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.

## Convergence of discontinuous Galerkin technique. Adaptive time stepping.

roposition

(Convergence of discontinuous Galerkin technique) Assume that the condition ( Generic parabolic PDE setup ) takes place and is a solution of the problem ( Generic parabolic PDE problem ). Let be a solution of the problem ( Discontinuous Galerkin time-discretization ) applied consecutively to the time intervals . Then

Proof

We introduce a mapping with the following properties: thus, for each the equals at but does not equal at . We proceed to prove existence and uniqueness of such . Let be an orthonormal basis of and for some sequence . For every fixed and we need to satisfy Thus, for each fixed we have equations for unknowns . To complete the proof of existence of it remains to show that the homogenous system of equations has only the trivial solution. Let then, by , for some and, by , but also and both the terms and do not change sign on . Thus, we must have on . Existence of has been established.

It follows from such result that if is a -polynomial of degree then would approximate it exactly.

According to the proposition ( Integral form of Taylor decomposition ), for a function we have where . By polynomial approximation up to degree , we must have Indeed, if one of the derivatives is not zero then we cannot have the polynomial approximation with the same formula. Thus where or Consequently We use the formula ( Cauchy inequality ), Then, by orthogonality of , and by properties of the operator listed in the condition ( Generic parabolic PDE setup ),

For being a solution of the problem ( Generic parabolic PDE problem ), be a solution of the problem ( Discontinuous Galerkin time-discretization ) and being the polynomial approximation defined in we decompose the error We already have the desired estimate for . It remains to estimate . Note that both and are solutions of the equation of the problem ( Discontinuous Galerkin time-discretization ), hence, for we have Consequently We calculate the part By definition of there is orthogonality of to -variable polynomials up to degree . Thus, we perform integration by parts. Now the integral is zero by the orthogonality because thus is a -degree -variable polynomial. The is zero at by definition of .

Therefore, we arrived to We set and evaluate the part We use the formula ( Cauchy inequality ). We arrived to or Hence Thus Note that by definition of the components and . and we use the estimate

 Notation. Index. Contents.