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

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 gradients and boundary layers which can be very challenging to recover using the conventional 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 functions 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 transient 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 timedependent 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 partition of unity finite element method.