From c1bd4a0955af82f6ef71cdf93ab0e95a712e0f87 Mon Sep 17 00:00:00 2001 From: bangerth Date: Fri, 1 Jun 2012 12:26:26 +0000 Subject: [PATCH] Add one large comment on adaptive meshes. git-svn-id: https://svn.dealii.org/trunk@25593 0785d39b-7218-0410-832d-ea1e28bc413d --- deal.II/examples/step-23/doc/intro.dox | 2 +- deal.II/examples/step-23/doc/results.dox | 70 ++++++++++++++++++++++++ 2 files changed, 71 insertions(+), 1 deletion(-) diff --git a/deal.II/examples/step-23/doc/intro.dox b/deal.II/examples/step-23/doc/intro.dox index 84d1014b0b..c8feb1dba8 100644 --- a/deal.II/examples/step-23/doc/intro.dox +++ b/deal.II/examples/step-23/doc/intro.dox @@ -104,7 +104,7 @@ For all these reasons, for the present program, we choose to use the Rothe method for discretization, i.e. we first discretize in time and then in space. We will not actually use adaptive meshes at all, since this involves a large amount of additional code, but we will comment -at the appropriate places what it would involve to add this feature. +on this some more in the results section below.

Rothe's method!

diff --git a/deal.II/examples/step-23/doc/results.dox b/deal.II/examples/step-23/doc/results.dox index 02fa8d7058..0fda35af8c 100644 --- a/deal.II/examples/step-23/doc/results.dox +++ b/deal.II/examples/step-23/doc/results.dox @@ -121,4 +121,74 @@ If you want to explore a bit, try out some of the following things: system_rhs, false); @endcode + +
  • deal.II being a library that supports adaptive meshes it would of course be + nice if this program supported change the mesh every few time steps. Given the + structure of the solution — a wave that travels through the domain — + it would seem appropriate if we only refined the mesh where the wave currently is, + and not simply everywhere. It is intuitively clear that we should be able to + save a significant amount of cells this way. (Though upon further thought one + realizes that this is really only the case in the initial stages of the simulation. + After some time, for wave phenomena, the domain is filled with reflections of + the initial wave going in every direction and filling every corner of the domain. + At this point, there is in general little one can gain using local mesh + refinement.) + + To make adaptively changing meshes possible, there are basically two routes. + The "correct" way would be to go back to the weak form we get using Rothe's + method. For example, the first of the two equations to be solved in each time + step looked like this: + \f{eqnarray*} + (u^n,\varphi) + k^2\theta^2(\nabla u^n,\nabla \varphi) &=& + (u^{n-1},\varphi) - k^2\theta(1-\theta)(\nabla u^{n-1},\nabla \varphi) + + + k(v^{n-1},\varphi) + + k^2\theta + \left[ + \theta (f^n,\varphi) + (1-\theta) (f^{n-1},\varphi) + \right]. + \f} + Now, note that we solve for $u^n$ on mesh ${\mathbb T}^n$, and + consequently the test functions $\varphi$ have to be from the space + $V_h^n$ as well. As discussed in the introduction, terms like + $(u^{n-1},\varphi)$ then require us to integrate the solution of the + previous step (which may have been computed on a different mesh + ${\mathbb T}^{n-1}$) against the test functions of the current mesh, + leading to a matrix $M^{n,n-1}$. This process of integrating shape + functions from different meshes is, at best, awkward. It can be done + but because it is difficult to ensure that ${\mathbb T}^{n-1}$ and + ${\mathbb T}^{n}$ differ by at most one level of refinement, one + has to recursively match cells from both meshes. It is feasible to + do this, but it leads to lengthy and not entirely obvious code. + + The second approach is the following: whenever we change the mesh, + we simply interpolate the solution from the last time step on the old + mesh to the new mesh, using the SolutionTransfer class. In other words, + instead of the equation above, we would solve + \f{eqnarray*} + (u^n,\varphi) + k^2\theta^2(\nabla u^n,\nabla \varphi) &=& + (I^n u^{n-1},\varphi) - k^2\theta(1-\theta)(\nabla I^n u^{n-1},\nabla \varphi) + + + k(I^n v^{n-1},\varphi) + + k^2\theta + \left[ + \theta (f^n,\varphi) + (1-\theta) (f^{n-1},\varphi) + \right], + \f} + where $I^n$ interpolates a given function onto mesh ${\mathbb T}^n$. + This is a much simpler approach because, in each time step, we no + longer have to worry whether $u^{n-1},v^{n-1}$ were computed on the + same mesh as we are using now or on a different mesh. Consequently, + the only changes to the code necessary are the addition of a function + that computes the error, marks cells for refinement, sets up a + SolutionTransfer object, transfers the solution to the new mesh, and + rebuilds matrices and right hand side vectors on the new mesh. Neither + the functions building the matrices and right hand sides, nor the + solvers need to be changed. + + While this second approach is, strictly speaking, + not quite correct in the Rothe framework (it introduces an addition source + of error, namely the interpolation), it is nevertheless what + almost everyone solving time dependent equations does. We will use this + method in step-31, for example. -- 2.39.5