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.
 6 Variational inequalities.
 A. Stationary variational inequalities.
 B. Evolutionary variational inequalities.
 a. Strong and variational formulations for evolutionary problem.
 b. Existence and uniqueness for evolutionary problem.
 c. Penalized evolutionary problem.
 d. Proof of existence for evolutionary problem.
 VIII. Bibliography
 Notation. Index. Contents.

## Penalized evolutionary problem.

roblem

(Evolutionary penalized problem) For a bounded set with smooth boundary, time interval and given functions , find a function satisfying the relationships where the operation is given by the definition ( Bilinear form B 2 ).

Remark

(Preliminary reduction 2) In the section ( Convergence of finite elements for generic parabolic operator section ) we saw how a change of the function transforms the problem to insure that the condition ( Assumption of coercivity 1 ) is met. In particular, if then and in light of the proposition ( Energy estimates for the bilinear form B )-2 the may be chosen so that the bilinear form would satisfy the condition ( Assumption of coercivity 1 ).

Proposition

(Existence and uniqueness for penalized problem 2) If the coefficients of satisfy the definition ( Elliptic differential operator ), the condition ( Assumption of coercivity 1 ) is satisfied and then the problem ( Evolutionary penalized problem ) has a unique solution.

Proof

(Uniqueness) Suppose and are two solutions of the problem ( Evolutionary penalized problem ). Then we deduce from the problem We introduce the notation , subtract and put Since we derive and conclude the uniqueness as in the proof of the proposition ( Existence and uniqueness for evolutionary problem ).

Proof

(Existence) We define to be the solution of and define the sequence as follows We denote and derive from : We add and obtain or We integrate both sides over : or Note that hence Therefore Let is the point where the (or we would modify the argument for a sequence that approaches the ). Then considering at we obtain Thus or Thus, for sufficiently small But then, using such result and we increase the and expand it to . Thus for some . Then from we also conclude Then implies that and by passing to the limit, is the solution of the problem ( Evolutionary penalized problem ).

Proposition

(A priory estimate for penalized solution 1) Under conditions of the proposition ( Existence and uniqueness for evolutionary problem ) the solution of the problem ( Evolutionary penalized problem ) satisfies the relationship where the is independent of .

Proof

We act similarly to the proof of the proposition ( Parabolic regularity 1 ). We deduce from the equation of the problem ( Evolutionary penalized problem ): Note that we used the condition on of the proposition ( Existence and uniqueness for evolutionary problem ) so that ). We introduce the following convenience notations so that We have By the condition ( Symmetric principal part ) We substitute , and into : We move the terms around and integrate over . Then, similarly to the proof of the proposition ( Parabolic regularity 1 ), the -term estimates from below by ellipticity, all the RHS estimates from above and we obtain the statement of the proposition.

 Notation. Index. Contents.