1) = 0 113 1 x 2. By hand, solve the following differential equation using the Galerkin method with the three shape functions from problem 1 (Do this problem by hand as practice for the midterm next week). (1 point for each of the steps below). a. Find the analytic solution of this boundary value problem b. Write this problem in weak form c. Define the conditions on the trial and weighting functions in the weak form statement of the problem d. Find the numerical solution of the Galerkin equation using weighting and trial func- tions constructed from the sum of shape functions that you defined in problem 1. e. Evaluate your numerical and exact solutions at x = 0, and x = ²/ 1. Write down the equation for the linear piecewise shape functions ₁, 2, and 3 shown in the figure below that are used to construct trial functions on two finite element subdomains of the and their derivatives for the following finite element subdomains (1 point for each one, and 1 point for checking that ₂ € H¹). φ(x) $1(x) 1 -1 $2(x) 3(x) (u,xx-3= 0 on (-1,1) u(-1) = 1 u(1) = 0 113 1 x 2. By hand, solve the following differential equation using the Galerkin method with the three shape functions from problem 1 (Do this problem by hand as practice for the midterm next week). (1 point for each of the steps below). a. Find the analytic solution of this boundary value problem b. Write this problem in weak form c. Define the conditions on the trial and weighting functions in the weak form statement of the problem d. Find the numerical solution of the Galerkin equation using weighting and trial func- tions constructed from the sum of shape functions that you defined in problem 1. e. Evaluate your numerical and exact solutions at x = 0, and x = ²/

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a The analytic solution of this boundary value problem is ux 1 3x2 b The ... View the full answer