## 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,
_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();
}``````

### 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,

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:

$\parallel f \parallel_\infty=\sup_\Omega(|f|)$

#### 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;``````