The two-dimensional Laplace equation

PROJECT # 2
[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 ;
   0UUT510E, Return date: 20 May 2021
   1