# Approximation of a differential operator by difference operators

An approximation of the differential operator by parameter-dependent operators such that the result of their application to a function is determined by the values of this function on some discrete set of points — a grid — which become more exact as its parameter (mesh, step of the grid) tends to zero.

Let , be a differential operator which converts any function of a class of functions into a function of a linear normed space . Let be the domain of definition of the functions in , and let there be some discrete subset in — a grid — which "becomes more dense" as . Consider the set of all functions defined on the grid only and coinciding with in the points of the grid. A difference operator is defined as any operator that converts the grid functions in into functions in . One says that the operator , , represents an order approximation to the differential operator on if for any function

as . Occasionally, an approximation is understood to be the equality

in the sense of some weak convergence. The approximation of a differential operator by difference operators is used for an approximate computation of the function from the table of values of the function and for the approximation of a differential equation by difference equations.

There are two principal methods for constructing operators approximating .

In the first method is defined as the result of applying the differential operator to a function in , obtained by some interpolation formula from the grid function .

The second method is as follows. In the domain of definition of a function in one introduces a grid , and considers the linear space of grid functions defined on . The operator is constructed as the product of two operators: an operator which converts the function into the grid function in , i.e. into a table of approximate values of , and an operator which extends from to the entire domain . For instance, in order to approximate the differential operator

one constructs the grid consisting of points , ,

and a grid consisting of the points

The values of the operator at the points are defined by the equations:

Thereafter, the definition of is piecewise linearly extended outside with possible breaks at the points , , only.

Let the norm in F be defined by the formula

Then, on the class of functions with a bounded third derivative, for and the operator represents an order 1, respectively 2, approximation to . On the class of functions with bounded second derivatives, the representation is of order 1 only, for any .

The task of approximating a differential operator by finite-difference operators is sometimes conditionally considered as solved if a method is found for the construction of the grid function

determined at the points of only, while the task of completing the function everywhere on is ignored. In such a case the approximation is defined by considering the space as normed, and by assuming, for the grid and for the norm, that for any function , the function , which coincides with at the points of , satisfies the equation

The operator is understood to be an operator from in , and one says that represents an order approximation to on if, for ,

In order to construct an operator which is an approximation to of given order on sufficiently smooth functions, one often replaces each derivative contained in the expression by its finite-difference approximation, basing oneself on the following fact. For any integers and for any , , in the equation

it is possible, by using the method of undetermined coefficients and Taylor's formula, to select numbers not depending on , so that for any function with () bounded derivatives, an inequality of the type

where depends only on and , is valid. As an example, suppose one constructs an approximating operator for the Laplace operator ,

if is the closed square , and is its interior . Assume that where is a natural number, and construct the grid with points

which belong to . The points

then belong to , for integers and . Since

can be approximated with second-order accuracy on a space of sufficiently smooth functions by the finite-difference operator if one puts, at the points of :

where and are the values of the functions and at the point .

There are also other methods of constructing operators which are approximations to the operator on the space of solutions of the differential equation , and which satisfy additional conditions.

#### References

 [1] A.F. Filippov, "On stability of difference equations" Dokl. Akad. Nauk SSSR , 100 : 6 (1955) pp. 1045–1048 (In Russian) [2] I.S. Berezin, N.P. Zhidkov, "Computing methods" , Pergamon (1973) (Translated from Russian)