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 and interpolation operator in the section ( Finite element ). In this section we provide generic estimates for the difference .

The is a bounded closed subset of with non-empty interior and smooth boundary. The condition means that includes evaluation of derivatives up to -th order.

Proposition

(Boundedness of interpolation operator) Let and . Then

Proof

According to the definition ( Interpolant )

We introduce the notations

Proposition

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

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

2. ,

3. .

If either or then we have for

Proof

Note that thus, by the proposition ( Properties of interpolant )-3, We estimate We use We apply the proposition ( Boundedness of interpolation operator ). We apply the proposition ( Sobolev inequality 2 ), this is where we need the restrictions or . We apply the proposition ( Bramble-Hilbert lemma ).

We need to extract the dependency on in explicit form. We apply the so called "homogeneity argument". We have . We suppress dependency of on other parameters in the notation: for any function .We are going to perform the change of variables and track the parameter as it arises from every term.

We set and calculate Let be a finite element connected to the finite element via the following relationships: and let , be the dual bases for and respectively, connected by the same relationships. We have Thus, and are connected by the relationship . Hence, we apply the result to and conclude from that with a constant independent from .

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

Proof

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

Proposition

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

Proof

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

 Notation. Index. Contents.