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.
 D. Finite elements for Heat equation with Neumann boundary conditions.
 E. Relaxed boundary conditions for approximation spaces.
 a. Elliptic problem with relaxed boundary approximation.
 b. Parabolic problem with relaxed boundary approximation.
 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.

## Elliptic problem with relaxed boundary approximation.

e consider the problem ( Poisson equation with Dirichlet boundary condition ).

Definition

(Nitsche bilinear form) We introduce the bilinear form and the norm where the is a positive constant and is the parameter controlling the precision of the finite dimensional approximation.

Let be a solution of the problem ( Poisson equation with Dirichlet boundary condition ). According to the proposition ( Green formula ) and the boundary conditions

Condition

(Approximation in Nitsche norm) We assume that the finite dimensional spaces satisfy the following conditions

1. .

2.

3.

Problem

(Poisson equation weak formulation 2) Find the solution that satisfies the condition

Note that

Proposition

(Nitsche form energy estimates) For a fixed and sufficiently smooth we have Assuming that the condition ( Approximation in Nitsche norm ) is satisfied, there exists numbers and such that

Proof

The first estimate is evident from the definitions.

We calculate the second estimate as follows: We use the formula ( Cauchy inequality with epsilon ): We use the condition ( Approximation in Nitsche norm )-2 as follows: hence we set and apply the above inequality twice: We choose then

Proposition

(Galerkin convergence 3) Assume that satisfies the condition ( Approximation in Nitsche norm ) and are the solutions of the problems ( Poisson equation with Dirichlet boundary condition ) and ( Poisson equation weak formulation 2 ) respectively. We have for

Proof

We estimate According to the proposition ( Nitsche form energy estimates ) we have We use the equality : and use the proposition ( Nitsche form energy estimates ) again: We also have according to the condition ( Approximation in Nitsche norm )-3. These results conclude the proof.

Notation

(Solution operator for elliptic problem) We introduce the notation for the mapping from the function of the elliptic problem ( Poisson equation with Dirichlet boundary condition ) into its solution . The operator does the same for a particular approximation of the elliptic problem in used in the context.

Condition

(Properties of solution operator) We assume the and have the following properties:

1. is selfadjoint, positive semidefinite on and positive definite on .

2. There is a positive integer such that

Proposition

The solution operators and for the problems ( Poisson equation with Dirichlet boundary condition ) and ( Galerkin approximation 1 ) satisfy the condition ( Properties of solution operator ).

Proof

According to the definition of as the solution operator for ( Galerkin approximation 1 ) we have Since for any we have can apply the above equality to both positions in the scalar product : Hence we have the selfadjointness. We substitute : thus is positive semidefinite on . In addition, for Hence, is positive definite on .

The condition is a consequence of the propositions ( Galerkin convergence 1 ) and ( Elliptic regularity for Poisson equation ).

Proposition

The solution operators and for the problems ( Poisson equation with Dirichlet boundary condition ) and ( Poisson equation weak formulation 2 ) satisfy the condition ( Properties of solution operator ).

Proof

The proof is similar to the previous proposition. Use the proposition ( Galerkin convergence 3 ) instead of ( Galerkin convergence 1 ).

 Notation. Index. Contents.