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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.
a. Weak and strong formulations for stationary variational inequality problem.
b. Existence and uniqueness for coercive stationary problem.
c. Penalized stationary problem.
d. Proof of existence for stationary problem.
e. Estimate of penalization error for stationary problem.
f. Monotonicity of solution of stationary problem.
g. Existence and uniqueness for non-coercive stationary problem.
B. Evolutionary variational inequalities.
VIII. Bibliography
Notation. Index. Contents.

Estimate of penalization error for stationary problem.


roposition

(Penalization error) Assume that

1. the coefficients of $B$ satisfy the definition ( Elliptic differential operator ),

2. the condition ( Assumption of coercivity 1 ) holds,

3. MATH ,

4. MATH ,

5. $\psi\geq0$ on $\partial U$

then for the solution $u_{\varepsilon}$ of the problem ( Stationary penalized problem ) and the solution $u$ of the problem ( Stationary variational inequality problem ) the following estimate holds MATH

Proof

We set MATH in the problem ( Stationary penalized problem ): MATH

and note that MATH thus we obtain MATH or MATH We estimate the RHS using the propositions ( Energy estimates for the bilinear form B ) and ( Cauchy inequality ) and we estimate the LHS using the condition ( Assumption of coercivity 1 ). We obtain MATH Using such estimate we revisit MATH , use the same tools again and obtain MATH

We introduce the quantity MATH Thus it remains to show that MATH

Note that MATH . Indeed, MATH also, since MATH and $\psi\geq0$ on $\partial U$ MATH We set $v=r_{\varepsilon}$ in the problem ( Stationary penalized problem ) MATH and set MATH in the problem ( Stationary variational inequality problem ): MATH We add $\left( \&\right) $ and MATH and obtain MATH Hence MATH But $u\in K$ hence MATH MATH Thus MATH and by the condition ( Assumption of coercivity 1 ) and the propositions ( Energy estimates for the bilinear form B )-1, MATH Together with MATH this implies the statement of the proposition.





Notation. Index. Contents.


















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