From 8849683d787caf8582d11b41531a345b422c3d49 Mon Sep 17 00:00:00 2001 From: wolf Date: Mon, 17 Sep 2001 09:43:36 +0000 Subject: [PATCH] New example. git-svn-id: https://svn.dealii.org/trunk@5018 0785d39b-7218-0410-832d-ea1e28bc413d --- .../chapter-2.step-by-step/navbar.html | 6 + .../step-11.data/img1.gif | Bin 0 -> 917 bytes .../step-11.data/img10.gif | Bin 0 -> 292 bytes .../step-11.data/img2.gif | Bin 0 -> 885 bytes .../step-11.data/img3.gif | Bin 0 -> 161 bytes .../step-11.data/img4.gif | Bin 0 -> 592 bytes .../step-11.data/img5.gif | Bin 0 -> 791 bytes .../step-11.data/img6.gif | Bin 0 -> 90 bytes .../step-11.data/img7.gif | Bin 0 -> 670 bytes .../step-11.data/img8.gif | Bin 0 -> 575 bytes .../step-11.data/img9.gif | Bin 0 -> 674 bytes .../step-11.data/intro.html | 176 ++++++++++++++++++ .../step-11.data/intro.tex | 113 +++++++++++ .../tutorial/chapter-2.step-by-step/toc.html | 6 + 14 files changed, 301 insertions(+) create mode 100644 deal.II/doc/tutorial/chapter-2.step-by-step/step-11.data/img1.gif create mode 100644 deal.II/doc/tutorial/chapter-2.step-by-step/step-11.data/img10.gif create mode 100644 deal.II/doc/tutorial/chapter-2.step-by-step/step-11.data/img2.gif create mode 100644 deal.II/doc/tutorial/chapter-2.step-by-step/step-11.data/img3.gif create mode 100644 deal.II/doc/tutorial/chapter-2.step-by-step/step-11.data/img4.gif create mode 100644 deal.II/doc/tutorial/chapter-2.step-by-step/step-11.data/img5.gif create mode 100644 deal.II/doc/tutorial/chapter-2.step-by-step/step-11.data/img6.gif create mode 100644 deal.II/doc/tutorial/chapter-2.step-by-step/step-11.data/img7.gif create mode 100644 deal.II/doc/tutorial/chapter-2.step-by-step/step-11.data/img8.gif create mode 100644 deal.II/doc/tutorial/chapter-2.step-by-step/step-11.data/img9.gif create mode 100644 deal.II/doc/tutorial/chapter-2.step-by-step/step-11.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 22fbbbee79..23f5670888 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 @@ -83,6 +83,12 @@ Step 10

+ +
  • +

    + Step 11 +

    +
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    Introduction

    + +

    +In this example, we demonstrate solving a simple Laplace problem using +higher order mappings, as well as using some stranger sort of +constraints and a variant of the SparsityPattern +class. As was noted in the introduction to the step-10 example program, it is difficult to find +problems where it actually makes a difference in the convergence order +whether we use simple linear mappings of the unit cells to the cells +in real space, or higher order mappings. Such problems are not exotic +(in fact, only the work with such problems generated the interest in +implementing higher order mappings in deal.II), and include for +example the numerical solution of the Euler equations of inviscid gas +flow, but they are too complicated to be made the subject of an +example program. We therefore restrict our attention to a problem +which exhibits the same order of convergence, but profits +significantly from higher order mappings nevertheless, by a reduced +size of the error. +

    + +

    +The problem we will be considering is the solution of Laplace's problem with +Neumann boundary conditions only: +
    +\begin{align*}-\Delta u &= f &&\text{in $\Omega$ }, +\\ +\partial_n u &= g && \text{on $\partial\Omega$ }. +\end{align*} +
    +It is well known that if this problem is to have a solution, then the forces +need to satisfy the compatibility condition +
    +\begin{gather*}\int_\Omega f\; dx + \int_{\partial\Omega} g\; ds = 0. +\end{gather*} +
    +We will consider the special case that $\Omega$ +is the circle of radius 1 +around the origin, and f=-2, g=1. This choice satisfies the compatibility +condition. + +

    +The compatibility condition allows a solution of the above equation, but it +nevertheless retains an ambiguity: since only derivatives of the solution +appear in the equations, the solution is only determined up to a constant. For +this reason, we have to pose another condition for the numerical solution, +which fixes this constant. + +

