From a603459e4df5de66408a31b069cc26b64d6c7c04 Mon Sep 17 00:00:00 2001 From: wolf Date: Sun, 27 Feb 2000 22:12:47 +0000 Subject: [PATCH] Various updates. git-svn-id: https://svn.dealii.org/trunk@2498 0785d39b-7218-0410-832d-ea1e28bc413d --- .../chapter-2.step-by-step/navbar.html | 2 +- .../step-7.data/intro.tex | 150 ++++++++++++++++++ .../tutorial/chapter-2.step-by-step/toc.html | 7 +- 3 files changed, 155 insertions(+), 4 deletions(-) create mode 100644 deal.II/doc/tutorial/chapter-2.step-by-step/step-7.data/intro.tex diff --git a/deal.II/doc/tutorial/chapter-2.step-by-step/navbar.html b/deal.II/doc/tutorial/chapter-2.step-by-step/navbar.html index f9ba46a792..fa1be6a1f6 100644 --- a/deal.II/doc/tutorial/chapter-2.step-by-step/navbar.html +++ b/deal.II/doc/tutorial/chapter-2.step-by-step/navbar.html @@ -62,7 +62,7 @@

  • - Step 7 (still missing) + Step 7

    • diff --git a/deal.II/doc/tutorial/chapter-2.step-by-step/step-7.data/intro.tex b/deal.II/doc/tutorial/chapter-2.step-by-step/step-7.data/intro.tex new file mode 100644 index 0000000000..f94f281102 --- /dev/null +++ b/deal.II/doc/tutorial/chapter-2.step-by-step/step-7.data/intro.tex @@ -0,0 +1,150 @@ +\documentclass{article} + +\begin{document} + +In this example program, we will consider two aspects: +\begin{itemize} +\item Verification of correctness of the program; +\item Nonhomogeneous Neumann boundary conditions for the Helmholtz equation. +\end{itemize} + +\paragraph{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, \texttt{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 $u_h$ the finite element field, and $K$ is an element of the +triangulation: +\begin{eqnarray*} + {\| u-u_h \|}_{L_1(K)} &=& \int_K |u-u_h| \; dx, + \\ + {\| u-u_h \|}_{L_2(K)} &=& \left( \int_K |u-u_h|^2 \; dx \right)^{1/2}, + \\ + {\| u-u_h \|}_{L_\infty(K)} &=& \max_{x \in K} |u(x) - u_h(x)|, + \\ + {| u-u_h |}_{H^1(K)} &=& \left( \int_K |\nabla(u-u_h)|^2 \; dx \right)^{1/2}, + \\ + {\| u-u_h \|}_{H^1(K)} &=& \left( {\| u-u_h \|}^2_{L_2(K)} + +{| u-u_h |}^2_{H^1(K)} \right)^{1/2}. +\end{eqnarray*} +All these norms and seminorms 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 particual 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 superconvergence might occur. + +The function \texttt{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 $(e_i)$ contains the local $L_2$ norms, then +$$ + E = \| {\mathbf e} \| = \left( \sum_i e_i^2 \right)^{1/2} +$$ +is the global $L_2$ 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 explicitely 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. + + +\paragraph{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'': +$$ + -\Delta u + u = f, +$$ +on the square $[-1,1]^2$, augmented by boundary conditions +$$ + u = g_1 +$$ +on some part $\Gamma_1$ of the boundary $\Gamma$, and +$$ + {\mathbf n}\cdot \nabla u = g_2 +$$ +on the rest $\Gamma_2 = \Gamma \backslash \Gamma_1$. + +We choose the right hand side function $f$ such that the exact solution is +$$ + u(x) = \sum_{i=1}^3 \exp\left(-\frac{|x-x_i|^2}{\sigma^2}\right) +$$ +where the centers $x_i$ 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 $g_1$ such that it resembles the exact values of $u$. Likewise, we choose +$g_2$ 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 $u\in H^1_g=\{v\in H^1: v|_{\Gamma_1}=g_1\}$ such +that +$$ + {(\nabla u, \nabla v)}_\Omega + {(u,v)}_\Omega + = + {(f,v)}_\Omega + {(g_2,v)}_{\Gamma_2} +$$ +for all test functions $v\in H^1_0=\{v\in H^1: v|_{\Gamma_1}=0\}$. 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 + +\left(\varphi_i, \varphi_j\right)_K, + \\ + f_i^K &=& \left(f,\varphi_i\right)_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 \texttt{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 \texttt{FEFaceValues} that performs these tasks for integrations on +faces of cells. One provides it with a quadrature formula for a mannifold 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. The details of how this is done are shown in the +following program. + + +\end{document} + +%%% Local Variables: +%%% mode: latex +%%% TeX-master: t +%%% End: diff --git a/deal.II/doc/tutorial/chapter-2.step-by-step/toc.html b/deal.II/doc/tutorial/chapter-2.step-by-step/toc.html index f0116803de..8aa9965d36 100644 --- a/deal.II/doc/tutorial/chapter-2.step-by-step/toc.html +++ b/deal.II/doc/tutorial/chapter-2.step-by-step/toc.html @@ -92,9 +92,10 @@ At present, the following programs exist:
    Step 7
    -
    What's new: This example is still - missing. It will contain some stuff about rates of convergence - for different finite elements. +
    What's new: Helmholtz + equation. Non-homogeneous Neumann boundary conditions and + boundary integrals. Verification of correctness of computed + solutions.
    Step 8
    -- 2.39.5