Convergence in any norm with Dirichlet boundary conditions

In Homework 1, exercises 5-6, you may have observed that the 1-norm of \(B=A^{-1}\) is \({\cal O}(h^{-1})\) but that somehow the global error is second order in the same norm \(\|E\|_1 = {\cal O}(h^2)\). This seems contradictory at first glance, since we have previously used (in class) the estimate

\[\|E\| \le \|B\|\|\tau\|\]

which in the 1-norm gives \(\|E\|_1 \le {\cal O}(h^{-1}){\cal O}(h^2) = {\cal O}(h)\) (here we have used the grid-function norm for \(\tau\) and the ordinary induced matrix norm for \(B\)). However, this is only an upper bound, and it may be too pessimistic. We can get a better estimate if we are more careful:

\[\begin{align*} E & = -B \tau \\ & = \sum_{j=0}^{m+1} B_j \tau_j \\ & = B_0 \tau_0 + B_{m+1}\tau_{m+1} + \sum_{j=1}^m B_j \tau_j\end{align*}\]

Taking norms and using the triangle inequality, we obtain

\[\begin{align*} \|E\| & \le \|B_0 \tau_0\| + \|B_{m+1}\tau_{m+1}\| + \sum_{j=1}^m \|B_j \tau_j\|\end{align*}\]

Now, observe that the Dirichlet boundary conditions are satisfied exactly by the numerical scheme, so \(\tau_0 = \tau_{m+1} = 0\). Thus the first two terms above vanish and we obtain

\[\begin{align*} \|E\| & \le \sum_{j=1}^m \|B_j \tau_j\| \\ & \le \sum_{j=1}^m |\tau_j| \|B_j\| \\ & = \sum_{j=1}^m {\cal O}(h^2) {\cal O}(h) = {\cal O}(h^2)\end{align*}\]

in any (grid-function) norm! Here we have used the fact that \(\|B_j\|={\cal O}(h)\) in any grid-function norm, and the fact that \(m={\cal O}(h^{-1})\).

Convergence in the 1-norm with a Neumann boundary condition

Now suppose that we are given a Neumann boundary condition at \(x=0\). Then the boundary condition will not be satisfied exactly and \(\tau_0\ne0\). Suppose that we use a second-order discretization of the boundary condition so that \(\tau_0 = {\cal O}(h^2)\).

Now, using the definition of the \(p\)-grid-function norm for \(p=1\), we have

\[\begin{align*} \|E\| & \le h\sum_{i=1}^{m+1}|B_{i0} \tau_0| + h\sum_{j=1}^m \sum_{i=1}^{m+1}|B_{ij} \tau_j| \\ & \le h|\tau_0|\sum_{i=1}^{m+1}|B_{i0}| + h|\tau_j|\sum_{j=1}^m \sum_{i=1}^{m+1}|B_{ij}| \\ & = {\cal O}(h^3) \sum_{i=0}^{m+1}{\cal O}(1) + {\cal O}(h^3) \sum_{j=1}^m \sum_{i=1}^{m+1} {\cal O}(h) \\ & = {\cal O}(h^2) + {\cal O}(h^2) = {\cal O}(h^2).\end{align*}\]

Thus we still obtain 2nd order convergence, despite the behavior of \(\|B\|_1\).

Induced grid-function norms

What’s going on here? We wish to estimate the grid-function norm of \(E\). Therefore, in the bound

\[\|E\| \le \|B\|\|\tau\|\]

we should use the grid-function norm of \(\tau\) and an induced grid-function norm of \(B\), which is properly defined as

\[\|B\|_1 = h \max_j \sum_i |B_{ij}|\] which in our case yields \(\|B\|_1={\cal O}(1)\).