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:

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

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
Unresolved include directive in modules/reference/pages/Integrals/norms.adoc - include::../../../../codes/mystokes.cpp[]

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.

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.

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:

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

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;