A partition of unity finite element method for three-dimensional transient diffusion problems with sharp gradients

Mustapha Malek, Nouh Izem, M. Shadi Mohamed, Mohammed Seaid, Omar Laghrouche

Research output: Contribution to journalArticlepeer-review

13 Citations (Scopus)
11 Downloads (Pure)

Abstract

An efficient partition of unity finite element method for three-dimensional transient diffusion problems is presented. A class of multiple exponential functions independent of time variable is proposed to enrich the finite element approximations. As a consequence of this procedure, the associated matrix for the linear system is evaluated once at the first time step and the solution is obtained at subsequent time step by only updating the right-hand side of the linear system. This results in an efficient numerical solver for transient diffusion equations in three space dimensions. Compared to the conventional finite element methods with h-refinement, the proposed approach is simple, more efficient and more accurate. The performance of the proposed method is assessed using several test examples for transient diffusion in three space dimensions. We present numerical results for a transient diffusion equation with known analytical solution to quantify errors for the new method. We also solve time-dependent diffusion problems in complex geometries. We compare the results obtained using the partition of unity finite element method to those obtained using the standard finite element method. It is shown that the proposed method strongly reduces the necessary number of degrees of freedom to achieve a prescribed accuracy.
Original languageEnglish
Pages (from-to)702-717
Number of pages16
JournalJournal of Computational Physics
Volume396
Early online date8 Jul 2019
DOIs
Publication statusPublished - 1 Nov 2019

Keywords

  • finite element method
  • Partition of unity method
  • Three-dimensional diffusion problems
  • Enrichment functions
  • Heat conduction
  • Sharp gradients

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