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 <a
+co-dimension one curves and surfaces $\Gamma$ embedded in $\Omega$ (read <a
href="https://en.wikipedia.org/wiki/Trace_operator">this wikipedia article</a>
for further details on the trace operator).
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
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:
(\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
src="https://www.dealii.org/images/steps/developer/step-60.C_interpolation.png"
alt=""> </p>
-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.
<h3>The testcase</h3>
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
$\Omega$.
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