Exam2

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Course: Numerical Modelling of Differential Problems Prof. E. Miglio - January 12th 2022 Surname Name Matricola INSTRUCTIONS: Duration of the exam: 2.5 hours. Upload the files using beep.metid.polimi.it. Exercise 1 (11 points) Consider the following differential problem: −uxx = sin(πx), in Ω = (0, 1) u(0) =u(1) = 0. (1) 1. ( 2 points) Compute the exact solution. 2. ( 4 points) Use FEniCS to solve the problem using linear and quadratic finite element. 3. ( 1 point ) Compare (on a plot) the numerical solutions obtained at point 2. with the exact one obtained at point 1. 4. ( 4 points) Plot the error as a function of h (take 5 values) for the linear and quadratic FE. Comment the plots; are they consistent with the theory ? Exercise 2 (11 points) Consider the following steady Stokes problem    −∆u +∇p = 0 in Ω = [0 , 1]2 ∇· u = 0 (2) with u = [y(y− 1), 0]T on the boundaries. 1. ( 4 points) Write the weak formulation of the problem. 2. ( 3 points) Propose a suitable couple of finite elements for velocity and pressure (with motivation). 3. ( 4 points) Solve the problem using FEniCS. Exercise 3 (11 points) Consider the following system ∂u ∂t +A∂u ∂x = 0, in [0, 10], (3) where u = [u1(x), u2(x)] and A = [ 4 5 6 5 ] (4) ˆ (2 points) Verify that the system is hyperbolic. ˆ (3 points) Discuss the boundary conditions in x = 0 ed x = 10. ˆ (3 points) Write the diagonal form of the system. ˆ (3 points) Write the upwind method for the diagonal system.