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Reference documentation for deal.II version 9.0.0
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Reference documentation for deal.II version 9.0.0
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#include <deal.II/base/polynomials_bdm.h>
Public Member Functions | |
| PolynomialsBDM (const unsigned int k) | |
| void | compute (const Point< dim > &unit_point, std::vector< Tensor< 1, dim > > &values, std::vector< Tensor< 2, dim > > &grads, std::vector< Tensor< 3, dim > > &grad_grads, std::vector< Tensor< 4, dim > > &third_derivatives, std::vector< Tensor< 5, dim > > &fourth_derivatives) const |
| unsigned int | n () const |
| unsigned int | degree () const |
| std::string | name () const |
Static Public Member Functions | |
| static unsigned int | compute_n_pols (unsigned int degree) |
Private Attributes | |
| const PolynomialSpace< dim > | polynomial_space |
| std::vector< Polynomials::Polynomial< double > > | monomials |
| unsigned int | n_pols |
| Threads::Mutex | mutex |
| std::vector< double > | p_values |
| std::vector< Tensor< 1, dim > > | p_grads |
| std::vector< Tensor< 2, dim > > | p_grad_grads |
| std::vector< Tensor< 3, dim > > | p_third_derivatives |
| std::vector< Tensor< 4, dim > > | p_fourth_derivatives |
This class implements the Hdiv-conforming, vector-valued Brezzi-Douglas-Marini ( BDM ) polynomials described in Brezzi and Fortin's Mixed and Hybrid Finite Element Methods (refer to pages 119 - 124).
The BDM polynomial space contain the entire \((P_{k})^{n}\) space (constructed with PolynomialSpace Legendre polynomials) as well as part of \((P_{k+1})^{n}\) (ie. \((P_{k})^{n} \subset BDM_{k} \subset (P_{k+1})^{n}\)). Furthermore, \(BDM_{k}\) elements are designed so that \(\nabla \cdot q \in P_{k-1} (K)\) and \(q \cdot n |_{e_{i}} \in P_{k}(e_{i})\). More details of two and three dimensional \(BDM_{k}\) elements are given below.
\( BDM_{k} = \{\mathbf{q} | \mathbf{q} = p_{k} (x,y) + r \; \text{curl} (x^{k+1}y) + s \; \text{curl} (xy^{k+1}), p_{k} \in (P_{k})^{2} \}\).
Note: the curl of a scalar function is given by \(\text{curl}(f(x,y)) = \begin{pmatrix} f_{y}(x,y) \\ -f_{x}(x,y) \end{pmatrix}\).
The basis used to construct the \(BDM_{1}\) shape functions is
\begin{align*} \phi_0 = \begin{pmatrix} 1 \\ 0 \end{pmatrix}, \phi_1 = \begin{pmatrix} -\sqrt{3}+2\sqrt{3}x \\ 0 \end{pmatrix}, \phi_2 = \begin{pmatrix} -\sqrt{3}+2\sqrt{3}y \\ 0 \end{pmatrix}, \phi_3 = \begin{pmatrix} 0 \\ 1 \end{pmatrix}, \phi_4 = \begin{pmatrix} 0 \\ -\sqrt{3}+2\sqrt{3}x \end{pmatrix}, \phi_5 = \begin{pmatrix} 0 \\ -\sqrt{3}+2\sqrt{3}y \end{pmatrix}, \phi_6 = \begin{pmatrix} x^2 \\ -2xy \end{pmatrix}, \phi_7 = \begin{pmatrix} 2xy \\ -y^2 \end{pmatrix}. \end{align*}
The dimension of the \(BDM_{k}\) space is \((k+1)(k+2)+2\), with \(k+1\) unknowns per edge and \(k(k-1)\) interior unknowns.
\( BDM_{k} = \{\mathbf{q} | \mathbf{q} = p_{k} (x,y,z) + \sum_{i=0}^{k} ( r_{i} \; \text{curl} \begin{pmatrix} 0\\0\\xy^{i+1}z^{k-i} \end{pmatrix} + s_{i} \; \text{curl} \begin{pmatrix} yz^{i+1}x^{k-i}\\0\\0 \end{pmatrix} + t_{i} \; \text{curl} \begin{pmatrix}0\\zx^{i+1}y^{k-i}\\0\end{pmatrix}) , p_{k} \in (P_{k})^{3} \}\).
Note: the 3D description of \(BDM_{k}\) is not unique. See Mixed and Hybrid Finite Element Methods page 122 for an alternative definition.
The dimension of the \(BDM_{k}\) space is \(\dfrac{(k+1)(k+2)(k+3)}{2}+3(k+1)\), with \(\dfrac{(k+1)(k+2)}{2}\) unknowns per face and \(\dfrac{(k-1)k(k+1)}{2}\) interior unknowns.
Definition at line 100 of file polynomials_bdm.h.
| PolynomialsBDM< dim >::PolynomialsBDM | ( | const unsigned int | k | ) |
Constructor. Creates all basis functions for BDM polynomials of given degree.
Definition at line 28 of file polynomials_bdm.cc.
| void PolynomialsBDM< dim >::compute | ( | const Point< dim > & | unit_point, |
| std::vector< Tensor< 1, dim > > & | values, | ||
| std::vector< Tensor< 2, dim > > & | grads, | ||
| std::vector< Tensor< 3, dim > > & | grad_grads, | ||
| std::vector< Tensor< 4, dim > > & | third_derivatives, | ||
| std::vector< Tensor< 5, dim > > & | fourth_derivatives | ||
| ) | const |
Compute the value and the first and second derivatives of each BDM polynomial at unit_point.
The size of the vectors must either be zero or equal n(). In the first case, the function will not compute these values.
If you need values or derivatives of all tensor product polynomials then use this function, rather than using any of the compute_value, compute_grad or compute_grad_grad functions, see below, in a loop over all tensor product polynomials.
Definition at line 55 of file polynomials_bdm.cc.
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inline |
Return the number of BDM polynomials.
Definition at line 207 of file polynomials_bdm.h.
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inline |
Return the degree of the BDM space, which is one less than the highest polynomial degree.
Definition at line 215 of file polynomials_bdm.h.
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Return the name of the space, which is BDM.
Definition at line 223 of file polynomials_bdm.h.
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static |
Return the number of polynomials in the space BDM(degree) without requiring to build an object of PolynomialsBDM. This is required by the FiniteElement classes.
Definition at line 408 of file polynomials_bdm.cc.
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private |
An object representing the polynomial space used here. The constructor fills this with the monomial basis.
Definition at line 160 of file polynomials_bdm.h.
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private |
Storage for monomials. In 2D, this is just the polynomial of order k. In 3D, we need all polynomials from degree zero to k.
Definition at line 166 of file polynomials_bdm.h.
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private |
Number of BDM polynomials.
Definition at line 171 of file polynomials_bdm.h.
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mutableprivate |
A mutex that guards the following scratch arrays.
Definition at line 176 of file polynomials_bdm.h.
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mutableprivate |
Auxiliary memory.
Definition at line 181 of file polynomials_bdm.h.
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mutableprivate |
Auxiliary memory.
Definition at line 186 of file polynomials_bdm.h.
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mutableprivate |
Auxiliary memory.
Definition at line 191 of file polynomials_bdm.h.
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mutableprivate |
Auxiliary memory.
Definition at line 196 of file polynomials_bdm.h.
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mutableprivate |
Auxiliary memory.
Definition at line 201 of file polynomials_bdm.h.
1.8.14