Operators

1. Operations

You can use the usual operations and logical operators.

Feel++ Keyword

Math Object

Description

+

\( f+g\)

tensor sum

-

\( f-g\)

tensor substraction

*

\( f*g\)

tensor product

/

\( f/g\)

tensor tensor division (\(g\) scalar field)

<

\( f<g\)

element wise less

⇐

\( f\leq g\)

element wise less or equal

>

\( f>g\)

element wise greater

>=

\( f \geq g\)

element wise greater or equal

==

\( f==g\)

element wise equal

!=

\( f \neq g\)

element wise not equal

-

\( -g\)

element wise unary minus

&&

\( f\) and \(g\)

element wise logical and

||

\( f\) or \(g\)

element wise logical or

!

\( !g\)

element wise logical not

2. Differential Operators

Feel++ finit element language use test and trial functions. Keywords are different according to the kind of the manipulated function.
Usual operators are for test functions.
t-operators for trial functions.
v-operators to get an evaluation.

Suppose that \( f \in X_h \) reads

\[f=\sum_{i=0}^{\mathcal{N}} f_i \phi_i\]

where \( X_h = \mathrm{span}\{ \phi_i, i=1,\ldots,\mathcal{N}\} \) is a finite element space.

Feel++ Keyword

Math Object

Description

Rank

Dimension

id(f)

\( \{\phi_i\} \)

test function

rank\( (f(\overrightarrow{x})) \)

\( m \times p \)

idt(f)

\( \{\phi_i\} \)

trial function

rank\( (f(\overrightarrow{x}))\)

\( m \times p \)

idv(f)

\( f \)

evaluation function

rank\( (f(\overrightarrow{x})) \)

\( m \times p \)

grad(f)

\( \nabla f \)

gradient of test function

rank\( (f(\overrightarrow{x}))+1 \)

\(m \times n \) \(p=1\)

gradt(f)

\( \nabla f \)

grdient of trial function

rank\( (f(\overrightarrow{x}))+1 \)

\(m \times n \) \(p=1\)

gradv(f)

\( \nabla f \)

evaluation function gradient

rank\((f(\overrightarrow{x}))+1\)

\(m \times n \) \(p=1\)

div(f)

\( \nabla\cdot f \)

divergence of test function

rank\( (f(\overrightarrow{x}))-1 \)

\( 1 \times 1 \)

divt(f)

\( \nabla\cdot f \)

divergence of trial function

rank\( (f(\overrightarrow{x}))-1 \)

\( 1 \times 1 \)

divv(f)

\( \nabla\cdot f \)

evaluation function divergence

rank\( (f(\overrightarrow{x}))-1 \)

\( 1 \times 1 \)

curl(f)

\( \nabla\times f \)

curl of test function

1

\( n \times 1 \) \( m=n \)

curlt(f)

\( \nabla\times f \)

curl of trial function

1

\( n \times 1 \) \( m=n \)

curlv(f)

\( \nabla\times f \)

evaluation function curl

1

\( n \times 1 \) \( m=n \)

laplacian(f)

\( \Delta f \)

laplacian of test function

0

\( 1 \times 1 \) \( m=p=1 \)

laplaciant(f)

\( \Delta f \)

laplacian of trial function

0

\( 1 \times 1 \) \( m=p=1 \)

laplacianv(f)

\( \Delta f \)

laplacian of the function \(f\)

0

\( 1 \times 1 \) \( m=p=1 \)

hess(f)

\( \nabla^2 f \)

hessian of test function

2

\( n \times n \) \( m=p=1 \)

trace(f)

\( \mathrm{trace}( f ) \)

trace of test matrix field

0

\( 1 \times 1 \) \( m=p=d \)

tracet(f)

\( \mathrm{trace}( f ) \)

trace of trial matrix field

0

\( 1 \times 1 \) \( m=p=d \)

tracev(f)

\( \mathrm{trace}( f ) \)

trace of matrix field \( f \)

0

\( 1 \times 1 \) \( m=p=d \)

normal(f)

\( f \cdot \overrightarrow{N} \)

normal component of test function

rank\((f(\overrightarrow{x}))-1\)

normalt(f)

\( f \cdot \overrightarrow{N} \)

normal component of trial function

rank\((f(\overrightarrow{x}))-1\)

normalv(f)

\( f \cdot \overrightarrow{N} \)

normal component of function \( f \)

rank\((f(\overrightarrow{x}))-1\)

dn(f)

\( \nabla f \cdot \overrightarrow{N} \)

normal derivative of test function

0

\( 1 \times 1 \) \( m=p=1 \)

dn(f)

\( \nabla f \ \overrightarrow{N} \)

normal derivative of test function

1

\( m \times 1 \) \(p=1 \)

dnt(f)

\( \nabla f \cdot \overrightarrow{N} \)

normal derivative of trial function

0

\( 1 \times1 \) \(m=p=1\)

dnt(f)

