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.

## Convergence of approximate solution. Energy norm argument.

et be the solution of the problem ( Variational toy problem ) and be the solution of the problem ( Approximate toy problem ). We seek conditions for the convergence as in some sence.

We introduce the norm By definition of and the formula ( Holder inequality ) we have We estimate Let . Note that and, by subtracting the problems ( Variational toy problem ) and ( Approximate toy problem ), satisfy . Hence, we continue Thus, or for any .

Proposition

(Toy approximation in energy norm) Let be the solution of the problem ( Variational toy problem ) and be the solution of the problem ( Approximate toy problem ). We have

 Notation. Index. Contents.