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.

## Parabolic problem with relaxed boundary approximation.

roblem

(Partially inverted semi discrete parabolic problem) Assuming existence of the operator as in the condition ( Properties of solution operator ) we pose the problem of finding such that

Proposition

(Partial inversion lemma) Let for where the operator is nonnegative for some scalar product . Then for the corresponding norm we have

Proof

We apply the operation to the relationship and obtain We also have and . Hence, using nonnegativeness of We integrate: Let , , then It remains to note that

Proposition

(Galerkin convergence 4) Assume that the condition ( Properties of solution operator ) is satisfied and let and be the solutions of the problems ( Heat equation with Dirichlet boundary condition ) and ( Partially inverted semi discrete parabolic problem ) respectively. We have for .

Proof

We introduce and calculate We want to get rid of all terms and all time derivatives. We substitute the equation from the problem ( Partially inverted semi discrete parabolic problem ): According to the definition ( Solution operator for elliptic problem ), we have : According to the problem ( Heat equation with Dirichlet boundary condition ), we have : We now apply the proposition ( Partial inversion lemma ) with : where By the condition ( Properties of solution operator ), In exactly the same way

 Notation. Index. Contents.