Finite Differences #

Consider the one-dimensional Poisson equation with homogeneous Dirichlet conditions $$-\frac{d^2 u}{d x^2}=f(x),~~~x\in(0,1)$$ with Dirichlet boundary conditions $$u(0)=u(1) = 0.$$

1. Discretise the Poisson equation by finite differences using an equidistant mesh size $h=1/N$ and $N+1$ grid points.
2. Write the finite difference approximation from 1. in matrix-vector form $Au=b$. Therefore, define the entries of the matrix $A\in\mathbb{R}^{N+1\times N+1}$.
3. Write the finite difference approximation as $Au=b$, where $A\in\mathbb{R}^{N-1\times N-1}$ and $b\in\mathbb{R}^{N-1}$, by substituting the values for $u(0)$ and $u(1)$.

Eigenvalues and eigenvectors #

When analysing the properties of numerical algorithms it is often helpful to know about the spectrum (eigenvectors) of the operator being treated. A non-zero vector x is an eigenvector of a matrix A if and only if there exists a scalar $\lambda$ such that $$Ax=\lambda x.$$ The scalar $\lambda$ is the corresponding eigenvalue.

Show that the discretised sine, i.e. $u_i = sin(k\pi ih)$, is an eigenvector with eigenvalue $\lambda=(4/h^2)\sin^2(k\pi h/2)$ of the finite difference matrix $A$ in the previous exercise.

You may find the following trigonometric identities useful: $$ \sin(a+b)+\sin(a-b) = 2\sin(a)\cos(b)$$ $$ \cos(2x) = 1-2\sin^2(x)$$