Besides these topics, again a variety of improvements and tricks will be
shown.
+
<h3>Verification of correctness</h3>
There has probably never been a
formulation. The equation that we want to solve here is the Helmholtz equation
"with the nice sign":
@f[
- -\Delta u + u = f,
+ -\Delta u + \alpha u = f,
@f]
-on the square $[-1,1]^2$, augmented by boundary conditions
+on the square $[-1,1]^2$ with $\alpha=1$, augmented by boundary conditions
@f[
u = g_1
@f]
{\mathbf n}\cdot \nabla u = g_2
@f]
on the rest $\Gamma_2 = \Gamma \backslash \Gamma_1$.
-In our particular testcase, we will use $\Gamma_1=\Gamma \cap\{\{x=1\} \cup \{y=1\}\}$.
+In our particular testcase, we will use $\Gamma_1=\Gamma \cap\{\{x=1\}
+\cup \{y=1\}\}$.
+(We say that this equation has the "nice sign" because the operator
+$-\Delta + \alpha I$ with the identity $I$ is a positive definite
+operator; the <a
+href="https://en.wikipedia.org/wiki/Helmholtz_equation">equation with
+the "bad sign"</a> is $-\Delta u - \alpha u$ and results from modeling
+time-harmonic processes. The operator is not necessarily positive
+definite, and this leads to all sorts of issues we need not discuss
+here.)
Because we want to verify the convergence of our numerical solution $u_h$,
we want a setup so that we know the exact solution $u$. This is where