integrate (4rheolef)

Integrate an expression over a domain by using a quadrature formulae. There are three main usages of the integrate function, depending upon the type of the expression. (i) When the expression is a numerical one, it leads to a numerical value. (ii) When the expression involves a symbolic test-function see test(2), the result is a linear form, represented by the field class. (iii) When the expression involves both symbolic trial- and test-functions see test(2), the result is a bilinear form, represented by the field class.

 template <class T, class M, class Expr>
 T integrate (const geo_basic<T,M>& omega, Expr expr,
   quadrature_option_type qopt)

 template <class T, class M, class Expr>
 field integrate (const geo_basic<T,M>& omega, VFExpr expr,
   quadrature_option_type qopt)

 template <class T, class M, class Expr>
 form integrate (const geo_basic<T,M>& omega, VFExpr expr,
   form_option_type fopt)

  Float f (const point& x);
  ...
  quadrature_option_type qopt;
  Float value = integrate (omega, f, qopt);
  field lh = integrate (omega, f*v, qopt);

The last argument specifies the quadrature formulae used for the computation of the integral. The expression can be any function, classs-function or any linear or nonlinear field expression see field(2).

In the case of a linear form, the domain is optional: by default it is the full domain definition of the test function.

  field l1h = integrate (f*v, qopt);

When the integration is perfomed on a subdomain, this subdomain simply replace the first argument and a domain name could also be used:

  field l2h = integrate (omega["boundary"], f*v, qopt);
  field l3h = integrate ("boundary", f*v, qopt);

The quadrature formulae is required, except when a test and/or trial function is provided in the expression to integrate. In that case, the quadrature formulae is deduced from the space containing the test (or trial) function. When a test function is suppied, let k be its polynomial degree. Then the default quadrature is choosen to be exact at least for 2*k+1 polynoms. When both a test and trial functions are suppied, let k1 and k2 be their polynomial degrees. Then the default quadrature is choosen to be exact at least for k1+k2+1 polynoms. Also, when the expression is a constant, the quadrature function is optional: in that case, the constant is also optional and the following call:

  Float meas = integrate (omega);

is valid and returns the measure of the domain.

test(2), test(2), field(2)