here are at least four major reasons to
use finite element method.
1. Justification of finite element method's convergence does not rely on
normality of the matrix of the problem and handles nonsmooth data. See the
section (
Remark on
stability of financial problems
) for one discussion of relevance.
2. Finite element method features cure for dimensionality curse. MonteCarlo
is useful, but production tools do not need to be based on MonteCarlo. See
the section (
Sparse tensor
product
).
3. Finite element method features technique for separation of a given problem
into several concurrent problems. This means that everything may be done in
real time using GPU cores. See the sections
(
Parallel subspace
correction method
),(
Laplace
quadrature
),(
Stable space
splittings
).
4. There are techniques for using apriori information about solution for
improvement of convergence. This means that one can push memory and speed
limitations imposed by hardware. See the sections
(
Preconditioning
),(
Adaptive
approximation
). Such techniques are especially powerful when combined with
multiscale constructions, see the section
(
Wavelet analysis
).
Unlike finite differences, however, finite elements do not have general
recipes. The dimensionality, boundedness and shape of the domain affects
application of Sobolev inequalities and approximation properties of the mesh,
the type of boundary conditions dictates the form of variational formulation,
the smoothness of coefficients affects the energy estimates and regularity
results. Each component requires a special trick. The theory of finite
elements is a collection of such tricks that successfully provides resolution
for PDE problems of finance on casebycase basis. In this chapter we study
example problems without American feature and cover several parts of the
toolkit.
American feature and free boundary conditions are treated in the section
(
Variational inequalities
).
The references for this chapter are
[Thomee]
and
[Brenner]
.
