VARIATIONAL CRIMES IN THE FINITE ELEMENT METHOD

Abstract

This chapter focuses on variational crimes in the finite element method. The finite element method is nearly a special case of the Rayleigh-Ritz technique. The convenience and effectiveness of the finite element technique is regarded as conclusively established; it has brought a revolution in the calculations of structural mechanics, and other applications are rapidly developing. The chapter reviews the modifications of the Ritz procedure which have been made to achieve an efficient finite element system. On a regular mesh one could regard the system of Ritz-finite element equations KQ = F as a finite difference scheme, and then the patch test would be equivalent to the formal consistency of the difference equations with the correct differential equation. The chapter discusses the convergence theory for non-conforming elements. It is impossible for a polynomial to satisfy a condition like u = 0 on a general curved boundary. Therefore, some alteration in the boundary condition will be necessary. The most important possibility is to change the domain.