Content of present website is being moved to www.lukoe.com/finance . Registration of www.opentradingsystem.com will be discontinued on 2020-08-14.
Quantitative Analysis
Parallel Processing
Numerical Analysis
C++ Multithreading
Python for Excel
Python Utilities
Services
Author
Printable PDF file
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.
a. H-tilde spaces.
b. Convergence of finite elements with nonsmooth initial condition.
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 with nonsmooth initial condition.


roblem

(Parabolic problem with nonsmooth initial condition) For a bounded set MATH with $C^{\infty}$ boundary $\partial U$ and positive $T$ find the MATH such that MATH where MATH .

Proposition

(Solution operator for PP with nonsmooth IC) Using the notation of the definition ( Eigenfunctions of Laplacian ) the solution of the problem ( Parabolic problem with nonsmooth initial condition ) is given by the expression MATH

Proposition

(Instant smoothness of solution operator for PP 1)

1. If MATH then MATH for any $s\geq0$ and $t>0$ .

2. If MATH then for any $q\geq0,~p\geq0$ , $0\leq s\leq q$ and $t>0$ we have MATH

Proof

We calculate MATH We estimate the expression MATH We substitute $\lambda_{k}t=\tau$ MATH Let MATH then MATH We continue the evaluation of MATH : MATH

Proposition

(Partial inversion lemma 2) Let MATH where the operator $T_{h}$ is nonnegative MATH and symmetric for some scalar product MATH . Then for the corresponding norm MATH we have

1. MATH

2. MATH

3. MATH

4. For $\varepsilon>0$ MATH

Proof

(1) Note that MATH and by the symmetry of $T_{h}$ MATH We apply the operation MATH to the equality $T_{h}e_{t}+e=\rho$ : MATH

or, by the proposition ( Cauchy inequality ), MATH Therefore, MATH and, after integration, MATH It remains to note that MATH and MATH .

Proof

(2) We apply the operation MATH to the equality $T_{h}e_{t}+e=\rho$ : MATH or MATH Since MATH we derive MATH We combine the above with the identities MATH to obtain MATH and integrate MATH We use the proposition ( Cauchy inequality with epsilon ) MATH We make MATH small, divide by $t$ and arrive to MATH We apply the statement (1) and conclude (2).

Problem

(Partially inverted semi discrete parabolic problem 1) Assuming existence of the operator $T_{h}$ as in the condition ( Properties of solution operator ) we pose the problem of finding MATH such that MATH

Proposition

(Galerkin convergence 5) Let $T$ be the solution operator of the problem ( Poisson equation with Dirichlet boundary condition ) and the operator MATH is given and satisfies the condition ( Properties of solution operator ). Assume that spaces $S_{h}$ satisfy the condition ( Finite dimensional approximation 1 ). Let $u_{h}$ be the solution of the problem ( Partially inverted semi discrete parabolic problem 1 ) and $u$ be the solution of the problem ( Parabolic problem with nonsmooth initial condition ). Assume that MATH for some MATH . Then we have MATH

Proof

Let $\tilde{u}_{h}$ be a solution of the problem ( Partially inverted semi discrete parabolic problem 1 ) with $g_{h}=P_{h}g$ . Then MATH and, for MATH being the solution operator of the problem ( Partially inverted semi discrete parabolic problem 1 ), MATH It may be shown that there is an estimate for MATH similar to the one of the proposition ( Instant smoothness of solution operator for PP 1 ), hence MATH by the conditions of this proposition, the condition ( Finite dimensional approximation 1 ) and proposition ( Characterization of H-tilde spaces ) MATH Hence, it is enough to prove the statement for $g_{h}=P_{h}g$ .

We intend to apply the proposition ( Partial inversion lemma 2 )-3. We introduce $e=u_{h}-u$ and recall from the proof of the proposition ( Galerkin convergence 4 ) that MATH Hence, MATH according to the condition ( Properties of solution operator )-2 MATH and by the proposition ( Instant smoothness of solution operator for PP 1 ) MATH Similarly, MATH Finally, for any MATH MATH by symmetry of $T_{h}$ as in the condition ( Properties of solution operator )-1, MATH because $T_{h}w\in S_{h}$ . Hence, MATH The statement now follows from the proposition ( Partial inversion lemma 2 )-3.

Proposition

(Galerkin convergence 6) Let $T$ be the solution operator of the problem ( Poisson equation with Dirichlet boundary condition ) and the operator MATH is given and satisfies the condition ( Properties of solution operator ). Let $u_{h}$ be the solution of the problem ( Partially inverted semi discrete parabolic problem 1 ) and $u$ be the solution of the problem ( Parabolic problem with nonsmooth initial condition ). Assume that MATH Then we have

1. MATH

2. MATH

Proof

(1) We introduce $e=u_{h}-u$ and $w=te$ and recall from the proof of the proposition ( Galerkin convergence 4 ) that MATH hence MATH We have MATH MATH We apply the proposition ( Partial inversion lemma 2 )-4 with MATH : MATH We set MATH such that MATH then MATH thus MATH We estimate the term MATH as follows. We integrate the relationships over $\left( 0,t\right) $ MATH to obtain MATH Then according to the proposition ( Partial inversion lemma 2 )-3, MATH Thus from MATH : MATH We combine this with MATH and obtain MATH Thus MATH We estimate the components as follows. MATH According the condition ( Properties of solution operator )-2 MATH and according to the proposition ( Instant smoothness of solution operator for PP 1 ) MATH We estimate the other terms similarly: MATH MATH Therefore MATH as claimed in the proposition for $r=2$ .

We now prove the statement for all $r$ as follows.

We introduce the error operator MATH by the relationship MATH The following identity holds MATH Indeed, by direct substitution we have MATH MATH We use the property MATH : MATH MATH The MATH because MATH acts MATH . Thus $P_{h}$ does not move the result of MATH . Also, MATH . MATH

We now estimate every component of the RHS in the identity MATH . According to the proposition ( Galerkin convergence 5 ), MATH and according to the proposition ( Instant smoothness of solution operator for PP 1 ) MATH The evaluation of other components is similar.

Proof

(2) We perform induction over $k$ . The $k=0$ is covered in the statement (1). Assume that the result holds for $k-1$ . We introduce the notation MATH We have MATH MATH We apply the proposition ( Partial inversion lemma 2 )-4 to the above result: MATH In the above expression the terms depending on MATH are estimated by induction assumption and the MATH terms are estimated by the condition ( Properties of solution operator )-2 and the proposition ( Instant smoothness of solution operator for PP 1 ). Then the statement follows by the same means as in the proof of (1).





Notation. Index. Contents.


















Copyright 2007