LaplaceProblem<dim>::
estimate_smoothness (Vector<float> &smoothness_indicators) const
{
- // The first thing we need to do is to
- // define the Fourier vectors $\vec k$ for
- // which we want to compute Fourier
- // coefficients of the solution on each
- // cell. In 2d, we pick those vectors $\vec
- // k=(\pi i, \pi j)^T$ for which
- // $\sqrt{i^2+j^2}\le N$, with $i,j$
- // integers and $N$ being the maximal
- // polynomial degree we use for the finite
- // elements in this program. The 3d case is
- // handled analogously. 1d and dimensions
- // higher than 3 are not implemented, and
- // we guard our implementation by making
- // sure that we receive an exception in
- // case someone tries to compile the
- // program for any of these dimensions.
+ // The first thing we need to do is
+ // to define the Fourier vectors
+ // ${\bf k}$ for which we want to
+ // compute Fourier coefficients of
+ // the solution on each cell. In
+ // 2d, we pick those vectors ${\bf
+ // k}=(\pi i, \pi j)^T$ for which
+ // $\sqrt{i^2+j^2}\le N$, with
+ // $i,j$ integers and $N$ being the
+ // maximal polynomial degree we use
+ // for the finite elements in this
+ // program. The 3d case is handled
+ // analogously. 1d and dimensions
+ // higher than 3 are not
+ // implemented, and we guard our
+ // implementation by making sure
+ // that we receive an exception in
+ // case someone tries to compile
+ // the program for any of these
+ // dimensions.
//
- // We exclude $\vec k=0$ to avoid problems
- // computing $|\vec k|^{-mu}$ and $\ln
- // |\vec k|$. The other vectors are stored
+ // We exclude ${\bf k}=0$ to avoid problems
+ // computing $|{\bf k}|^{-mu}$ and $\ln
+ // |{\bf k}|$. The other vectors are stored
// in the field <code>k_vectors</code>. In
// addition, we store the square of the
// magnitude of each of these vectors (up
// Next, we need to assemble the matrices
// that do the Fourier transforms for each
// of the finite elements we deal with,
- // i.e. the matrices ${\cal F}_{\vec k,j}$
+ // i.e. the matrices ${\cal F}_{{\bf k},j}$
// defined in the introduction. We have to
// do that for each of the finite elements
// in use. Note that these matrices are
std::vector<Table<2,std::complex<double> > >
fourier_transform_matrices (fe_collection.size());
- // In order to compute them, we of course
- // can't perform the Fourier transform
- // analytically, but have to approximate it
- // using quadrature. To this end, we use a
- // quadrature formula that is obtained by
- // iterating a 2-point Gauss formula as
- // many times as the maximal exponent we
- // use for the term $e^{i\vec k\cdot \vec
+ // In order to compute them, we of
+ // course can't perform the Fourier
+ // transform analytically, but have
+ // to approximate it using
+ // quadrature. To this end, we use
+ // a quadrature formula that is
+ // obtained by iterating a 2-point
+ // Gauss formula as many times as
+ // the maximal exponent we use for
+ // the term $e^{i{\bf k}\cdot{\bf
// x}$:
QGauss<1> base_quadrature (2);
QIterated<dim> quadrature (base_quadrature, N);
// respective matrix ${\cal F}$ to the
// right size, and integrate each entry of
// the matrix numerically as ${\cal
- // F}_{\vec k,j}=\sum_q e^{i\vec k\cdot\vec
- // x}\varphi_j(\vec x_q) w_q$, where $x_q$
+ // F}_{{\bf k},j}=\sum_q e^{i{\bf k}\cdot
+ // {\bf x}\varphi_j({\bf x}_q)
+ // w_q$, where $x_q$
// are the quadrature points and $w_q$ are
// the quadrature weights. Note that the
// imaginary unit $i=\sqrt{-1}$ is obtained
// only fit our exponential decay of
// Fourier coefficients to the largest
// coefficients for each possible value
- // of $|\vec k|$. To this end, we
+ // of $|{\bf k}|$. To this end, we
// create a map that for each magnitude
- // $|\vec k|$ stores the largest $|\hat
- // U_{\vec k}|$ found so far, i.e. we
+ // $|{\bf k}|$ stores the largest $|\hat
+ // U_{{\bf k}}|$ found so far, i.e. we
// overwrite the existing value (or add
// it to the map) if no value for the
- // current $|\vec k|$ exists yet, or if
+ // current $|{\bf k}|$ exists yet, or if
// the current value is larger than the
// previously stored one:
std::map<unsigned int, double> k_to_max_U_map;
// of vectors as integers, since this
// way we do not have to deal with
// round-off-sized differences between
- // different values of $|\vec k|$.
+ // different values of $|{\bf k}|$.
// As the final task, we have to
// calculate the various contributions
// to the formula for $\mu$. We'll only
// take those Fourier coefficients with
// the largest magnitude for a given
- // value of $|\vec k|$ as explained
+ // value of $|{\bf k}|$ as explained
// above:
double sum_1 = 0,
sum_ln_k = 0,
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