The Logarithmic finite element method in a multigrid setting

Abstract

AbstractThe Logarithmic finite element (“LogFE”) method is a novel finite element approach for solving boundary‐value problems proposed in [1]. In contrast to the standard Ritz‐Galerkin formulation, the shape functions are given on the logarithmic space of the deformation function, which is obtained by the exponentiation of the linear combination of the shape functions given by the degrees of freedom.Unlike many existing multigrid formulations, the LogFE method allows for a very smooth interpolation between nodal values on the coarse grid. It can thus avoid problems with regard to locking and convergence that appear in multigrid applications using only linear interpolation, especially for larger corsening factors.We illustrate the use of the LogFE method as a coarse grid algorithm, in conjunction with an atomistic finite element method on the fine grid, for calculating the mechanical response of super carbon nanotubes. (© 2016 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)