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.
 F. Convergence of finite elements applied to nonsmooth data.
 a. H-tilde spaces.
 b. Convergence of finite elements with nonsmooth initial condition.
 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.

Convergence of finite elements with nonsmooth initial condition.

roblem

(Parabolic problem with nonsmooth initial condition) For a bounded set with boundary and positive find the such that where .

Proposition

(Solution operator for PP with nonsmooth IC) Using the notation of the definition ( Eigenfunctions of Laplacian ) the solution of the problem ( Parabolic problem with nonsmooth initial condition ) is given by the expression

Proposition

(Instant smoothness of solution operator for PP 1)

1. If then for any and .

2. If then for any , and we have

Proof

We calculate We estimate the expression We substitute Let then We continue the evaluation of :

Proposition

(Partial inversion lemma 2) Let where the operator is nonnegative and symmetric for some scalar product . Then for the corresponding norm we have

1.

2.

3.

4. For

Proof

(1) Note that and by the symmetry of We apply the operation to the equality :

or, by the proposition ( Cauchy inequality ), Therefore, and, after integration, It remains to note that and .

Proof

(2) We apply the operation to the equality : or Since we derive We combine the above with the identities to obtain and integrate We use the proposition ( Cauchy inequality with epsilon ) We make small, divide by and arrive to We apply the statement (1) and conclude (2).

Problem

(Partially inverted semi discrete parabolic problem 1) Assuming existence of the operator as in the condition ( Properties of solution operator ) we pose the problem of finding such that

Proposition

(Galerkin convergence 5) Let be the solution operator of the problem ( Poisson equation with Dirichlet boundary condition ) and the operator is given and satisfies the condition ( Properties of solution operator ). Assume that spaces satisfy the condition ( Finite dimensional approximation 1 ). Let be the solution of the problem ( Partially inverted semi discrete parabolic problem 1 ) and be the solution of the problem ( Parabolic problem with nonsmooth initial condition ). Assume that for some . Then we have

Proof

Let be a solution of the problem ( Partially inverted semi discrete parabolic problem 1 ) with . Then and, for being the solution operator of the problem ( Partially inverted semi discrete parabolic problem 1 ), It may be shown that there is an estimate for similar to the one of the proposition ( Instant smoothness of solution operator for PP 1 ), hence by the conditions of this proposition, the condition ( Finite dimensional approximation 1 ) and proposition ( Characterization of H-tilde spaces ) Hence, it is enough to prove the statement for .

We intend to apply the proposition ( Partial inversion lemma 2 )-3. We introduce and recall from the proof of the proposition ( Galerkin convergence 4 ) that Hence, according to the condition ( Properties of solution operator )-2 and by the proposition ( Instant smoothness of solution operator for PP 1 ) Similarly, Finally, for any by symmetry of as in the condition ( Properties of solution operator )-1, because . Hence, The statement now follows from the proposition ( Partial inversion lemma 2 )-3.

Proposition

(Galerkin convergence 6) Let be the solution operator of the problem ( Poisson equation with Dirichlet boundary condition ) and the operator is given and satisfies the condition ( Properties of solution operator ). Let be the solution of the problem ( Partially inverted semi discrete parabolic problem 1 ) and be the solution of the problem ( Parabolic problem with nonsmooth initial condition ). Assume that Then we have

1.

2.

Proof

(1) We introduce and and recall from the proof of the proposition ( Galerkin convergence 4 ) that hence We have We apply the proposition ( Partial inversion lemma 2 )-4 with : We set such that then thus We estimate the term as follows. We integrate the relationships over to obtain Then according to the proposition ( Partial inversion lemma 2 )-3, Thus from : We combine this with and obtain Thus We estimate the components as follows. According the condition ( Properties of solution operator )-2 and according to the proposition ( Instant smoothness of solution operator for PP 1 ) We estimate the other terms similarly: Therefore as claimed in the proposition for .

We now prove the statement for all as follows.

We introduce the error operator by the relationship The following identity holds Indeed, by direct substitution we have We use the property : The because acts . Thus does not move the result of . Also, .

We now estimate every component of the RHS in the identity . According to the proposition ( Galerkin convergence 5 ), and according to the proposition ( Instant smoothness of solution operator for PP 1 ) The evaluation of other components is similar.

Proof

(2) We perform induction over . The is covered in the statement (1). Assume that the result holds for . We introduce the notation We have We apply the proposition ( Partial inversion lemma 2 )-4 to the above result: In the above expression the terms depending on are estimated by induction assumption and the terms are estimated by the condition ( Properties of solution operator )-2 and the proposition ( Instant smoothness of solution operator for PP 1 ). Then the statement follows by the same means as in the proof of (1).

 Notation. Index. Contents.