Partition of Unity Finite Element Analysis of Nonlinear Transient Diffusion Problems Using p -Version Re finement

摘要

We propose a high -order enriched partition of unity finite element method for linear and nonlinear time -dependent diffusion problems. The solution of this class of problems often exhibits non -smooth features such as steep gradi- ents and boundary layers which can be very challenging to recover using the con- ventional low -order finite element methods. A class of steady-state exponential functions has been widely used for enrichment and its performance to numerically solve these challenges has been demonstrated. However, these enrichment func- tions have been used only in context of the standard h -version refinement or the so-called q -version refinement. In this paper we demonstrate that the p -version refinement can also be a very attractive option in terms of the efficiency and the accuracy in the enriched partition of unity finite element method. First, the tran- sient diffusion problem is integrated in time using a semi -implicit scheme and the semi -discrete problem is then integrated in space using the p -version enriched finite elements. Numerical results are presented for three test examples of time - dependent diffusion problems in both homogeneous and heterogeneous media. The computed results show the significant improvement when using the p -version refined enriched approximations in the finite element analysis. In addition, these results support our expectations for a robust and high -order accurate enriched par- tition of unity finite element method.