Page 103 - Standard Handbook Of Petroleum & Natural Gas Engineering

92 Mathematics

The three primary advantages of the finite element approach over finite
difference methods are [9]:

1. Easy handling of irregularly shaped regions.
2. Variation in size of elements over a region, allowing smaller elements where
strong variations occur.
3. Larger elements can produce comparable accuracy to smaller mesh elements
of a finite difference grid, which is especially useful in handling elliptic
partial differential matrices.
Other methods for solving PDE's include Monte Carlo, spectral, and varia-
tional. Spectral methods in particular converge more rapidly than finite dif-
ference methods, but do not handle problems involving irregular geometries or
discontinuities well.

APPLIED STATISTICS
See References 23-28 for additional information.
A discrete random variable is one that may take on only distinct, usually integer,
values. A continuous random variable is one that may take on any value within a
continuum of values.

Moments

The moments describe the characteristics of a sample or distribution function.
The mean, which locates the average value on the measurement axis, is the first
moment of values measured about the origin. The mean is denoted by p for
the population and X for the sample and is given for a continuous random
variable by

For a discrete random variable, the mean is given by

and if each observation is given equal weight, f(X) = l/n and

- 1"
x or p = -Cxi
n i=l
The variance is the second moment about the mean and indicates the closeness
of values to the mean. It is denoted by o2 (population) or Sz (sample) and is
given for a continuous random variable by