NAME

       basis_option - options of the finite element space

DESCRIPTION

       The  basis_option  class  is  used  to set options for the space constructor of the finite
       element space (see space(2)).  These options are directly transmitted to the  basis  class
       when  computing  the  polynomial  basis  (see  basis(2) and basis(1)).  There are two main
       options: Lagrange nodes and raw polynomial basis.

LAGRANGE NODES

       There are two possible node sets.  The equispaced, refers to an equispaced set of Lagrange
       node during interpolation.  The warburton one, refers to a non-equispaced optimized set of
       Lagrange nodes, suitable for high order polynomials (see Warburton, 2006, J. Eng.  Math.).
       With  this  choice,  the  interpolation  error  is  dramatically  decreased for high order
       polynomials.  The default is the equispaced node set, for backward compatibility purpose.

CONTINUOUS FEATURE

       Some elements are possibly continuous.  For instance, Pk elements family,  with  k  >=  1,
       possibly  have  continuous  interelements reconnections and thus, could define a piecewise
       polynomial and globaly continuous function.  These basis families  can  be  used  both  as
       continuous  or  discontinuous  elements.   Nevertheless,  this  choice should be done when
       building the basis: the information is transmitted during basis  construction  by  sending
       the  basis_option  class and setting appropriately this feature with the set_continuous(c)
       member function.

RAW POLYNOMIAL BASIS

       The raw (or initial) basis is used for building the Lagrange basis,  via  the  Vandermonde
       matrix  and  its  inverse.   Its choice do not change the definition of the FEM basis, but
       only the way it is builded.  There are three possible raw basis.  The  monomial  basis  is
       the  simplest  choice,  suitable  for  low order polynomials (less than five).  For higher
       order polynomial, the Vandermonde matrix becomes ill-conditionned and its inverse leads to
       errors  in double precision.  The dubiner basis (see Dubiner, 1991 J. Sci. Comput.)  leads
       to better condition number.  The bernstein basis could also be used as an alternative  raw
       basis.  The default is the dubiner raw basis.

STRING OPTION REPRESENTATION

       The is_option(string) and set(string) members leads to easy setting of combined options at
       run time.  By this way, options can be specified, together with  basis  basename,  on  the
       command line or from a file.

       The  stamp()  member  returns  an unique string.  This string is used for specifying basis
       options, e.g. on command line or in files.  This string is empty when all options are  set
       to  default  values.  Otherwise, it returns a comma separated list of options, enclosed by
       braces, specifying only non-default options.  For instance, combining Warburton  node  set
       and  Dubiner  raw  polynomials leads to "{warburton}".  Also, combining Warburton node set
       and Bernstein raw polynomials leads to "{warburton,bernstein}".

       Note that the continuous or discontinuous feature is not specified by the stamp()  string:
       it  will  be  specified  into  the  basis  basename,  by  appending  a  "d"  letter, as in
       "P6d{warburton}".

NOTES

       There are two distinct kind of polynomial basis: the raw basis and the finite element one.
       (see  basis(2)  and  basis(1)).   When  using  the Pk Lagrange finite element basis, these
       options are used to transform from one raw (initial) polynomial basis to the Lagrange one,
       based  on  a node set.  When using an alternative finite element basis, e.g.  the spectral
       Sk or the Bernstein Bk, these options do not have any effect.

IMPLEMENTATION

       class basis_option {
       public:
       // typedefs:

         typedef size_t size_type;

         typedef enum {
           equispaced = 0,
           warburton  = 1,
           fekete     = 2,
           max_node   = 3
         } node_type; // update also node_name[]

         typedef enum {
           monomial           = 0,
           bernstein          = 1,
           dubiner            = 2,
           max_raw_polynomial = 3
         } raw_polynomial_type; // update also raw_polynomial_name[]

         static const node_type           default_node           = basis_option::equispaced;
         static const raw_polynomial_type default_raw_polynomial = basis_option::dubiner;

       // allocators:

         basis_option(
           node_type           nt = default_node,
           raw_polynomial_type pt = default_raw_polynomial);

         basis_option (const basis_option& sopt);
         basis_option& operator= (const basis_option& sopt);

       // accessors & modifiers:

         node_type           get_node() const;
         raw_polynomial_type get_raw_polynomial() const;
         std::string         get_node_name() const;
         std::string         get_raw_polynomial_name() const;
         bool                is_continuous() const;
         bool                is_discontinuous() const;

         void set_node           (node_type type);
         void set_raw_polynomial (raw_polynomial_type type);
         void set                (std::string option_name);
         void set_node           (std::string node_name);
         void set_raw_polynomial (std::string raw_polynomial_name);
         void set_continuous     (bool c = true);
         void set_discontinuous  (bool c = false);

         bool is_node_name           (std::string name) const;
         bool is_raw_polynomial_name (std::string name) const;
         bool is_option_name         (std::string name) const;

         std::string stamp() const;

       // data:
       protected:
         node_type            _node;
         raw_polynomial_type  _poly;
         bool                 _is_continuous;
       };

SEE ALSO

       space(2), basis(2), basis(1), basis(2), basis(1)

COPYRIGHT

       Copyright (C) 2000-2018 Pierre Saramito <Pierre.Saramito@imag.fr> GPLv3+: GNU GPL  version
       3  or  later  <http://gnu.org/licenses/gpl.html>.   This is free software: you are free to
       change and redistribute it.  There is NO WARRANTY, to the extent permitted by law.