The use of the covering condition in different formal elliptic boundary value problems tackled by the finite element methods: An interface problem

Among the existing tools to analyze and model the partial differential equations, the Lopatinskii\u27s Theorem (1953) has really made a breakthrough. This paper is concerned with, first, to use the Covering Condition that is based on this statement to verify the well-posed quality of the Elliptic Boundary Value Problems tackled by the main conforming and nonconforming finite element theory. It is understood in the sense that it prepares for the well-posed Abstract Variational Problem, which in its turn, give senses to the Discrete Variational Problem, used by finite element methods. Next, this study builds up a stable and convergent system of approximation of a given space V, the domain of the Elliptic Boundary Value Problem. This system consists of a family of finite elements (regular or not), a Gelfand Triple, V ( H = H\u27 ( V\u27, (These spaces are Slobodeckii - Sobolev spaces), a continuous monomorphism M (or epimorphism), defined as a product of an extension mapping Fh, and a restriction rh. Altogether, the Elliptic Boundary Value Problem and this system constitute an Interface Problem between the Elliptic Boundary Value Problem and FEM (Finite Element Method). If this interface satisfies respectively the Lopatinskii - Sapiro Condition, or the Lax - Milgram Lemma, we call it, the First Interface Problem, or Second Interface Problem. If it satisfies at the same time the two conditions, it is called Complete Interface Problem. It is shown that this system fits for different Dirichlet, Neumann, Robin Problems... The study of error convergence reveals the internal interaction between the two faces of this Interface Problem..