From 09601ec57fd1b2b18723539b230ad50413aeed0a Mon Sep 17 00:00:00 2001 From: wolf Date: Wed, 22 Mar 2000 15:06:21 +0000 Subject: [PATCH] Finish intro. git-svn-id: https://svn.dealii.org/trunk@2613 0785d39b-7218-0410-832d-ea1e28bc413d --- .../step-7.data/intro.html | 303 +++++++++++++++++- .../step-7.data/intro/img1.gif | Bin 0 -> 90 bytes .../step-7.data/intro/img10.gif | Bin 0 -> 1201 bytes .../step-7.data/intro/img11.gif | Bin 0 -> 431 bytes .../step-7.data/intro/img12.gif | Bin 0 -> 453 bytes .../step-7.data/intro/img13.gif | Bin 0 -> 451 bytes .../step-7.data/intro/img14.gif | Bin 0 -> 233 bytes .../step-7.data/intro/img15.gif | Bin 0 -> 734 bytes .../step-7.data/intro/img16.gif | Bin 0 -> 836 bytes .../step-7.data/intro/img17.gif | Bin 0 -> 1198 bytes .../step-7.data/intro/img18.gif | Bin 0 -> 810 bytes .../step-7.data/intro/img19.gif | Bin 0 -> 389 bytes .../step-7.data/intro/img2.gif | Bin 0 -> 5293 bytes .../step-7.data/intro/img20.gif | Bin 0 -> 335 bytes .../step-7.data/intro/img21.gif | Bin 0 -> 173 bytes .../step-7.data/intro/img22.gif | Bin 0 -> 1794 bytes .../step-7.data/intro/img3.gif | Bin 0 -> 202 bytes .../step-7.data/intro/img4.gif | Bin 0 -> 987 bytes .../step-7.data/intro/img5.gif | Bin 0 -> 391 bytes .../step-7.data/intro/img6.gif | Bin 0 -> 157 bytes .../step-7.data/intro/img7.gif | Bin 0 -> 123 bytes .../step-7.data/intro/img8.gif | Bin 0 -> 435 bytes .../step-7.data/intro/img9.gif | Bin 0 -> 331 bytes 23 files changed, 300 insertions(+), 3 deletions(-) create mode 100644 deal.II/doc/tutorial/chapter-2.step-by-step/step-7.data/intro/img1.gif create mode 100644 deal.II/doc/tutorial/chapter-2.step-by-step/step-7.data/intro/img10.gif create mode 100644 deal.II/doc/tutorial/chapter-2.step-by-step/step-7.data/intro/img11.gif create mode 100644 deal.II/doc/tutorial/chapter-2.step-by-step/step-7.data/intro/img12.gif create mode 100644 deal.II/doc/tutorial/chapter-2.step-by-step/step-7.data/intro/img13.gif create mode 100644 deal.II/doc/tutorial/chapter-2.step-by-step/step-7.data/intro/img14.gif create mode 100644 deal.II/doc/tutorial/chapter-2.step-by-step/step-7.data/intro/img15.gif create mode 100644 deal.II/doc/tutorial/chapter-2.step-by-step/step-7.data/intro/img16.gif create mode 100644 deal.II/doc/tutorial/chapter-2.step-by-step/step-7.data/intro/img17.gif create mode 100644 deal.II/doc/tutorial/chapter-2.step-by-step/step-7.data/intro/img18.gif create mode 100644 deal.II/doc/tutorial/chapter-2.step-by-step/step-7.data/intro/img19.gif create mode 100644 deal.II/doc/tutorial/chapter-2.step-by-step/step-7.data/intro/img2.gif create mode 100644 deal.II/doc/tutorial/chapter-2.step-by-step/step-7.data/intro/img20.gif create mode 100644 deal.II/doc/tutorial/chapter-2.step-by-step/step-7.data/intro/img21.gif create mode 100644 deal.II/doc/tutorial/chapter-2.step-by-step/step-7.data/intro/img22.gif create mode 100644 deal.II/doc/tutorial/chapter-2.step-by-step/step-7.data/intro/img3.gif create mode 100644 deal.II/doc/tutorial/chapter-2.step-by-step/step-7.data/intro/img4.gif create mode 100644 deal.II/doc/tutorial/chapter-2.step-by-step/step-7.data/intro/img5.gif create mode 100644 deal.II/doc/tutorial/chapter-2.step-by-step/step-7.data/intro/img6.gif create mode 100644 deal.II/doc/tutorial/chapter-2.step-by-step/step-7.data/intro/img7.gif create mode 100644 deal.II/doc/tutorial/chapter-2.step-by-step/step-7.data/intro/img8.gif create mode 100644 deal.II/doc/tutorial/chapter-2.step-by-step/step-7.data/intro/img9.gif diff --git a/deal.II/doc/tutorial/chapter-2.step-by-step/step-7.data/intro.html b/deal.II/doc/tutorial/chapter-2.step-by-step/step-7.data/intro.html index cf39f30d76..1331c732d1 100644 --- a/deal.II/doc/tutorial/chapter-2.step-by-step/step-7.data/intro.html +++ b/deal.II/doc/tutorial/chapter-2.step-by-step/step-7.data/intro.html @@ -1,6 +1,303 @@

Introduction

-

-Still to be written. -

+

+In this example 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 +non-trivial finite element program that worked right from the start. It is +therefore necessary to find ways to verify whether a computed solution is +correct or not. Usually, this is done by choosing the set-up of a simulation +such that we know the exact continuous solution and evaluate the difference +between continuous and computed discrete solution. If this difference +converges to zero with the right order of convergence, this is already a good +indication of correctness, although there may be other sources of error +persisting which have only a small contribution to the total error or are of +higher order. + +

