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 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.
 a. Variational formulation, essential and natural boundary conditions.
 b. Ritz-Galerkin approximation.
 c. Convergence of approximate solution. Energy norm argument.
 d. Approximation in L2 norm. Duality argument.
 e. Example of finite dimensional subspace construction.
 f. Adaptive approximation.
 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.

## Example of finite dimensional subspace construction. n this section we construct an example subspace and prove the condition ( Energy approximation ).

We introduce a "partition" set ("mesh")  and define to be a class of functions such that for all sets of numbers and .

We define to be a set of functions from such that The set is a basis of . Indeed, it is linearly independent and for any we have Therefore, for a function we define an approximation and proceed to estimate We introduce the convenience notations We estimate We make a linear change of variables that maps into : thus We continue and introduce the convenience notation thus Note that by construction of . Hence, there is a such that then we apply the formula ( Holder inequality ) We substitute the last result into the estimate We change the order of integration We sum the estimate for and obtain Notation. Index. Contents.
 Copyright 2007