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 , , such that for some functions and . The is assumed to be a bounded domain with a boundary. The functions and are assumed to satisfy the compatibility conditions of the proposition ( Parabolic regularity 2 ). The form of the operator is given by the formula ( Operator L 2 ) and the coefficients of satisfy the definition ( Elliptic differential operator ).

Problem

(Variational Dirichlet problem for generic parabolic operator) Find such that for any . The form of 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 such that We introduce the notation . We choose the number so that the estimate

 (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 .

Problem

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

Definition

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

Remark

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

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

Definition

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

Condition

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

1. 2. There is an integer such that for

3.

Proposition

Let be defined by the problem and the spaces satisfy the condition ( Finite dimensional approximation 1 ). Then the operator satisfies the condition ( Properties of solution operator 2 ).

Proof

According to we have and according to the formula ( Energy estimate 2 ) This also establishes that and iff . According to , hence 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 . According to the formula ( Energy estimate 2 ) Note that hence and by the proposition ( Energy estimates for the bilinear form B ) Therefore, for any and according to the condition ( Finite dimensional approximation 1 ) this implies We now estimate We introduce the function such that then for any . We apply the proposition ( Energy estimates for the bilinear form B ): and substitute the recent result : We apply the condition ( Finite dimensional approximation 1 ) with to the term where we have the freedom of choosing : We invoke the proposition ( Boundary elliptic regularity ) for with : : Thus The results taken with the proposition ( Boundary elliptic regularity ) applied to : constitute It remains to prove the part of the condition ( Properties of solution operator 2 )-2. We introduce the notation for the time derivative of the form so that the differentiation of is written as According to the formula ( Energy estimate 2 ) where we introduced an arbitrary . Thus, may be regarded as . Each term has the following energy estimates We derive We have the inequality of the form . We introduce and we are given that the domain of is such that . In this context and it must lie between the roots of the quadratic polynomial for permitted values of . From this analysis there is constant such that Hence, we continue and apply the result and the condition ( Finite dimensional approximation 1 ) The is a solution of the problem Hence by the proposition ( Energy estimates for the bilinear form B ) and ( Boundary elliptic regularity ) . We conclude which is not quite the goal yet. To gain the extra power of we perform the following calculation.

Let is the adjoint of and for arbitrary let is such that We act as above: We integrate by parts in the last term We apply the condition ( Finite dimensional approximation 1 ) We apply the proposition ( Boundary elliptic regularity ) and the results and arrive to the remaining statement (2):

The statement (3) is a direct verification.

Proposition

Assume that the conditions ( Properties of solution operator 2 )-1,2 hold, is the solution of the problem ( Variational Dirichlet problem for generic parabolic operator ) and is defined in the problem ( Partially inverted problem ). Then we have the error estimate for .

Proof

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

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

 Notation. Index. Contents.