### 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:

• 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. H1 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

or semi norm:

Required parameters:

• _range = domain of integration

• _expr = mesurable function

Optional parameters:

• 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());
-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,

With test or trial function u

double errorH1 = normH1( _range=elements(mesh),
_expr=(u-g),

#### 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;