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.

## Finite elements for Heat equation with Dirichlet boundary conditions.

roblem

(Heat equation with Dirichlet boundary condition) Find the function , , such that for some functions and . The is assumed to be a bounded domain with a boundary. The functions and are assumed to satisfy the compatibility conditions of the proposition ( Parabolic regularity 2 ).

 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.
 Notation. Index. Contents.