From: bangerth Date: Mon, 4 Feb 2013 17:28:38 +0000 (+0000) Subject: Some expansion. X-Git-Url: https://gitweb.dealii.org/cgi-bin/gitweb.cgi?a=commitdiff_plain;h=d65917ea843961468aced6868bd740a38bc962f5;p=dealii-svn.git Some expansion. git-svn-id: https://svn.dealii.org/trunk@28223 0785d39b-7218-0410-832d-ea1e28bc413d --- diff --git a/deal.II/doc/doxygen/tutorial/toc-list.html b/deal.II/doc/doxygen/tutorial/toc-list.html index 4a56b86be2..92a02748ae 100644 --- a/deal.II/doc/doxygen/tutorial/toc-list.html +++ b/deal.II/doc/doxygen/tutorial/toc-list.html @@ -4,7 +4,7 @@ Step-by-Step - + @@ -51,7 +51,8 @@ Computations on successively refined grids. Reading a grid from disk. Some optimizations. Using assertions. Non-constant coefficient in - the elliptic operator. Preconditioning the CG solver for the + the elliptic operator (yielding the extended Poisson + equation). Preconditioning the CG solver for the linear system of equations. diff --git a/deal.II/examples/step-5/doc/intro.dox b/deal.II/examples/step-5/doc/intro.dox index 251d677560..7102702df4 100644 --- a/deal.II/examples/step-5/doc/intro.dox +++ b/deal.II/examples/step-5/doc/intro.dox @@ -25,7 +25,7 @@ them are:
  • Finite element programs usually use extensive amounts of computing time, so some optimizations are sometimes necessary. We will show some of them. -
  • On the other side, finite element programs tend to be rather +
  • On the other hand, finite element programs tend to be rather complex, so debugging is an important aspect. We support safe programming by using assertions that check the validity of parameters and %internal states in a debug mode, but are removed @@ -35,3 +35,35 @@ them are: preconditioned iterative solvers for the linear systems of equations. + +The equation to solve here is as follows: +@f{align*} + -\nabla \cdot a(\mathbf x) \nabla u(\mathbf x) &= 1 \qquad\qquad & \text{in}\ \Omega, + \\ + u &= 0 \qquad\qquad & \text{on}\ \partial\Omega. +@f} +If $a(\mathbf x)$ was a constant coefficient, this would simply be the Poisson +equation. However, if it is indeed spatially variable, it is a more complex +equation (often referred to as the "extended Poisson equation"). Depending on +what the variable $u$ refers to it models a variety of situations with wide +applicability: + +- If $u$ is the electric potential, then $-a\nabla u$ is the electric current + in a medium and the coefficient $a$ is the conductivity of the medium at any + given point. (In this situation, the right hand side of the equation would + be the electric source density and would usually be zero or consist of + localized, Delta-like, functions.) +- If $u$ is the vertical deflection of a thin membrane, then $a$ would be a + measure of the local stiffness. This is the interpretation that will allow + us to interpret the images shown in the results section below. + +Since the Laplace/Poisson equation appears in so many contexts, there are many +more interpretations than just the two listed above. + +When assembling the linear system for this equation, we need the weak form +which here reads as follows: +@f{align*} + (a \nabla \varphi, \nabla u) &= (\varphi, 1) \qquad \qquad \forall \varphi. +@f} +The implementation in the assemble_system function follows +immediately from this. diff --git a/deal.II/examples/step-5/doc/tooltip b/deal.II/examples/step-5/doc/tooltip index 4f09ebdbd7..51b43ab8ec 100644 --- a/deal.II/examples/step-5/doc/tooltip +++ b/deal.II/examples/step-5/doc/tooltip @@ -1 +1 @@ -Reading a grid from disk. Computations on successively refined grids. +Reading a grid from disk. Computations on successively refined grids. Variable coefficients.