ISSN 0012-2661, Differential Equations, 2014, Vol. 50, No. 7, pp. 955–970. c  Pleiades Publishing, Ltd., 2014. Original Russian Text c  M.R. Timerbaev, 2014, published in Differentsial’nye Uravneniya, 2014, Vol. 50, No. 7, pp. 963–978. NUMERICAL METHODS A Posteriori Error Estimates in the Finite Element Method for Elliptic BVP with Degeneration M. R. Timerbaev Kazan Federal University, Kazan, Russia e-mail: Marat.timerbaev@kpfu.ru Received February 13, 2014 Abstract—We consider a class of elliptic boundary value problems with degenerating coeffi- cients for which we construct FEM schemes with optimal convergence on the basis of multiplica- tive extraction of the singularity. For a scale of weighted Sobolev norms including the energy norm of the differential operator, we prove a posteriori estimates for the error of the discrete solutions. DOI: 10.1134/S0012266114070118 1. INTRODUCTION Numerous boundary value problems of elliptic type are stated in variational form: find a function u ∈ U such that a(u, v)= 〈f,v〉, v ∈ V. (1) Here (U, V ) is a dual, with respect to the bilinear form a, pair of Hilbert or Banach spaces of functions (possibly, distributions), f ∈ V ′ (V ′ is the dual space of V ), and the brackets 〈· , ·〉 stand for the duality between V ′ and V . In this statement, it is natural to require the continuity of the form a on U × V , |a(u, v)|≤ M ‖u‖ U ‖v‖ V , u ∈ U, v ∈ V. (2) The continuous bilinear forms and the linear continuous operators A : U → V ′ [we write A ∈ B(U → V ′ )] are in a one-to-one correspondence given by the relation a(u, v)= 〈Au, v〉 for u ∈ U and v ∈ V . Consequently, problem (1) is equivalent to the operator equation Au = f . The following assertion is a straightforward consequence of the general theory of linear continuous operators in Banach spaces. Theorem 1. If U and V are reflexive spaces, then the following assertions are equivalent. (a) The operator A is an isomorphism of U onto V ′ ; i.e., there exists an inverse operator A −1 ∈B(V ′ → U ). (b) The bilinear form a associated with the operator A satisfies the inf–sup condition inf u∈U sup v∈V a(u, v) ‖u‖ U ‖v‖ V = c a > 0. (3) (c) The adjoint operator A ′ is an isomorphism of V onto U ′ . (d) The following dual inf–sup condition is satisfied : inf v∈V sup u∈U a(u, v) ‖u‖ U ‖v‖ V = c ′ a > 0. If one of these assertions holds, then ‖A −1 ‖ V ′ →U = 1 c a = 1 c ′ a = ‖A ′−1 ‖ U ′ →V . 955