### 1. Mean value of a function

Let $f$ a bounded function on domain $\Omega$. You can evaluate the mean value of a function thanks to the `mean()` function :

$\bar{f}=\frac{1}{|\Omega|}\int_\Omega f=\frac{1}{\int_\Omega 1}\int_\Omega f$

#### 1.1. Interface

``  mean( _range, _expr, _quad, _geomap );``

Required parameters:

• `_range` = domain of integration

• `_expr` = mesurable function

Optional parameters:

• `_quad` = quadrature to use.

• Default = `integer` corresponding to the quadrature order, see integrate.

• `_geomap` = type of geometric mapping.

• Default = `GEOMAP_OPT`

#### 1.2. Example

Stokes example using `mean`
``````int main(int argc, char**argv )
{
Environment env( _argc=argc, _argv=argv,
_author="Feel++ Consortium",
_email="feelpp-devel@feelpp.org"));

// create the mesh
auto mesh = loadMesh(_mesh=new Mesh<Simplex< 2 > > );

// function space
auto Vh = THch<2>( mesh );

// element U=(u,p) in Vh
auto U = Vh->element();
auto u = U.element<0>();
auto p = U.element<1>();

// left hand side
auto a = form2( _trial=Vh, _test=Vh );
a = integrate(_range=elements(mesh),

a+= integrate(_range=elements(mesh),
_expr=-div(u)*idt(p)-divt(u)*id(p));

auto syms = symbols<2>();
auto u1 = parse( option(_name="functions.alpha").as<std::string>(), syms );
auto u2 = parse( option(_name="functions.beta").as<std::string>(), syms );
matrix u_exact = matrix(2,1);
u_exact = u1,u2;
auto p_exact = parse( option(_name="functions.gamma").as<std::string>(), syms );
auto f = -laplacian( u_exact, syms ) + grad( p_exact, syms ).transpose();
LOG(INFO) << "rhs : " << f;

// right hand side
auto l = form1( _test=Vh );
l = integrate(_range=elements(mesh),
_expr=trans(expr<2,1,5>( f, syms ))*id(u));
a+=on(_range=boundaryfaces(mesh), _rhs=l, _element=u,
_expr=expr<2,1,5>(u_exact,syms));

// solve a(u,v)=l(v)
a.solve(_rhs=l,_solution=U);

double mean_p = mean(_range=elements(mesh),_expr=idv(p))(0,0);
double mean_p_exact = mean(_range=elements(mesh),_expr=expr(p_exact,syms))(0,0);
double l2error_u = normL2( _range=elements(mesh), _expr=idv(u)-expr<2,1,5>( u_exact, syms ) );
double l2error_p = normL2( _range=elements(mesh), _expr=idv(p)-mean_p-(expr( p_exact, syms )-mean_p_exact) );
LOG(INFO) << "L2 error norm u: " << l2error_u;
LOG(INFO) << "L2 error norm p: " << l2error_p;

// save results
auto e = exporter( _mesh=mesh );
e->add( "u", u );
e->add( "p", p );
e->save();
}``````