Error analysis on bi-parameter finite elements

Abstract

A bi-parameter finite element is introduced in this paper, in which a new local grid refinement technique is presented for a class of second-order elliptic equations. The new approach is based on a bi-parameter finite element space defined, respectively, on the whole domain with size H and on a subdomain with size h(h ⪡ H). The stability and convergence analysis of the bi-parameter finite element approximation is studied. The behavior of approximation with respect to the parameters H and h is presented. In particular, the error estimates for the finite element Green's function expressed by H and h are derived.