From: Luca Heltai Date: Mon, 7 May 2018 20:32:59 +0000 (+0200) Subject: Fixes to JP comments X-Git-Tag: v9.0.0-rc4~7^2~1 X-Git-Url: https://gitweb.dealii.org/cgi-bin/gitweb.cgi?a=commitdiff_plain;h=ced9429d807e47897e1d4e24558f47c40ef22bfd;p=dealii.git Fixes to JP comments --- diff --git a/examples/step-60/doc/intro.dox b/examples/step-60/doc/intro.dox index 536843b455..6e9dd7cdda 100644 --- a/examples/step-60/doc/intro.dox +++ b/examples/step-60/doc/intro.dox @@ -41,7 +41,7 @@ The same is true, with a less regular range space (namely $H^{1/2}(\Gamma)$), when the dimension of $\Gamma$ is one less with respect to $\Omega$, and $\Gamma$ does not have a boundary. In this second case, the operator $\gamma$ is also known as the *trace* operator, and it is well defined for Lipschitz -co-dimension one curves and surfaces $\Gamma$ embedded in $\Omega$ (read on this wikipedia article for further details on the trace operator). @@ -52,8 +52,8 @@ and three dimensions. In this tutorial program we're not interested in further details on $\gamma$: we take the extension $\gamma$ for granted, assuming that the dimension of the -embedded domain (`dim`) is always smaller by one or equal w.r.t. the dimension -of the embedding domain $\Omega$ (`spacedim`). +embedded domain (`dim`) is always smaller by one or equal with respect to the +dimension of the embedding domain $\Omega$ (`spacedim`). We are going to solve the following differential problem: given a sufficiently regular function $g$ on $\Gamma$, find the solution $u$ to @@ -218,7 +218,7 @@ information on two non-matching grids. In particular, the integral that appears in the computation of a single entry of $C$, is computed on $\Gamma$. As usual in finite elements we split this integral into contributions from all cells of the triangulation used to -discretize $\Gamma$, we tranform the integral on $K$ to an integral on the +discretize $\Gamma$, we transform the integral on $K$ to an integral on the reference element $\hat K$, where $F_{K}$ is the mapping from $\hat K$ to $K$, and compute the integral on $\hat K$ using a quadrature formula: @@ -242,7 +242,7 @@ taken (as shown in the picture below): (\hat x_i)$, where $x_i$ is one of the quadrature points used for the integral on $K \subseteq \Gamma$. -- Find the cell of $\Omega$ in which $y_i$ lies, say it's $T$. +- Find the cell of $\Omega$ in which $y_i$ lies. We shall call this element $T$. - To evaluate the basis function use the inverse of the mapping $G_T$ that transforms the reference element $\hat T$ into the element $T$: $v_j(y_i) = \hat @@ -252,33 +252,33 @@ v_j \circ G^{-1}_{T} (y_i)$. src="https://www.dealii.org/images/steps/developer/step-60.C_interpolation.png" alt="">

-The three steps above can be computed using by calling, in turn: +The three steps above can be computed by calling, in turn, -- GridTools::find_active_cell_around_point, followed by +- GridTools::find_active_cell_around_point(), followed by -- Mapping::tranform_real_to_unit_cell +- Mapping::transform_real_to_unit_cell(). We then - construct a custom Quadrature formula, containing the point in the reference - cell + cell and then -- construct an FEValues object, with the given quadrature formula, and initialized - with the cell obtained in the first step. +- construct an FEValues object, with the given quadrature formula, and + initialized with the cell obtained in the first step. -This is what the deal.II function VectorTools::point_value does when evaluating a -finite element field (not just a single shape function) at an arbitrary point; but -this is would be inefficient in this case. +This is what the deal.II function VectorTools::point_value() does when +evaluating a finite element field (not just a single shape function) at an +arbitrary point; but this would be inefficient in this case. -A better solution is to use a convenient wrapper to perform the first three steps -on a collection of points: GridTools::compute_point_locations. If one is +A better solution is to use a convenient wrapper to perform the first three +steps on a collection of points: GridTools::compute_point_locations(). If one is actually interested in computing the full coupling matrix, then it is possible -to call the method NonMatching::create_coupling_mass_matrix, that performs the +to call the method NonMatching::create_coupling_mass_matrix(), that performs the above steps in an efficient way, reusing all possible data structures, and -gathering expensive steps together. This is the function we'll be using later -in this tutorial. +gathering expensive steps together. This is the function we'll be using later in +this tutorial. We solve the final saddle point problem by an iterative solver, applied to the -Schur complement $S$ (whose construction is described, for example, in step-20), and -we construct $S$ using LinearOperator classes. +Schur complement $S$ (whose construction is described, for example, in step-20), +and we construct $S$ using LinearOperator classes.

The testcase

@@ -307,7 +307,7 @@ reference interval $[0,1]$ to a curve in $\Omega$. If the curve is closed, then the results will be similar to running the same problem on a grid whose boundary is $\Gamma$. The program will happily run also -with a non closed $\Gamma$, although in those cases, the mathematical +with a non-closed $\Gamma$, although in those cases the mathematical formulation of the problem is more difficult, since $\Gamma$ will have a boundary by itself that has co-dimension two with respect to the domain $\Omega$. diff --git a/examples/step-60/step-60.cc b/examples/step-60/step-60.cc index a58d339990..3f03315b38 100644 --- a/examples/step-60/step-60.cc +++ b/examples/step-60/step-60.cc @@ -208,8 +208,6 @@ namespace Step60 { public: - // @sect3{Parameters} - // // The `Parameters` class is derived from ParameterAcceptor. This allows us // to use the ParameterAcceptor::add_parameter() method in its constructor. // @@ -623,7 +621,8 @@ namespace Step60 } - + // @sect3{Set up} + // // The function `DistributedLagrangeProblem::setup_grids_and_dofs()` is used // to set up the finite element spaces. Notice how `std_cxx14::make_unique` is // used to create objects wrapped inside `std::unique_ptr` objects. @@ -932,6 +931,8 @@ namespace Step60 coupling_matrix.reinit(coupling_sparsity); } + // @sect3{Assembly} + // // The following function creates the matrices: as noted before computing the // stiffness matrix and the rhs is a standard procedure. template @@ -969,6 +970,8 @@ namespace Step60 } } + // @sect3{Solve} + // // All parts have been assembled: we solve the system // using the Schur complement method template