NMDP120124

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Numerical Modeling of Differential Problems Instructors: Prof. E. Miglio, Dr. M. Gambarini January 12th 2024 - Duration of the exam: 2.5 hours. Exercise 1 (12 points) Consider the elliptic equation −∆u = f, (1) defined on the domain Ω = (0 , 1) × (0, 1). The exact solution is given by uex(x, y) = 2(1 + y) (3 + x)2 + (1 + y)2 . (2) 1. ( 2 pts) Compute the forcing term f. 2. ( 2 pts) Define the Dirichlet boundary conditions on each edge on the boundary according to the exact solution. 3. ( 1 pts) Is the provided exact solution smooth in Ω? 4. ( 3 pts) Compute the approximate solution using Firedrake and P2 finite elements. 5. ( 4 pts) Which is the expect order of convergence? Verify experimentally that the expected order of convergence is achieved. Exercise 2 (12 points) Consider the advection-diffusion problem    −µ∆u + b · ∇u = f in Ω = (0, 1)2, ∇u · n = gN on ΓN : {(x, y) ∈ ∂Ω, x = 0}, u = 0 on ∂Ω \ ΓN , with µ = 10−3, b = [y(y − 1), 0], f = 1, gN = 1. 1. ( 3 pts) Write the weak formulation of the problem. 2. ( 3 pts) Write a function solve AD to solve the problem. The function must take as inputs a mesh and the degree of the finite element space to be used to compute the solution. Use the function to solve the problem with a mesh of triangles with N = 10 and comment the result. 3. ( 2 pts) Add (on paper) the SUPG/GLS stabilizations to the weak formulation. 4. ( 2 pts ) Add the SUPG/GLS stabilizations to function solve AD, adding to the inputs a parameter which allows switching between the two. 5. ( 2 pts ) Solve the problem on the same mesh using the SUPG and GLS stabilizations with P1 elements. Compute the L2 norm of the difference between uSU P G h and uGLS h and comment the result. Exercise 3 (9 points) Consider the advection equation ∂u ∂t + a ∂u ∂x = 0, (3)…