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.
 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.

## Ritz-Galerkin approximation.

et be an -dimensional subspace of the class . We introduce the following problem.

Problem

(Approximate toy problem) For find such that

We propose as an approximation of the solution of the problem ( Variational toy problem ). Since is a solution of a finite dimensional problem, its existence follows from uniqueness considerations and is easily proven.

Proposition

There exists a unique solution of the problem ( Approximate toy problem ).

Proof

Let be a basis of . We seek the solution of the form for a numerical sequence . It suffices to have The above is a linear system of algebraic equations for the definition of with a square matrix. Hence, it suffices to prove that the system can have only the trivial solution. To do so we perform integration by parts as in the proof of the proposition ( Equivalence of toy problems ) and arrive to the conclusion that must satisfy From and we derive that . Consequently, from and we obtain .

 Notation. Index. Contents.