88 Advanced Topics in Applied Mathematics
Cauchy principal values are discussed in many books on this subject
(seePozrikidis,2002,Hartmann,1989,Brebbia,1978).
In the example, if ∂u/∂n is given on the boundary, u on the bound-
ary will be unknown. To solve for u, we move the interior point to the
boundary and obtain a Fredholm equation of the second kind.
Compared to other numerical methods such as the Finite Element
Method (FEM) and the Finite Difference Method (FDM), the num-
ber of unknowns in the system of equations in the Boundary Element
Method is equal to the number of discrete points on the boundary and
not in the domain. In the 2D case, the Boundary Element Method has
N equations and the other two methods may have up to N
2
equations.
Thus, highly accurate solutions can be obtained by BEM using mini-
mal computer time. On the negative side, BEM requires considerable
analysis from the user, and it is not readily available in the form of an
all-purpose software.
2.19 PROPER ORTHOGONAL DECOMPOSITION (POD)
Proper orthogonal decomposition (POD) is a method for approxi-
mating data distributed in time and space using a finite number of
orthogonal basis functions of the space variables, which are selected
to minimize the expectation value of the standard deviation of the
error. Generally, we are used to selecting basis functions a prior such
as in a Fourier series. The POD method provides the best sequence of
basis functions depending on the data. In the literature POD appears
under various names: principal component analysis, Karhunen-Loève
transformation, and singular value decomposition.
Although the applications of this method in turbulence and other
dynamical systems often involve vector valued data, we discuss POD
using a scalar function
v = v(x, t), x in , t in [0, T]. (2.191)