The patch test and the general convergence criteria of the finite element method

Abstract

A synthetic description of continuous structural models followed by some general ideas on hybrid discrete models is first presented. An approximation theorem is then established which leads to a general expression of an upper bound of the discretization error, equally dependent on the strain and displacement interpolation errors, as well as on the exact and approximate solutions. The last part of the paper is devoted to the justification of the patch test with the help of such expression. It becomes clear that passing the patch test is not a necessary condition for convergence and that the simple patch test is sufficient for accuracy and not merely for convergence analysis.