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Quantitative Analysis
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Numerical Analysis
C++ Multithreading
Python for Excel
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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.

Weak and strong formulations for stationary variational inequality problem.


roblem

(Stationary variational inequality problem) For a bounded set MATH with smooth boundary and given functions MATH and MATH find a function MATH satisfying the relationships MATH where the operation $B$ is given by the definition ( Bilinear form B ) and the class of functions $K$ is defined by MATH We always assume that $K\not =\not 0 $ .

We denote MATH the solution operator MATH MATH

Problem

(Strong formulation of stationary problem) For a bounded set MATH with smooth boundary and given functions MATH and MATH find a function MATH satisfying the relationships MATH in $U$ and MATH where the operator $L$ is defined by the formula ( Operator L ).

Proposition

For a function MATH the formulations ( Strong formulation of stationary problem ) and ( Stationary variational inequality problem ) are equivalent.

Proof

See the remarks in the section ( Variational inequalities ) and the section ( Elliptic PDE ).

Problem

(Weak formulation of stationary problem) For a bounded set MATH with smooth boundary and given functions MATH and MATH find a function MATH satisfying the relationships MATH where the operation $B$ is given by the definition ( Bilinear form B ) and the class of functions $K$ is defined by MATH

Proposition

Assume that MATH Then the formulations ( Stationary variational inequality problem ) and ( Weak formulation of stationary problem ) are equivalent.

Proof

If MATH and MATH then MATH and MATH follows. The prove in the opposite direction is the same.





Notation. Index. Contents.


















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