SIMPLE SOLUTIONS

# FORM(2RHEOLEF) - Linux man page online | System calls

Representation of a finite element bilinear form.

Chapter
rheolef-6.7
form(2rheolef) rheolef-6.7 form(2rheolef)

## NAME

form - representation of a finite element bilinear form

## DESCRIPTION

The form class groups four sparse matrix, associated to a bilinear form on two finite ele‐ ment spaces: a: U*V ----> IR (u,v) |---> a(u,v) The operator A associated to the bilinear form is defined by: A: U ----> V' u |---> A(u) where u and v are fields (see field(2)), and A(u) is such that a(u,v)=<A(u),v> for all u in U and v in V and where <.,.> denotes the duality product between V and V'. Since V is a finite dimensional spaces, the duality product is the euclidian product in IR^dim(V). Since both U and V are finite dimensional spaces, the linear operator can be represented by a matrix. The form class is represented by four sparse matrix in csr format (see csr(2)), associated to unknown and blocked degrees of freedom of origin and destination spaces (see space(2)).

## EXAMPLE

The operator A associated to a bilinear form a(.,.) by the relation (Au,v) = a(u,v) could be applied by using a sample matrix notation A*u, as shown by the following code: geo omega("square"); space V (omega,"P1"); form a (V,V,"grad_grad"); field uh = interpolate (fct, V); field vh = a*uh; cout << v; The form-field vh=a*uh operation is equivalent to the following matrix-vector operations: vh.set_u() = a.uu()*uh.u() + a.ub()*uh.b(); vh.set_b() = a.bu()*uh.u() + a.bb()*uh.b();

## ALGEBRA

Forms, as matrices (see csr(2)), support linear algebra: Adding or substracting two forms writes a+b and a-b, respectively, and multiplying a form by a field uh writes a*uh. Thus, any linear combination of forms is available.

## WEIGHTED FORM

A weighted form is a form with an extra weight function w(x), e.g.: / | a(uh,vh) = | grad(uh).grad(vh) w(x) dx | / Omega In the present implementation, w can be any field, function or class-function or any non‐ linear field expression (see field(2)). As the integration cannot be performed exactly in general, a quadrature formula can be supplied. This feature is extensively used when solving nonlinear problems.