Weighted Residual Formulation of Approximate Methods

Abstract

Boundary integral equations were until recently, considered to be a different type of analytical method, somewhat unrelated to other approximate methods such as finite differences or finite elements. They became known in Europe through a series of Russian authors amongst them Mikhlin [1], Kupradze [2] and Smirnov [3] and a predecessor of this work — Kellogg [4] — who applied integral equations to solve potential problems in 1953. Work in other numerical methods was also greatly influenced by the work of Kantorovich-Krylov [5], Courant and Hilbert [6], Collatz [7] and Morse and Fesbach [8], This work had important consequences for the development of finite elements in the 1960’s although this is not clearly understood by many researchers. Another important development in approximate analysis was the investigation of mixed formulations which allows for the inclusion of different types of governing equations and subsidiary conditions in a variational type statement. These formulations are nowadays called Hu-Washizu and their exposition for structural mechanics can be seen in the book by Washizu [8].