A partition of unity FEM for time-dependent diffusion problems using multiple enrichment functions

Abstract

An enriched partition of unity FEM is developed to solve time-dependent diffusion problems. In the present formulation, multiple exponential functions describing the spatial and temporal diffusion decay are embedded in the finite element approximation space. The resulting enrichment is in the form of a local asymptotic expansion. Unlike previous works in this area where the enrichment must be updated at each time step, here, the temporal decay in the solution is embedded in the asymptotic expansion. Thus, the system matrix that is evaluated at the first time step may be decomposed and retained for every next time step by just updating the right-hand side of the linear system of equations. The advantage is a significant saving in the computational effort where, previously, the linear system must be reevaluated and resolved at every time step. In comparison with the traditional finite element analysis with p-version refinements, the present approach is much simpler, more efficient, and yields more accurate solutions for a prescribed number of DoFs. Numerical results are presented for a transient diffusion equation with known analytical solution. The performance of the method is analyzed on two applications: the transient heat equation with a single source and the transient heat equation with multiple sources. The aim of such a method compared with the classical FEM is to solve time-dependent diffusion applications efficiently and with an appropriate level of accuracy.