* respectively. Furthermore, the extension of these polynomials to higher
* degrees $n>3$ is constructed by adding additional nodes inside the unit
* interval, again ensuring better conditioning. The nodes are computed as
- * the roots of the Jacobi polynomials for $\alpha=\beta=2$, which are
- * orthogonal against the generating function $x^2(1-x)^2$ with the Hermite
+ * the roots of the Jacobi polynomials for $\alpha=\beta=4$, which are
+ * orthogonal against the square of the generating function $x^2(1-x)^2$
+ * with the Hermite
* property. Then, these polynomials are constructed in the usual way as
- * Lagrange polynomials with double roots at $x=0$ and $x=1$. For example at
- * $n=4$, all of $p_0, p_1, p_3, p_4$ get an additional root at $x=0.5$
+ * Lagrange polynomials with double roots at $x=0$ and $x=1$. For example
+ * with $n=4$, all of $p_0, p_1, p_3, p_4$ get an additional root at $x=0.5$
* through the factor $(x-0.5)$. In summary, this basis is dominated by
* nodal contributions, but it is not a nodal one because the second and
* second to last polynomials that are non-nodal, and due to the presence of
* set such that the sum of all polynomials with unit weight represents the
* constant function 1, similarly to Lagrange polynomials.
*
- * The basis only contains Hermite information at <code>degree>=3</code>,
+ * The basis only contains Hermite information for <code>degree>=3</code>,
* but it is also implemented for degrees between 0 and two. For the linear
* case, the usual hat functions are implemented, whereas the polynomials
* for <code>degree=2</code> are $p_0(x)=(1-x)^2$, $p_1(x)=2x(x-1)$, and
- * $p_2(x)=x^2$, in accordance with the construction principle for degree 3
- * that allows a non-zero of $p_0$ and $p_2$.
+ * $p_2(x)=x^2$, in accordance with the construction principle for degree 3.
*
* These two relaxations improve the condition number of the mass matrix
* (i.e., interpolation) significantly, as can be seen from the following
* <tr>
* <th>n=5</th>
* <th>1.875e+04</th>
- * <th>19.37</th>
+ * <th>15.99</th>
* </tr>
* <tr>
* <th>n=6</th>
* <th>6.033e+04</th>
- * <th>18.99</th>
+ * <th>16.34</th>
* </tr>
* <tr>
* <th>n=10</th>
* <th>9.756e+05</th>
- * <th>25.65</th>
+ * <th>20.70</th>
* </tr>
* <tr>
* <th>n=15</th>
* <th>9.431e+06</th>
- * <th>36.47</th>
+ * <th>27.91</th>
* </tr>
* <tr>
* <th>n=25</th>
* <th>2.220e+08</th>
- * <th>62.28</th>
+ * <th>43.54</th>
* </tr>
* <tr>
* <th>n=35</th>
* <th>2.109e+09</th>
- * <th>91.50</th>
+ * <th>59.51</th>
* </tr>
* </table>
*
//
// | x 0 x x x x x |
// | 0 x x x . . . x x x |
- // | x x x x x x x |
- // | x x x x x x x |
+ // | x x x 0 0 x x |
+ // | x x 0 x 0 x x |
// | . . . |
// M = | . . . |
// | . . . |
- // | x x x x x x x |
+ // | x x 0 0 x x x |
// | x x x x . . . x x 0 |
// | x x x x x 0 x |
//
// We find the inner points as the zeros of the Jacobi polynomials
- // with alpha = beta = 2 which is the polynomial with the kernel
- // (1-x)^2 (1+x)^2, the two polynomials achieving zero value and zero
- // derivative at the boundary.
+ // with alpha = beta = 4 which is the polynomial with the kernel
+ // (1-x)^4 (1+x)^4. Since polynomials (1-x)^2 (1+x)^2 are contained
+ // in every interior polynomial (bubble function), their product
+ // leads us to the orthogonality condition of the Jacobi(4,4)
+ // polynomials.
std::vector<double> jacobi_roots =
- jacobi_polynomial_roots<double>(degree - 3, 2, 2);
+ jacobi_polynomial_roots<double>(degree - 3, 4, 4);
AssertDimension(jacobi_roots.size(), degree - 3);
// iteration from variable support point N with secant method
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projects / dealii.git / commitdiff