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3.3 Numerical quadrature
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Formula 
Table 
Tunable 
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Unless the coefficients of the differential equation (here
and
)
are singular and require a special treatment somewhere in the solution
domain, the precision of the quadrature depends mainly on the choice of
the basis and test functions: it is indeed sufficient to simply preserve
the convergence rate that is guaranteed by the FEM discretization.
Writing the numerical quadrature as a truncated sum
table 3.3.0#tab.1 shows how a judicious choice of the abscissas
and weights
yields an approximation of
that integrates exactly polynomials with a degree up to
.
Table 3.3.0#tab.1:
Quadrature abscissas and weights for the interval
quadrature 
terms 



midpoint 
2 
0 
1 
p=2 
trapezoidal 
2 
1 
1 
p=2 
Gaussian 
2 

1 
p=4 
Gaussian 
3 
0 
8/9 
p=6 



5/9 

Gaussian 
4 


p=8 






For example, a superposition of
terms (where the integrand is
evaluated on the specified location
and weighted
by the factor
) is required using a Gaussian quadrature, when
four evaluations of the integrand in every mesh interval are used
to compute exact overlap integrals involving the product of (two)
cubic test and basis functions.
In the case of linear rooftop functions, the midpoint and trapezoidal
rules are both equally precise and require only one evaluation per interval;
although this is slightly more expensive with one extra evaluation per
interval, both rules can nicely be combined into a socalled
tunable integration [8]

(3) 
As will become clear below, this allows to gradually modify the piecewise
linear FEM discretization (obtained with
) into either an equivalent
FD discretization (with
) or a piecewise constant FEM approximation
(with
).
Apart from the academic interest, this feature can be used to change the
slope of the numerical convergence and even to stabilize an approximation
that is marginally unstable.
Armed with a variety of quadrature rules, it is now possible to implement
the linear FEM scheme sketched in sect.3.2 using a tunable integration
(3.3#eq.3) to evaluate the matrix elements in (3.2#eq.5).
Since rooftop functions reach only as far as their
nearest neighbors,
all the matrix elements
with
vanish, except
for the overlap at the boundaries.
Choosing a sequential numbering of the unknowns
,
this results in a tridiagonal matrix, with two extra elements in the
upperright and lower left corner taking care of the periodicity.
It is a good exercise for the reader to verify that all the matrix elements
can be written in terms of the integrals
provided that the advection and diffusion coefficients are constant
across the intervals as will be assumed here for pedagogical reasons.
In a real code constructed for an inhomogeneous medium, each formula
is replaced by
function calls returning the value of the integrand
(where
,
are evaluated on quadrature points) and the
weighted sum (3.3#eq.1) is simply accumulated in the corresponding
matrix element.
SYLLABUS Previous: 3.2 An engineer's formulation
Up: 3 FINITE ELEMENT METHOD
Next: 3.4 Implementation and solution
