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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.
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.
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 for generic parabolic operator.


roblem

(Dirichlet problem for generic parabolic operator) Find the function MATH , MATH , MATH such that MATH for some functions MATH and MATH . The $U$ is assumed to be a bounded domain with a $C^{\infty}$ boundary. The functions $f$ and $g$ are assumed to satisfy the compatibility conditions of the proposition ( Parabolic regularity 2 ). The form of the operator $L$ is given by the formula ( Operator L 2 ) and the coefficients of $L$ satisfy the definition ( Elliptic differential operator ).

Problem

(Variational Dirichlet problem for generic parabolic operator) Find MATH such that MATH for any MATH . The form of $B$ is given by the definition ( Bilinear form B 2 ).

Problem

(Time dependent elliptic problem) Under assumptions of the definition ( Dirichlet problem for generic parabolic operator ) find the function MATH such that MATH We introduce the notation MATH . We choose the number $\kappa$ so that the estimate

MATH (Energy estimate 2)
of the proposition ( Energy estimates for the bilinear form B ) holds and the proposition ( Existence of weak solution for elliptic Dirichlet problem 1 ) holds for $\mu\geq\kappa$ .

Problem

(Time dependent elliptic variational problem) Find $u$ such that MATH for any MATH We introduce the notation MATH .

Definition

(Solution operator for generic elliptic problem) We denote MATH the solution operator for the problem ( Time dependent elliptic problem ): MATH

Remark

Let $u$ be the solution of the problem ( Dirichlet problem for generic parabolic operator ). We introduce the function MATH then

MATH (Transformation of parabolic problem 1)
where MATH . Consequently
MATH (Transformation of parabolic problem 2)

Definition

(Partially inverted problem) We assume existence of approximate solution MATH for the problem ( Solution operator for generic elliptic problem ). Then we introduce the problem MATH and consider the function $u_{h}$ , MATH to be the approximation of $u$ .

Condition

(Properties of solution operator 2) We assume that the operator MATH has the following properties

1. MATH MATH 2. There is an integer $r$ such that for $2\leq s\leq r$ MATH

3. MATH

Proposition

Let $T_{h}$ be defined by the problem MATH and the spaces $S_{h}$ satisfy the condition ( Finite dimensional approximation 1 ). Then the operator $T$ satisfies the condition ( Properties of solution operator 2 ).

Proof

According to MATH we have MATH and according to the formula ( Energy estimate 2 ) MATH This also establishes that MATH and MATH iff $T_{h}\chi=0$ . According to MATH , MATH hence MATH This completes the proof of the condition ( Properties of solution operator 2 )-1.

We proceed to prove the condition ( Properties of solution operator 2 )-2. We introduce the notation MATH . According to the formula ( Energy estimate 2 ) MATH Note that MATH hence MATH and by the proposition ( Energy estimates for the bilinear form B ) MATH Therefore, for any $\chi\in S_{h}$ MATH and according to the condition ( Finite dimensional approximation 1 ) this implies MATH We now estimate MATH We introduce the function $\psi$ such that MATH then MATH for any $\chi\in S_{h}$ . We apply the proposition ( Energy estimates for the bilinear form B ): MATH and substitute the recent result MATH : MATH We apply the condition ( Finite dimensional approximation 1 ) with $s=2$ to the term MATH where we have the freedom of choosing $\chi$ : MATH We invoke the proposition ( Boundary elliptic regularity ) for $\psi$ with $m=0$ : MATH : MATH Thus MATH The results MATH taken with the proposition ( Boundary elliptic regularity ) applied to $e$ : MATH constitute MATH It remains to prove the MATH part of the condition ( Properties of solution operator 2 )-2. We introduce the notation $B^{\prime}$ for the time derivative of the form $B_{\kappa}$ so that the differentiation of MATH is written as MATH According to the formula ( Energy estimate 2 ) MATH where we introduced an arbitrary $\chi\in S_{h}$ . Thus, $e_{t}+\chi$ may be regarded as MATH . Each term has the following energy estimates MATH MATH MATH We derive MATH We have the inequality of the form MATH . We introduce MATH and we are given that the domain of MATH is such that $f\geq0$ . In this context $x\geq0$ and it must lie between the roots of the quadratic polynomial for permitted values of $y,z$ . From this analysis there is constant $C_{1}$ such that MATH Hence, we continue MATH and apply the result MATH and the condition ( Finite dimensional approximation 1 ) MATH The MATH is a solution of the problem MATH Hence by the proposition ( Energy estimates for the bilinear form B ) and ( Boundary elliptic regularity ) MATH . We conclude MATH which is not quite the goal yet. To gain the extra power of $h$ we perform the following calculation.

Let $B_{\kappa}^{\ast}$ is the adjoint of $B_{\kappa}$ and for arbitrary $\varphi$ let $\psi$ is such that MATH We act as above: MATH We integrate by parts in the last term MATH We apply the condition ( Finite dimensional approximation 1 ) MATH We apply the proposition ( Boundary elliptic regularity ) MATH and the results MATH MATH and arrive to the remaining statement (2): MATH MATH

The statement (3) is a direct verification.

Proposition

Assume that the conditions ( Properties of solution operator 2 )-1,2 hold, $u$ is the solution of the problem ( Variational Dirichlet problem for generic parabolic operator ) and $u_{h}$ is defined in the problem ( Partially inverted problem ). Then we have the error estimate MATH for $t\in(0,T]$ .

Proof

We introduce the quantity MATH and calculate the error equation exactly as in the proof of the proposition ( Galerkin convergence 4 ), (change MATH ): MATH We apply the proposition ( Partial inversion lemma ): MATH and estimate every term on the right.

According to the condition ( Properties of solution operator 2 )-2 MATH MATH and we apply the condition ( Properties of solution operator 2 )-2 again MATH Hence, MATH where we used the proposition ( Parabolic regularity 1 ).





Notation. Index. Contents.


















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