I. Basic math.
 II. Pricing and Hedging.
 III. Explicit techniques.
 IV. Data Analysis.
 V. Implementation tools.
 VI. Basic Math II.
 VII. Implementation tools II.
 1 Calculational Linear Algebra.
 2 Wavelet Analysis.
 3 Finite element method.
 A. Tutorial introduction into finite element method.
 B. Finite elements for Poisson equation with Dirichlet boundary conditions.
 C. Finite elements for Heat equation with Dirichlet boundary conditions.
 D. Finite elements for Heat equation with Neumann boundary conditions.
 E. Relaxed boundary conditions for approximation spaces.
 F. Convergence of finite elements applied to nonsmooth data.
 G. Convergence of finite elements for generic parabolic operator.
 4 Construction of approximation spaces.
 5 Time discretization.
 6 Variational inequalities.
 VIII. Bibliography
 Notation. Index. Contents.

## Finite element method.

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 non-smooth data. See the section ( Remark on stability of financial problems ) for one discussion of relevance.

2. Finite element method features cure for dimensionality curse. Monte-Carlo is useful, but production tools do not need to be based on Monte-Carlo. 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 apriory 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 powerfull 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 case-by-case 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 inequalitites ).

The references for this chapter are [Thomee] and [Brenner] .

 A. Tutorial introduction into finite element method.
 B. Finite elements for Poisson equation with Dirichlet boundary conditions.
 C. Finite elements for Heat equation with Dirichlet boundary conditions.
 D. Finite elements for Heat equation with Neumann boundary conditions.
 E. Relaxed boundary conditions for approximation spaces.
 F. Convergence of finite elements applied to nonsmooth data.
 G. Convergence of finite elements for generic parabolic operator.
 Notation. Index. Contents.