The misconception in finite element method and revision from quasi-conforming

Abstract

The theory and application of present finite element method have misconceptions. These misconceptions include the insufficient understanding of functional space of finite element and the formulation of weak form derivatives. These misconceptions block the development of finite element method, which lead to the difficulty and failure in solving some problems. The theory and application of quasi-conforming finite element method are displayed as a reference. A sequence of functions is defined to demonstrate the functional space of finite element. The independence principle of elements is given to solve the "conforming problem". The polynomial basis rather than "shape function" is emphasized. The formulation of approach functions of finite element is discussed, which should convergence to the corresponding Taylor series. It is proved that the weak form of strain-displacement equation is necessary and sufficient condition of weak form of equilibrium equation. The string net weak derivative is proposed, which expands the connotation of weak form derivative in finite element method.