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3.1 Mathematical background
Slide : [
Weighted residuals 
Variational form 
Partial integration 
Approximation 
BC 
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To approximate a set of linear partial differential equations

(1) 
for an unknown
that is
continuously defined with
derivatives in the volume
and is
subject to the boundary conditions

(2) 
a mathematician involves first the socalled

Weighted residuals.
 Having defined a scalar product
and a norm
,
the calculation essentially amounts to the minimization of a residual
vector

(3) 
which is carried out using tools from the variational calculus.

Variational principle.
 A quadratic form is constructed for that purpose by choosing a
test function
in a subspace
that is
``sufficiently general'' and satisfies the boundary conditions.
The linear equation (3.1#eq.1) can then be written as an
equivalent variational problem

(4) 

Integration by parts.
 If
is differentiable at least once in
, the regularity required by
the forthcoming discretization can often be relaxed by partial integrations.
Using
for illustration, Leibniz' rule states that

(5) 
Integrating over the volume
and using Gauss' divergence theorem
yields a generalized formula for partial integration:

(6) 
For the special case where
, this is known as Green's
formula

(7) 
The last (surface) term can sometimes be imposed to zero (or to a finite
value) when applying socalled natural boundary conditions.

Numerical approximation.
 It turns out that the formulation as a variational problem is general
enough that the solution
of (3.1#eq.4) remains a
converging approximation of (3.1#eq.1) even when the subspaces
are restricted to finite, a priori nonorthogonal,
but still complete sets
of functions. The overlap integrals involving nonorthogonal functions
are often evaluated with a numerical quadrature that can be easily be
handled by the computer.
In general, the discretized solution
is expanded in basis
functions
, which
either reflect a property of the solution (e.g. the operator
Green's function in the Method of Moments),
or which are simple and localized enough so that they yields cheap inner
products
and sparse linear systems (e.g. the
rooftop function for the linear Finite Element Method).
Different choices can also be made for the test functions
:
among the most popular are the Galerkin method, where test and basis
functions are both chosen from the same subspace
and the method of
collocation, which consists in taking Dirac functions
that lead to pointwise evaluations of the integrand on the mesh
.

Boundary conditions.
 If natural boundary conditions are not already sufficient,
essential boundary conditions have to be imposed either by choosing
the functional space
(e.g. let
and
overlap in a periodic domain
as in the next section) or by replacing one or several
equations of the linear system (e.g. use higher order finite differences
to preserve the convergence rate as in exercise 3.02).
SYLLABUS Previous: 3 FINITE ELEMENT METHOD
Up: 3 FINITE ELEMENT METHOD
Next: 3.2 An engineer's formulation
