Exam1

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Numerical Modeling of Differential Problems Prof. Edie Miglio - August 23, 2021 Duration of the exam: 2.5 hours Exercise 1 (14+2 points) Consider the problem −∆u =f in Ω = (0, 1)2, ∇u· n +u =gN on ∂Ω, (1) where f(x,y ) =π2 ( cos(2π(y +x)) + cos(2π(y−x)) + cos(2πy) + cos(2πx) ) . 1. ( 2pts) Write the weak formulation of the problem. 2. ( 4pts) Discuss the well posedness of the (weak) problem. Does a solution exist? Is this unique? 3. ( 2pts) Consider the exact solution u(x,y ) = (cos(xπ) cos(yπ))2. • Compute the gradient∇u(x,y ). • Show that∇u(x,y )· n = 0 on ∂Ω and that gN(x,y ) = (cos(xπ) cos(yπ))2 on ∂Ω. 4. ( 2pts) Solve the problem using FEniCS and plot the solution. 5. ( 4pts) Study the accuracy of the FE discretization. 6. (Extra points 2pts) Prove your answer for point 2. To do that you may use the following version of the Poincar´ e-Friedrichs inequality ∥v∥2 L2(Ω)≤C ( ∥∇v∥2 L2(Ω) +∥v∥2 L2(∂Ω) ) for all v∈H1(Ω) and the following inequality ∥v∥L2(∂Ω)≤ ˜C∥v∥H 1(Ω) for all v∈H1(Ω), where C and ˜C are positive constants. Exercise 2 (10 points) Consider the following code: from fenics import * import matplotlib.pyplot as plt n = 10 degree = 1 mu = Constant(1e-3) b = Constant((1.0, 1.0)) f = Constant(1.0) gD = Constant(0.0) def solve_problem(n, degree, f, gD, b, mu): mesh = UnitSquareMesh(n, n, ’crossed’) V = FunctionSpace(mesh, ’CG’, degree) def boundary(x, on_boundary): 1 return on_boundary bc = DirichletBC(V, gD, boundary) u = TrialFunction(V) v = TestFunction(V) a = (mu * dot(grad(u), grad(v)) + dot(b, grad(u)) * v) * dx L = f * v * dx u = Function(V) solve(a == L, u, bc) return u, mesh u, mesh = solve_problem(n, degree, f, gD, b, mu) ax = plot(mesh) ax = plot(u) plt.colorbar(ax,fraction=0.045) plt.show() 1. ( 4pts) Study the code and identify the problem that this…