External Approximations by FE 2

it just stated that the error of non-conformity of the approximation must tend to zero when the mesh of FE is refined. So, the justification of the specific non-conformed FE was performed in work [22] for the model boundary value problems using the conventional technique for obtaining the estimates of the non-conformity errors.

In the above-mentioned works, the justification of the specific non-conformed FE was performed for specific types of boundary value problems, and the properties of the corresponding equations were extensively utilized. In such justifications, the proof technique usually assumed that a subspace of conformed FE was present in the space of non-conformed FE. This assumption reduced the applicability of the results obtained and prevented the development of general approaches to the construction of non-conformed approximations. In fact, the suggested criteria only helped to find out what kind of non-conformed functions can be added to the basic functions of conformed FE in order to improve accuracy on rough partitions of the domain.

Another approach to treat non-conformed approximations is based on the concept of the external approximations introduced in work [32]. The general theory of the external approximations of Sobolev spaces and variational equations has been developed in work [33] where were also given the examples of its application to the finite difference approximations and the approximations by  parts. In terms of the theory of external approximations some non-conformed FE have been studied in works [11, 28, 34].

Let us note that the problem of justification of non-conformed FE is closely related to the  problem of combination of approximations of different types on sub-domains, and with the