A new 3d–2d interior error estimate independent on the geometry of a linear elastic plate

Abstract

We study how three-dimensional linearized elasticity for thin plates can be approximated by a two-dimensional projection. The classical approach using formal asymptotic expansions in powers of the thickness in the Hellinger–Reissner formulation, only provides error estimates in the H¹ norm for the displacements, assuming at least L² regularity for the applied forces, plus additional regularity for some components. Here we make use of elliptic regularity theory. We prove a 3d–2d interior error estimate between the 3d displacement solution and its 2d projection. Moreover the constants involved in our estimate are independent on the particular geometry of the plate. Our approach yields an H² error estimate, assuming only L² regularity for the applied forces, which is optimal from the point of view of elliptic regularity theory. We also obtain interior W^k,p and C^k,α error estimates.