+In this example, we will not go into the theories of systematic software +verification which is a very complicated problem. Rather we will demonstrate +the tools which deal.II can offer in this respect. This is basically centered +around the functionality of a single function, integrate_difference. +This function computes the difference between a given continuous function and +a finite element field in various norms on each cell. At present, the +supported norms are the following, where u denotes the continuous function +and uh the finite element field, and K is an element of the +triangulation: +

+
+\begin{eqnarray*}{\Vert u-u_h \Vert}_{L_1(K)} &=& \int_K \vert u-u_h\vert \; dx,... +...ert}^2_{L_2(K)} ++{\vert u-u_h \vert}^2_{H^1(K)} \right)^{1/2}. +\end{eqnarray*} +

+
All these norms and semi-norms can also be evaluated with weighting functions, +for example in order to exclude singularities from the determination of the +global error. The function also works for vector-valued functions. It should +be noted that all these quantities are evaluated using quadrature formulas; +the choice of the right quadrature formula is therefore crucial to the +accurate evaluation of the error. This holds in particular for the $L_\infty$norm, where we evaluate the maximal deviation of numerical and exact solution +only at the quadrature points; one should then not try to use a quadrature +rule with points only at points where super-convergence might occur. + +

+The function integrate_difference evaluates the desired norm on each +cell K of the triangulation and returns a vector which holds these +values. From the local values, we can then obtain the global error. For +example, if the vector (ei) contains the local L2 norms, then +

+
+ + + +\begin{displaymath}E = \Vert {\mathbf e} \Vert = \left( \sum_i e_i^2 \right)^{1/2} +\end{displaymath} +
+
+

+is the global L2 error. + +

+In the program, we will show how to evaluate and use these quantities, and we +will monitor their values under mesh refinement. Of course, we have to choose +the problem at hand such that we can explicitly state the solution and its +derivatives, but since we want to evaluate the correctness of the program, +this is only reasonable. If we know that the program produces the correct +solution for one (or, if one wants to be really sure: many) specifically +chosen right hand sides, we can be rather confident that it will also compute +the correct solution for problems where we don't know the exact values. + +

+ +

+Non-homogeneous Neumann boundary conditions. +

The second, totally +unrelated, subject of this example program is the use of non-homogeneous +boundary conditions. These are included into the variational form using +boundary integrals which we have to evaluate numerically when assembling the +right hand side vector. + +

+Before we go into programming, let's have a brief look at the mathematical +formulation. The equation which we want to solve is Helmholtz's equation +``with the nice sign'': +

+
+ + + +\begin{displaymath}-\Delta u + u = f, +\end{displaymath} +
+
+

+on the square [-1,1]2, augmented by boundary conditions +

+
+ + + +u = g1 +
+
+

+on some part $\Gamma_1$ +of the boundary $\Gamma$, +and +

+
+ + + +\begin{displaymath}{\mathbf n}\cdot \nabla u = g_2 +\end{displaymath} +
+
+

+on the rest + +$\Gamma_2 = \Gamma \backslash \Gamma_1$. + +

+We choose the right hand side function f such that the exact solution is +

+
+ + + +\begin{displaymath}u(x) = \sum_{i=1}^3 \exp\left(-\frac{\vert x-x_i\vert^2}{\sigma^2}\right) +\end{displaymath} +
+
+

+where the centers xi of the exponentials are + + +$x_1=(-\frac 12,\frac 12)$, + + +$x_2=(-\frac 12,-\frac 12)$, +and + + +$x_3=(\frac 12,-\frac 12)$. +The half width is set to + +$\sigma=\frac 13$. + +

+We further choose + +$\Gamma_1=\Gamma \cap\{\{x=1\} \cup \{y=1\}\}$, +and there +set g1 such that it resembles the exact values of u. Likewise, we choose +g2 on the remaining portion of the boundary to be the exact normal +derivatives of the continuous solution. + +

+Using the above definitions, we can state the weak formulation of the +equation, which reads: find + +$u\in H^1_g=\{v\in H^1: v\vert _{\Gamma_1}=g_1\}$ +such +that +

+
+ + + +\begin{displaymath}{(\nabla u, \nabla v)}_\Omega + {(u,v)}_\Omega += +{(f,v)}_\Omega + {(g_2,v)}_{\Gamma_2} +\end{displaymath} +
+
+

+for all test functions + +$v\in H^1_0=\{v\in H^1: v\vert _{\Gamma_1}=0\}$. +The +boundary term + +${(g_2,v)}_{\Gamma_2}$ +has appeared by partial integration and +using + +$\partial_n u=g$ +on $\Gamma_2$ +and v=0 on $\Gamma_1$. +The cell +matrices and vectors which we use to build the global matrices and right hand +side vectors in the discrete formulation therefore look like this: +

+
+\begin{eqnarray*}A_{ij}^K &=& \left(\nabla \varphi_i, \nabla \varphi_j\right)_K +... +...ight)_K ++\left(g_2, \varphi_i\right)_{\partial K\cap \Gamma_2}. +\end{eqnarray*} +

+
Since the generation of the domain integrals has been shown in previous +examples several times, only the generation of the contour integral is of +interest here. It basically works along the following lines: for domain +integrals we have the FEValues class that provides values and +gradients of the shape values, as well as Jacobian determinants and other +information and specified quadrature points in the cell; likewise, there is a +class FEFaceValues that performs these tasks for integrations on +faces of cells. One provides it with a quadrature formula for a manifold with +dimension one less than the dimension of the domain is, and the cell and the +number of its face on which we want to perform the integration. The class will +then compute the values, gradients, normal vectors, weights, etc at the +quadrature points on this face, which we can then use in the same way as for +the domain integrals. 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