On projection-difference analogues of the identity operator

Abstract

In this note an answer is given to the question of A. B. Borisov, formulated during the Second Soviet School “Contemporary Problems of Numerical Analysis” (Khar’kov, October 1990). The question concerns the structure of projectivegrid analogues of the operator ∂/∂t − ∆; in a slightly reformulated form the problem leads to constructing the approximating subspaces with the optimal rate of approximation in the metric of W 1 2 such that the projective-grid analogue of the identity operator has the same five point stencil as the simplest projective-grid analogue of the Laplace operator. Below the negative answer to this question is given, i.e., it is shown that there exists no finite element subspace of the space W 1 2 whose approximation properties are optimal with respect to the rate and which generates an analogue of the unit operator with the five point “cross-type” stencil. Moreover, a full description of the projective-grid analogues of the identity operator is given. Let E2 be the two-dimensional Euclidean space, and Z2 the set of integervalued vectors in E2. We consider the prolongation operators of the grid functions of the form