NMDP1507241

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Numerical Modeling of Differential Problems Instructors: Prof. E. Miglio, Dr. M. Gambarini July 15th 2024 - Duration of the exam: 2.5 hours. Exercise 1 (12 points) Consider the problem ( −∇ · (K∇u) = f in Ω = (0, 1)2, u = 0 on ∂Ω, where K =  4 −2 −2 4  , (1) with exact solution uex = 1 4 y(1 − y) sin(πx) (2) and f = −π2y (y − 1) sin (πx) − π (2y − 1) cos (πx) + 2 sin (πx). (3) 1. ( 2 pts) Write the weak formulation of the problem. 2. ( 3 pts) Prove that the weak problem is well posed. In particular, estimate the constants appearing in the C´ ea lemma. 3. ( 2 pts) Compute a numerical solution uh of the problem with P2 triangular elements using Firedrake. 4. ( 2 pts) Verify the order of convergence in the H 1 norm with respect to the grid size. 5. ( 3 pts) Using the Firedrake command assemble, compute the numerical approximation of the value of a(uh − uex, uh). What theoretical property justifies this result? Exercise 2 (12 points) Consider a Stokes problem −∆u + ∇p = f , in Ω ∇ · u = 0, in Ω u = g, on ∂Ω (4) where Ω = [−1, 1]2, u = [u, v] is the velocity vector and p is the pressure. The exact solution is given by: u = 20xy3, v = 5x4 − 5y4, p = 60x2y − 20y3. (5) 1. ( 2 pts) Compute f and g (write the values of u and v on each edge of the square). 2. ( 3 pts) Write the continuous and the discrete weak formulation of (4). 3. ( 3 pts ) Using Firedrake solve (4) adopting P1 − P1 and P2 − P1 finite element couples. Comment on the obtained results. 4. ( 4 pts) Define the error as follows E = ∥u − uh∥H 1 + ∥p − ph∥L2 , (6) where uh and ph are the numerical solutions for the velocity and for the pressure. Verify that E is of order 2 with respect to h when using the P2 − P1 couple. 1 Exercise 3 (9 points) Consider the following system ∂u ∂t + A ∂u ∂x = 0, in [0, 10], (7) where…