    +For this, there are various possibilities: +

      +
    • Fix one node of the discretization to zero or any other fixed value. + This amounts to an additional condition + +uh(x0)=0. Although this is + common practice, it is not necessarily a good idea, since we know that the + solutions of Laplace's equation are only in H1, which does not allow for + the definition of point values because it is not a subset of the continuous + functions. Therefore, even though fixing one node is allowed for + discretitized functions, it is not for continuous functions, and one can + often see this in a resulting error spike at this point in the numerical + solution. +
    • Fixing the mean value over the domain to zero or any other value. This + is allowed on the continuous level, since + +$H^1(\Omega)\subset L^1(\Omega)$ +by Sobolev's inequality, and thus also on the continuous level since we + there only consider subsets of H1. +
    • Fixing the mean value over the boundary of the domain to zero or any + other value. This is also allowed on the continuous level, since + + +$H^{1/2}(\partial\Omega)\subset L^1(\partial\Omega)$, +again by Sobolev's + inequality. +
    We will choose the last possibility, since we want to demonstrate another +technique with it. + +

    +While this describes the problem to be solved, we still have to figure out how +to implement it. Basically, except for the additional mean value constraint, +we have solved this problem several times, using Dirichlet boundary values, +and we only need to drop the treatment of Dirichlet boundary nodes. The use of +higher order mappings is also rather trivial and will be explained at the +various places where we use it; in almost all conceivable cases, you will only +consider the objects describing mappings as a black box which you need to +worry about, because their only uses seem to be to be passed to places deep +inside the library where functions know how to handle them (i.e. in the +FEValues classes and their descendents). + +

    +The tricky point in this program is the use of the mean value +constraint. Fortunately, there is a class in the library which knows how to +handle such constraints, and we have used it quite often already, without +mentioning its generality. Note that if we assume that the boundary nodes are +spaced equally along the boundary, then the mean value constraint +
    +\begin{gather*}\int_{\partial \Omega} u(x) \; ds = 0 +\end{gather*} +
    +can be written as +
    +\begin{gather*}\sum_{i\in\partial\Omega_h} u_i = 0, +\end{gather*} +
    +where the sum shall run over all degree of freedom indices which are located +on the boundary of the computational domain. Let us denote by i0 that index +on the boundary with the lowest number (or any other conveniently chosen +index), then the constraint can also be represented by +
    +\begin{gather*}u_{i_0} = \sum_{i\in\partial\Omega_h\backslash i_0} -u_i. +\end{gather*} +
    +This, luckily, is exactly the form of constraints for which the +ConstraintMatrix class was designed. Note that we have used this +class in several previous examples for the representation of hanging nodes +constraints, which also have this form: there, the middle vertex shall have +the mean of the values of the adjacent vertices. In general, the +ConstraintMatrix class is designed to handle homogeneous constraints +of the form +
    +\begin{gather*}CU = 0 +\end{gather*} +
    +where C denotes a matrix, and U the vector of nodal values. + +

    +In this example, the mean value along the boundary allows just such a +representation, with C being a matrix with just one row (i.e. there is only +one constraint). In the implementation, we will create a +ConstraintMatrix object, add one constraint (i.e. add another row to +the matrix) referring to the first boundary node i0, and insert the weights +with which all the other nodes contribute, which in this example happens to be +just -1. + +

    +Later, we will use this object to eliminate the first boundary node from the +linear system of equations, reducing it to one which has a solution without +the ambiguity of the constant shift value. One of the problems of the +implementation will be that the explicit elimination of this node results in a +number of additional elements in the matrix, of which we do not know in +advance where they are located and how many additional entries will be in each +of the rows of the matrix. We will show how we can use an intermediate object +to work around this problem. + +