\( \nabla f \ \overrightarrow{N} \)

normal derivative of trial function

1

\( m \times 1 \) \(p=1\)

dnv(f)

\( \nabla f \cdot \ \overrightarrow{N} \)

evaluation of normal derivative

0

\( 1 \times 1 \) \(m=p=1\)

dnv(f)

\( \nabla f \ \overrightarrow{N} \)

evaluation of normal derivative

1

\( m \times 1 \) \(p=1\)

dx(f)

\( \nabla f \cdot \overrightarrow{i} \)

derivative of test function in \( x \)

0

\( 1 \times 1 \) \( m=p=1 \)

dy(f)

\( \nabla f \cdot \overrightarrow{j} \)

derivative of test function in \( y \)

0

\( 1 \times 1 \) \( m=p=1 \)

dz(f)

\( \nabla f \cdot \overrightarrow{k} \)

derivative of test function in \( z \)

0

\( 1 \times 1 \) \( m=p=1 \)

3. Two Valued Operators

Feel++ Keyword

Math Object

Description

Rank

Dimension

jump(f)

\( f=f_0\overrightarrow{N_0}+f_1\overrightarrow{N_1} \)

jump of test function

0

\( n \times 1 \) \( m=1 \)

jump(f)

\( \overrightarrow{f}=\overrightarrow{f_0}\cdot\overrightarrow{N_0}+\overrightarrow{f_1}\cdot\overrightarrow{N_1} \)

jump of test function

0

\( 1 \times 1 \) \( m=2 \)

jumpt(f)

\( f=f_0\overrightarrow{N_0}+f_1\overrightarrow{N_1} \)

jump of trial function

0

\( n \times 1 \) \( m=1 \)

jumpt(f)

\( \overrightarrow{f}=\overrightarrow{f_0}\cdot\overrightarrow{N_0}+\overrightarrow{f_1}\cdot\overrightarrow{N_1} \)

jump of trial function

0

\( 1 \times 1 \) \( m=2 \)

jumpv(f)

\( f=f_0\overrightarrow{N_0}+f_1\overrightarrow{N_1} \)

jump of function evaluation

0

\( n \times 1 \) \( m=1 \)

jumpv(f)

\( \overrightarrow{f}=\overrightarrow{f_0}\cdot\overrightarrow{N_0}+\overrightarrow{f_1}\cdot\overrightarrow{N_1} \)

jump of function evaluation

0

\( 1 \times 1 \) \( m=2 \)

average(f)

\( {f}=\frac{1}{2}(f_0+f_1) \)

average of test function

rank \( ( f(\overrightarrow{x})) \)

\( n \times n \) \(m=n\)

averaget(f)

\( {f}=\frac{1}{2}(f_0+f_1) \)

average of trial function

rank \( ( f(\overrightarrow{x})) \)

\(n \times n \) \(m=n\)

averagev(f)

\( {f}=\frac{1}{2}(f_0+f_1) \)

average of function evaluation

rank\( ( f(\overrightarrow{x})) \)

\( n \times n \) \(m=n\)

leftface(f)

\( f_0 \)

left test function

rank\( ( f(\overrightarrow{x})) \)

\( n \times n \) \( m=n \)

leftfacet(f)

\( f_0 \)

left trial function

rank\( ( f(\overrightarrow{x})) \)

\( n \times n \) \( m=n \)

leftfacev(f)

\( f_0 \)

left function evaluation

rank\( ( f(\overrightarrow{x})) \)

\( n \times n \) \( m=n \)

rightface(f)

\( f_1 \)

right test function

rank\( ( f(\overrightarrow{x})) \)

\( n \times n \) \( m=n \)

rightfacet(f)

\( f_1 \)

right trial function

rank\( ( f(\overrightarrow{x})) \)

\( n \times n \) \( m=n \)

rightfacev(f)

\( f_1 \)

right function evaluation

rank\( ( f(\overrightarrow{x})) \)

\( n \times n \) \( m=n \)

maxface(f)

\( \max(f_0,f_1) \)

maximum of right and left test function

rank\( ( f(\overrightarrow{x})) \)

\( n \times p \)

maxfacet(f)

\( \max(f_0,f_1) \)

maximum of right and lef trial function

rank\( ( f(\overrightarrow{x})) \)

\( n \times p \)

maxfacev(f)

\( \max(f_0,f_1) \)

maximum of right and left function evaluation

rank\( ( f(\overrightarrow{x})) \)

\( n \times p \)

minface(f)

\( \min(f_0,f_1) \)

minimum of right and left test function

rank\( ( f(\overrightarrow{x})) \)

\( n \times p \)

minfacet(f)

\( \min(f_0,f_1) \)

minimum of right and left trial function

rank\( ( f(\overrightarrow{x})) \)

\( n \times p \)

minfacev(f)

\( \min(f_0,f_1) \)

minimum of right and left function evaluation

rank\( ( f(\overrightarrow{x})) \)

\( n \times p \)