Grid function

Grids and grid functions. The composition of a difference scheme approximating a differential equation of interest amouts to performing the following operations ... 

Throughout the entire chapter, the functions u(x) of the continuous argument x G G are the elements of some functional space Hq- The space Hh comprises all of the grid functions yii(x), providing a possibility to replace within the framework of the finite difference method the space Hq by the space Hh of grid functions yh x). Recall that although the fixed notation is usually adopted, there is a wide variety of possible choices of the functional form of . ... 

Under the second approach mentioned above we proceed to the accurate account of the errors of difference methods in the space of grid functions. In the most cases the spaces involved appear to be finite-dimensional. 

After preliminary discussions of the simplest examples illustrating some ways of producing grids and, thereby, of forming the spaces Hh of grid functions we concentrate primarily on the problem of the difference approximation of differential operators. 

Example 5 Lv =. In that case the values of a grid function at... 

The next step is to introduce an operator L in the space Hq and an operator Lh carrying a grid function into a grid function Lh Vk on the grid Lh Hh Hh)-A grid function... 

Remark 1 We give below several examples of projectors Vh onto the set of grid functions ... 

The main goal of any approximate method is to solve an original (continnons) problem with a prescribed accuracy e > 0 in a finite number of operations. In order to clarify whether it is possible in principle to approximate a solution u of problem (35)-(36) by a solntion j/ , of problem (37) with any prescribed accuracy e > 0 depending on the step h[e), we follow established practice. This is concerned with further comparison of with u x) in the space of grid functions Hh. Let be a value of the function u x) on the grid u>i, so that Hh- The error of... 

In Section 1 we have already introduced two types of difference derivatives for grid functions the left and the right ones, which correspond to different formulae for difference differentiating of a product... 

For grid functions an analog of the Green formula can be obtained by the summation by parts formulae. This can be done by substituting... 

Any grid function f x) defined on the grid arranges itself into a... 

Let be the set of grid functions defined at the inner nodes of the grid W/j. The set so constructed is certainly linear. Once equipped with the inner product (y, v) = Vi h and associated norm y = / y, y),... 

Example 5 Consider now the third boundary-value problem (9). As in Example 2 of Section 1 it will be convenient to introduce the space = 0 , of the dimension A +1 consisting of all grid functions defined on the uniform ... 

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