100 points] The two-dimensional Laplace equation around a cylinder with a domain [0, 2π]×[1, 5] is given
= 0 (1)
The analytical solution is given by
ψ(r, θ) = U∞
In here R = 1 and U∞ = 1. The Drichlet boundary conditions:
ψ(1, θ) = 0 (4)
ψ(5, θ) =
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
- Implement the fully implicit finite element solution algorithm and use a direct solver (LU factorization) to solve.
- Plot Error function versus ∆r and ∆θ in a log-log scale. Compute the spatial convergence rate.
- Compare the error with the solutions of finite difference method.
The error function is given by
Error = kψi,j − ψanalytick2
[20 points] Solve the same problem with an unstructured quadrilateral/triangular elements.
Several useful MATLAB commands:
Crate a sparse matrix
A=sparse(i, j, s, m, n);
To solve a sparse linear system
x = A\b ;