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.
 4 Construction of approximation spaces.
 5 Time discretization.
 A. Change of variables for parabolic equation.
 B. Discontinuous Galerkin technique.
 a. Weak formulation with respect to time parameter.
 b. Discretization with respect to time parameter.
 c. Discretization for backward Kolmogorov equation.
 d. Existence and uniqueness for time-discretized problem.
 e. Convergence of discontinuous Galerkin technique. Adaptive time stepping.
 6 Variational inequalities.
 VIII. Bibliography
 Notation. Index. Contents.

## Discontinuous Galerkin technique.

ondition

(Generic parabolic PDE setup) Let be a Hilbert space and the Hilbert space is densely and compactly embedded in : where the refers to the duality with respect to -topology. For example, , , compare with the chapter ( Parabolic PDE ). Let be a mapping measurable with respect to the time parameter . For every the operator is self adjoint and, uniformly in , has the properties Let be mappings We are considering the equation

 (Generic parabolic PDE problem)
for some .

 a. Weak formulation with respect to time parameter.
 b. Discretization with respect to time parameter.
 c. Discretization for backward Kolmogorov equation.
 d. Existence and uniqueness for time-discretized problem.
 e. Convergence of discontinuous Galerkin technique. Adaptive time stepping.
 Notation. Index. Contents.