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.
 a. Weak formulation for Heat equation with Dirichlet boundary conditions.
 b. Spacial discretization for Heat equation with Dirichlet boundary conditions.
 c. Backward Euler time discretization for Heat equation with Dirichlet boundary conditions.
 d. Crank-Nicolson time discretization for the 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.

## Spacial discretization for Heat equation with Dirichlet boundary conditions.

roblem

(Galerkin approximation 2) We define the finite dimensional approximation to the solution of the problem ( Heat equation weak formulation 1 ) as the solution of the problem where the is some -based approximation of .

Proposition

There exists a solution of the problem ( Galerkin approximation 2 ).

Proof

Follow the procedure of the section ( Galerkin approximation for parabolic Dirichlet problem ). Under a reasonable choice of the basis for the space the problem ( Galerkin approximation 2 ) becomes a Cauchy problem for a system of ODEs that always have a solution.

Proposition

(Galerkin convergence 2) Let and be the solutions of the problems ( Galerkin approximation 2 ) and ( Heat equation with Dirichlet boundary condition ) respectively. Assume that . We have for :

1.

2.

Proof

(1) We introduce the notations so that We estimate, according to the proposition ( Ritz projection convergence 1 ), We put the over the relationship and conclude To estimate the we write and substitute the definition of : We want to remove everything involving . Hence we substitute the relationships and We substitute the relationship : or We have , hence, we substitute in the last relationship and obtain Consequently and we continue deriving the consequences: According to the initial conditions for and , and we use the proposition ( Ritz projection convergence 1 ) in the second term: The statement now follows from the obtained results

Proof

(2) We have According to the proposition ( Ritz projection convergence 1 ) We return to the relationship from the first part of this proof and substitute : We substitute and (proposition ( Cauchy inequality )): and we continue deriving consequences: and the second estimate follows after application of the proposition ( Ritz projection convergence 1 ) to the term .

 Notation. Index. Contents.