From: Wolfgang Bangerth Date: Tue, 21 Jan 2020 18:28:22 +0000 (-0700) Subject: Augment introduction to step-7. X-Git-Tag: v9.2.0-rc1~621^2 X-Git-Url: https://gitweb.dealii.org/cgi-bin/gitweb.cgi?a=commitdiff_plain;h=refs%2Fpull%2F9411%2Fhead;p=dealii.git Augment introduction to step-7. --- diff --git a/examples/step-7/doc/intro.dox b/examples/step-7/doc/intro.dox index 57f1c13ec2..3fee3fe029 100644 --- a/examples/step-7/doc/intro.dox +++ b/examples/step-7/doc/intro.dox @@ -10,6 +10,7 @@ In this program, we will mainly consider two aspects: Besides these topics, again a variety of improvements and tricks will be shown. +

Verification of correctness

There has probably never been a @@ -101,9 +102,9 @@ Before we go into programming, let's have a brief look at the mathematical 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] @@ -112,7 +113,16 @@ on some part $\Gamma_1$ of the boundary $\Gamma$, and {\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 equation with +the "bad sign" 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