Finite Element Method

  [100 points] The two-dimensional Laplace equation around a cylinder with a domain [0, 2π]×[1, 5] is given by. ∂ 2ψ ∂x2 + ∂ 2ψ ∂y2 = 0 (1) The analytical solution is given by ψ(r, θ) = U∞ r − R2 r sin(θ) (2) (3) In here R = 1 and U∞ = 1. The Drichlet boundary conditions: ψ(1, θ) = 0 (4) ψ(5, θ) = 5 − 1 5 sin(θ) (5) (6) Use the second-order accurate finite element discretization with uniform 80 × 40, 160 × 80 and 320 × 160 quadrilateral elements to solve the above Laplace equation. For this purpose 1. Implement the fully implicit finite element solution algorithm and use a direct solver (LU factorization) to solve. 2. Plot Error function versus ∆r and ∆θ in a log-log scale. Compute the spatial convergence rate. 3. Compare the error with the solutions of finite difference method. The error function is given by Error = kψi,j − ψanalytick2 √ imaxjmax (7) [20 points] Solve the same problem with an unstructured quadrilateral/triangular elements. Several useful MATLAB commands: Crate a sparse matrix i=[]; j=[]; s=[]; m=100; n=100; A=sparse(i, j, s, m, n); To solve a sparse linear system x = Ab ;