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.

## Approximation in L2 norm. Duality argument.

e will use the proposition ( Toy approximation in energy norm ) to approximate the difference in norm.

Let be the solution of the problem Let . We estimate Thus We assume that the following condition holds.

Condition

(Energy approximation) The spaces satisfy for any and some small .

Under such assumption we continue: We recall thus We use the proposition ( Toy approximation in energy norm ): and apply the condition ( Energy approximation ):

Proposition

Under condition ( Energy approximation ) the solutions and of the problem ( Variational toy problem ) and ( Approximate toy problem ) satisfy the error estimate

 Notation. Index. Contents.