Approximation Theory for the P-Version of the Finite Element Method in Three Dimensions Part II: Convergence of the P Version of the Finite Element Method

Already very early Prof. Oleg Zienkiewicz pointed to potential advantages of using higher order instead of low order shape functions for the finite element method [1]. The p-version as a systematic extension process of the finite element method leaves the mesh unchanged and increases the polynomial degree of the shape functions locally or globally [5]. It has turned out to be a very efficient discretization strategy for many linear elliptic problems, for example, the Poisson equation, the Lamé equations, the Reissner–Mindlin problem, shell discretizations etc [e.g. 2,6]. For this class of problems the p-version is in general superior to the classical h-version approach. Also, in the case of singularities, the p-version shows an exponential rate of convergence in energy norm in the preasymptotic range, when combined with a proper mesh design [5]. In recent years, the p-version has been further developed to solve nonlinear problems such as hyperelasticity, elastoplasticity, fluid dynamics, fluid-structure interaction [e.g. 4,9] and it has been demonstrated that it can be efficiently applied to industrial problems arising for example in sheet metal forming [7]. A very important advantage of p-FEM is its ability to use elements with a very large aspect ratio. This feature allows to model even very thin-walled structures in a strictly three-dimensional setting [3]. Moreover, the inherent independence of the geometric shape from the space of the Ansatz functions opens many possibility to a thight connection of CAD-models and a p-FEM computation. In combination with an error estimation being based on the inherent hierarchic nature of the p-FEM, a model adaptivity for best approximation of a given physical problem is also possible [8]. The presentation will give an overview over the achievements of the p-version during the last two decades, addressing also recent results related to nonlinear problems in Computational Mechanics.