From e238f33eabca9a1d8de3edf75854282ad0ac91eb Mon Sep 17 00:00:00 2001 From: bangerth Date: Tue, 13 Feb 2007 04:31:09 +0000 Subject: [PATCH] A first couple of paragraphs on the new smoothness indicator git-svn-id: https://svn.dealii.org/trunk@14467 0785d39b-7218-0410-832d-ea1e28bc413d --- deal.II/examples/step-27/doc/intro.dox | 46 ++++++++++++++++++++++++++ 1 file changed, 46 insertions(+) diff --git a/deal.II/examples/step-27/doc/intro.dox b/deal.II/examples/step-27/doc/intro.dox index c59cfdf609..159f90427d 100644 --- a/deal.II/examples/step-27/doc/intro.dox +++ b/deal.II/examples/step-27/doc/intro.dox @@ -1,2 +1,48 @@

Introduction

+ +

A simple indicator for smoothness

+ +One of the central pieces of the adaptive finite element method is that we +inspect the computed solution (a posteriori) with an indicator that tells us +which are the cells where the error is largest, and then refine them. In many +of the other tutorial programs, we use the KellyErrorEstimator class to get an +indication of the size of the error on a cell, although we also discuss more +complicated strategies in some programs, most importantly in @ref step_14 +"step-14". + +In any case, as long as the decision is only "refine this cell" or "do not +refine this cell", the actual refinement step is not particularly +challenging. However, here we have a code that is capable of hp refinement, +i.e. we suddenly have two choices whenever we detect that the error on a +certain cell is too large for our liking: we can refine the cell by splitting +it into several smaller ones, or we can increase the polynomial degree of the +shape functions used on it. How do we know which is the more promising +strategy? + +In short, the question does not appear to be settled in the literature at this +time. There are a number of more or less complicated schemes that address it, +but there is nothing like the KellyErrorEstimator that is universally accepted +as a good indicator of the error. Most proposals use the fact that it is +beneficial to increase the polynomial degree whenever the solution is locally +smooth whereas it is better to refine the mesh wherever it is rough. However, +the questions of how to determine the local smoothness of the solution as well +as the decision when a solution is smooth enough to allow for an increase in +$p$ are certainly big and important ones. + +We do not intend to enter a sophisticated proposal into the fray about answer +to the general question. However, to demonstrate our approach to hp finite +elements, we need a simple indicator that does generate some useful +information. Our approach here is simple: for a function $u(x)$ to be in the +Sobolev space $H^s(K)$ on a cell $K$, it has to satisfy the condition +@f[ + \int_K |\nabla^s u(x)|^2 \; dx < \infty. +@f] +Assuming that the cell $K$ is not degenerate, i.e. that the mapping from the +unit cell to cell $K$ is sufficiently regular, above condition is of course +equivalent to +@f[ + \int_{\hat K} |\nabla^s \hat u(\hat x)|^2 \; dx < \infty +@f] +where $\hat u(\hat x)$ is the function $u(x)$ mapped back onto the unit cell +$\hat K$. -- 2.39.5