* reasons, we will perform the calculation of these matrices and coefficients
* only on the reference cell $\hat K$. We only have to calculate the
* transformation matrices once this way. However, results are only applicable
- * if mapping from the reference cell to the actual cell is linear. We use
+ * if the mapping from the reference cell to the actual cell is affine. We use
* the class FESeries::Legendre to determine all coefficients $a_k$.
*
* A function is analytic, i.e., representable by a power series, if and only
* coefficients of some functions may have a repeating pattern of zero
* coefficients (i.e. for functions that are locally symmetric or
* antisymmetric about the midpoint of the element in any coordinate
- * direction). Thus this parameters allows to ingore small (in absolute
+ * direction). Thus this parameters allows to ignore small (in absolute
* value) coefficients within the linear regression fit. In case there are
* less than two nonzero coefficients, the returned value for this cell will
* be $\sigma=\infty$.
* Note that Legendre coefficients of some functions may have a repeating
* pattern of zero coefficients (i.e. for functions that are locally
* symmetric or antisymmetric about the midpoint of the element in any
- * coordinate direction). Thus this parameters allows to ingore small (in
+ * coordinate direction). Thus this parameters allows to ignore small (in
* absolute value) coefficients within the linear regression fit. In case
* there are less than two nonzero coefficients for a coordinate direction,
* this direction will be skipped. If all coefficients are zero, the
* FESeries::Fourier to determine all coefficients $a_{\bf k}$.
*
* If the finite element approximation on cell $K$ is part of the Hilbert
- * space $H^s(K)$, then the following integral must exit for both finite
+ * space $H^s(K)$, then the following integral must exist for both the finite
* element and spectral representation of our solution
* @f{eqnarray*}
* \| \nabla^s u_h({\bf x}) \|_{L^2(K)}^2 &=&
* {-\left(s + \frac d2 + \epsilon \right)} \right)
* @f]
*
- * The next step is that we have to estimate how fast these coefficients
+ * The next step is to estimate how fast these coefficients
* decay with $\|{\bf k}\|_2$. Thus, we perform a least-squares fit
* @f[
* \min_{\alpha,\sigma}
* A series expansion object @p fe_fourier has to be supplied, which needs
* to be constructed with the same FECollection object as the @p dof_handler.
*
- * The parameter @p smallest_abs_coefficient allows to ingore small (in
+ * The parameter @p smallest_abs_coefficient allows to ignore small (in
* absolute value) coefficients within the linear regression fit. In case
* there are less than two nonzero coefficients for a coordinate direction,
* this direction will be skipped. If all coefficients are zero, the
* provided to this function. Note that its size is $p+1$, where $p$ is the
* polynomial degree of the FE basis on a given element. The default
* implementation will use all Fourier coefficients in each coordinate
- * direction, i.e. set all elements of the vector to `true`.
+ * direction, i.e., set all the elements of the vector to `true`.
*
* For a provided solution vector @p solution defined on a DoFHandler
* @p dof_handler, this function returns a vector @p smoothness_indicators
* A series expansion object @p fe_fourier has to be supplied, which needs
* to be constructed with the same FECollection object as the @p dof_handler.
*
- * The parameter @p smallest_abs_coefficient allows to ingore small (in
+ * The parameter @p smallest_abs_coefficient allows to ignore small (in
* absolute value) coefficients within the linear regression fit. In case
- * there are less than two nonzero coefficients for a coordinate direction,
+ * there are fewer than two nonzero coefficients for a coordinate direction,
* this direction will be skipped. If all coefficients are zero, the
* returned value for this cell will be $\sigma=\infty$.
*
projects / dealii.git / commitdiff