1. Norms

Let \(f\) a bounded function on domain \(\Omega\).

1.1. L2 norms

Let \(f \in L^2(\Omega)\) you can evaluate the \(L^2\) norm using the normL2() function:

\[\parallel f\parallel_{L^2(\Omega)}=\sqrt{\int_\Omega |f|^2}\]

1.1.1. Interface

  normL2( _range, _expr, _quad, _geomap );

or squared norm:

  normL2Squared( _range, _expr, _quad, _geomap );

Required parameters:

  • _range = domain of integration

  • _expr = mesurable function

Optional parameters:

  • _quad = quadrature to use.

    • Default = _Q<integer>()

  • _geomap = type of geometric mapping.

    • Default = GEOMAP_OPT

1.1.2. Example

From doc/manual/laplacian/laplacian.cpp

  double L2error =normL2( _range=elements( mesh ),
                          _expr=( idv( u )-g ) );

From doc/manual/stokes/stokes.cpp

Stokes example using mean
int main(int argc, char**argv )
{
    Environment env( _argc=argc, _argv=argv,
                     _about=about(_name="mystokes",
                                  _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),
                  _expr=trace(gradt(u)*trans(grad(u))) );

    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();
}

1.2. H^1 norm

In the same idea, you can evaluate the H1 norm or semi norm, for any function \(f \in H^1(\Omega)\):

\[\begin{aligned} \parallel f \parallel_{H^1(\Omega)}&=\sqrt{\int_\Omega |f|^2+|\nabla f|^2}\\ &=\sqrt{\int_\Omega |f|^2+\nabla f * \nabla f^T}\\ |f|_{H^1(\Omega)}&=\sqrt{\int_\Omega |\nabla f|^2} \end{aligned}\]

where \(*\) is the scalar product \(\cdot\) when \(f\) is a scalar field and the frobenius scalar product \(:\) when \(f\) is a vector field.

1.2.1. Interface

  normH1( _range, _expr, _grad_expr, _quad, _geomap );

or semi norm:

  normSemiH1( _range, _grad_expr, _quad, _geomap );

Required parameters:

  • _range = domain of integration

  • _expr = mesurable function

  • _grad_expr = gradient of function (Row vector!)

Optional parameters:

  • _quad = quadrature to use.

    • Default = _Q<integer>()

  • _geomap = type of geometric mapping.

    • Default = GEOMAP_OPT

normH1() returns a float containing the H^1 norm.

1.2.2. Example

With expression:

  auto g = sin(2*pi*Px())*cos(2*pi*Py());
  auto gradg = 2*pi*cos(2* pi*Px())*cos(2*pi*Py())*oneX()
            -2*pi*sin(2*pi*Px())*sin(2*pi*Py())*oneY();
// There gradg is a column vector!
// Use trans() to get a row vector
  double normH1_g = normH1( _range=elements(mesh),
                     _expr=g,
                   _grad_expr=trans(gradg) );

With test or trial function u

  double errorH1 = normH1( _range=elements(mesh),
                    _expr=(u-g),
                  _grad_expr=(gradv(u)-trans(gradg)) );

1.3. \(L^\infty\) norm

You can evaluate the infinity norm using the normLinf() function:

\[\parallel f \parallel_\infty=\sup_\Omega(|f|)\]

1.3.1. Interface

  normLinf( _range, _expr, _pset, _geomap );

Required parameters:

  • _range = domain of integration

  • _expr = mesurable function

  • _pset = set of points (e.g. quadrature points)

Optional parameters:

  • _geomap = type of geometric mapping.

    • Default = GEOMAP_OPT

The normLinf() function returns not only the maximum of the function over a sampling of each element thanks to the _pset argument but also the coordinates of the point where the function is maximum. The returned data structure provides the following interface

  • value(): return the maximum value

  • operator()(): synonym to value()

  • arg(): coordinates of the point where the function is maximum

1.3.2. Example

  auto uMax = normLinf( _range=elements(mesh),
                        _expr=idv(u),
                        _pset=_Q<5>() );
  std::cout << "maximum value : " << uMax.value() << std::endl
            <<  "         arg : " << uMax.arg() << std::endl;