    +But now on for the implementation of the program solving this problem... +

    diff --git a/deal.II/doc/tutorial/chapter-2.step-by-step/step-11.data/intro.tex b/deal.II/doc/tutorial/chapter-2.step-by-step/step-11.data/intro.tex new file mode 100644 index 0000000000..79adfde4f5 --- /dev/null +++ b/deal.II/doc/tutorial/chapter-2.step-by-step/step-11.data/intro.tex @@ -0,0 +1,113 @@ +\documentclass{article} +\usepackage{amsmath} + +\begin{document} +The problem we will be considering is the solution of Laplace's problem with +Neumann boundary conditions only: +\begin{align*} + -\Delta u &= f &&\text{in $\Omega$}, + \\ + \partial_n u &= g && \text{on $\partial\Omega$}. +\end{align*} +It is well known that if this problem is to have a solution, then the forces +need to satisfy the compatibility condition +\begin{gather*} + \int_\Omega f\; dx + \int_{\partial\Omega} g\; ds = 0. +\end{gather*} +We will consider the special case that $\Omega$ is the circle of radius 1 +around the origin, and $f=-2$, $g=1$. This choice satisfies the compatibility +condition. + +The compatibility condition allows a solution of the above equation, but it +nevertheless retains an ambiguity: since only derivatives of the solution +appear in the equations, the solution is only determined up to a constant. For +this reason, we have to pose another condition for the numerical solution, +which fixes this constant. + +For this, there are various possibilities: +\begin{itemize} +\item Fix one node of the discretization to zero or any other fixed value. + This amounts to an additional condition $u_h(x_0)=0$. Although this is + common practice, it is not necessarily a good idea, since we know that the + solutions of Laplace's equation are only in $H^1$, which does not allow for + the definition of point values because it is not a subset of the continuous + functions. Therefore, even though fixing one node is allowed for + discretitized functions, it is not for continuous functions, and one can + often see this in a resulting error spike at this point in the numerical + solution. + +\item Fixing the mean value over the domain to zero or any other value. This + is allowed on the continuous level, since $H^1(\Omega)\subset L^1(\Omega)$ + by Sobolev's inequality, and thus also on the continuous level since we + there only consider subsets of $H^1$. + +\item Fixing the mean value over the boundary of the domain to zero or any + other value. This is also allowed on the continuous level, since + $H^{1/2}(\partial\Omega)\subset L^1(\partial\Omega)$, again by Sobolev's + inequality. +\end{itemize} +We will choose the last possibility, since we want to demonstrate another +technique with it. + +While this describes the problem to be solved, we still have to figure out how +to implement it. Basically, except for the additional mean value constraint, +we have solved this problem several times, using Dirichlet boundary values, +and we only need to drop the treatment of Dirichlet boundary nodes. The use of +higher order mappings is also rather trivial and will be explained at the +various places where we use it; in almost all conceivable cases, you will only +consider the objects describing mappings as a black box which you need to +worry about, because their only uses seem to be to be passed to places deep +inside the library where functions know how to handle them (i.e. in the +\texttt{FEValues} classes and their descendents). + +The tricky point in this program is the use of the mean value +constraint. Fortunately, there is a class in the library which knows how to +handle such constraints, and we have used it quite often already, without +mentioning its generality. Note that if we assume that the boundary nodes are +spaced equally along the boundary, then the mean value constraint +\begin{gather*} + \int_{\partial \Omega} u(x) \; ds = 0 +\end{gather*} +can be written as +\begin{gather*} + \sum_{i\in\partial\Omega_h} u_i = 0, +\end{gather*} +where the sum shall run over all degree of freedom indices which are located +on the boundary of the computational domain. Let us denote by $i_0$ that index +on the boundary with the lowest number (or any other conveniently chosen +index), then the constraint can also be represented by +\begin{gather*} + u_{i_0} = \sum_{i\in\partial\Omega_h\backslash i_0} -u_i. +\end{gather*} +This, luckily, is exactly the form of constraints for which the +\texttt{ConstraintMatrix} class was designed. Note that we have used this +class in several previous examples for the representation of hanging nodes +constraints, which also have this form: there, the middle vertex shall have +the mean of the values of the adjacent vertices. In general, the +\texttt{ConstraintMatrix} class is designed to handle homogeneous constraints +of the form +\begin{gather*} + CU = 0 +\end{gather*} +where $C$ denotes a matrix, and $U$ the vector of nodal values. + +In this example, the mean value along the boundary allows just such a +representation, with $C$ being a matrix with just one row (i.e. there is only +one constraint). In the implementation, we will create a +\texttt{ConstraintMatrix} object, add one constraint (i.e. add another row to +the matrix) referring to the first boundary node $i_0$, and insert the weights +with which all the other nodes contribute, which in this example happens to be +just $-1$. + +Later, we will use this object to eliminate the first boundary node from the +linear system of equations, reducing it to one which has a solution without +the ambiguity of the constant shift value. One of the problems of the +implementation will be that the explicit elimination of this node results in a +number of additional elements in the matrix, of which we do not know in +advance where they are located and how many additional entries will be in each +of the rows of the matrix. We will show how we can use an intermediate object +to work around this problem. + +But now on for the implementation of the program solving this problem... + +\end{document} 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 8a2f794707..ab7f8c4758 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 @@ -117,6 +117,12 @@ At present, the following programs exist: solve equations, but rather compute the value of pi to high accuracy. + +
    Step 11
    +
    What's new: Solving a Laplace problem with + higher order mappings. Using strange constraints and + intermediate representations of sparsity patterns. +
    -- 2.39.5