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.
4. Construction of approximation spaces.
A. Finite element.
B. Averaged Taylor polynomial.
a. Properties of averaged Taylor polynomial.
b. Remainder of averaged Taylor decomposition.
c. Estimates for remainder of averaged Taylor polynomial. Bramble-Hilbert lemma.
d. Bounds for interpolation error. Homogeneity argument.
C. Stable space splittings.
D. Frames.
E. Tensor product splitting.
F. Sparse tensor product. Cure for curse of dimensionality.
5. Time discretization.
6. Variational inequalities.
VIII. Bibliography
Notation. Index. Contents.

Bounds for interpolation error. Homogeneity argument.


e have introduced a notion of finite element MATH and interpolation operator $I_{\Omega}$ in the section ( Finite element ). In this section we provide generic estimates for the difference $v-I_{\Omega}v$ .

The $\Omega$ is a bounded closed subset of $\QTR{cal}{R}^{n}$ with non-empty interior and smooth boundary. The condition MATH means that MATH includes evaluation of derivatives up to $l$ -th order.

Proposition

(Boundedness of interpolation operator) Let MATH and MATH . Then MATH

Proof

According to the definition ( Interpolant ) MATH

We introduce the notations MATH

Proposition

(Estimate of interpolation error) Suppose the finite element MATH satisfies the following conditions:

1. $\Omega$ is star-shaped with respect to the ball MATH , MATH (see the definition ( Chunkiness parameter )),

2. MATH ,

3. MATH .

If either MATH or $p=1,m-l-n\geq0$ then we have for MATH MATH

Proof

Note that MATH thus, by the proposition ( Properties of interpolant )-3, MATH We estimate MATH We use MATH MATH We apply the proposition ( Boundedness of interpolation operator ). MATH We apply the proposition ( Sobolev inequality 2 ), this is where we need the restrictions MATH or $p=1,m-l-n\geq0$ . MATH We apply the proposition ( Bramble-Hilbert lemma ). MATH

We need to extract the dependency on $d$ in explicit form. We apply the so called "homogeneity argument". We have MATH . We suppress dependency of $C$ on other parameters in the notation: MATH for any function MATH .We are going to perform the change of variables $x=\hat{x}$ $d$ and track the parameter $d=diam~\Omega$ as it arises from every term.

We set MATH and calculate MATH MATH MATH Let MATH be a finite element connected to the finite element MATH via the following relationships: MATH and let MATH , MATH be the dual bases for MATH and MATH respectively, connected by the same relationships. We have MATH Thus, MATH and $I_{\Omega}v$ are connected by the relationship MATH . Hence, we apply the result MATH to MATH and conclude from $\left( \#\right) $ that MATH with a constant $C$ independent from $d$ .

Remark

For any parameter that scales one may assume that the parameter is equal to 1, complete the calculation and then recover the dependency on such parameter via the homogeneity argument.

Proposition

(Estimate of interpolation error 2) Under conditions of the proposition ( Estimate of interpolation error ) we have MATH

Proof

According to the proposition ( Estimate of interpolation error ) we have MATH We now apply the homogeneity argument as in the proof of the proposition ( Estimate of interpolation error ). MATH thus MATH

Proposition

(Estimate of interpolation error 3) Under conditions of the proposition ( Estimate of interpolation error ) we have MATH

Proof

We use the notation of the proof of the proposition ( Estimate of interpolation error ) and apply the proposition ( Sobolev inequality 2 ): MATH We apply the proposition ( Estimate of interpolation error ). MATH The rest follows by the homogeneity argument of the proof of the proposition ( Estimate of interpolation error ).





Notation. Index. Contents.


















Copyright 2007