Public Member Functions | Static Public Member Functions | Private Member Functions

Polynomials::Lobatto Class Reference

Inheritance diagram for Polynomials::Lobatto:
Inheritance graph
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List of all members.

Public Member Functions

 Lobatto (const unsigned int p=0)

Static Public Member Functions

static std::vector< Polynomial
< double > > 
generate_complete_basis (const unsigned int p)

Private Member Functions

std::vector< double > compute_coefficients (const unsigned int p)

Detailed Description

Lobatto polynomials of arbitrary degree on [0,1].

These polynomials are the integrated Legendre polynomials on [0,1]. The first two polynomials are the standard linear shape functions given by $l_0(x) = 1-x$ and $l_1(x) = x$. For $i\geq2$ we use the definition $l_i(x) = \frac{1}{\Vert L_{i-1}\Vert_2}\int_0^x L_{i-1}(t)\,dt$, where $L_i$ denotes the $i$-th Legendre polynomial on $[0,1]$. The Lobatto polynomials $l_0,\ldots,l_k$ form a complete basis of the polynomials space of degree $k$.

Calling the constructor with a given index k will generate the polynomial with index k. But only for $k\geq 1$ the index equals the degree of the polynomial. For k==0 also a polynomial of degree 1 is generated.

These polynomials are used for the construction of the shape functions of Nédélec elements of arbitrary order.

Author:
Markus Bürg, 2009

Definition at line 540 of file polynomial.h.


Constructor & Destructor Documentation

Polynomials::Lobatto::Lobatto ( const unsigned int  p = 0)

Constructor for polynomial of degree p. There is an exception for p==0, see the general documentation.


Member Function Documentation

static std::vector<Polynomial<double> > Polynomials::Lobatto::generate_complete_basis ( const unsigned int  p) [static]

Return the polynomials with index 0 up to degree. There is an exception for p==0, see the general documentation.

std::vector<double> Polynomials::Lobatto::compute_coefficients ( const unsigned int  p) [private]

Compute coefficients recursively.


The documentation for this class was generated from the following file:
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deal.II documentation generated on Mon May 21 2012 12:06:36 by doxygen 1.7.3