A residual a posteriori error estimate for partition of unity finite elements for three‐dimensional transient heat diffusion problems using multiple global enrichment functions

Abstract

SUMMARYIn this article, a study of residual based a posteriori error estimation is presented for the partition of unity finite element method (PUFEM) for three‐dimensional (3D) transient heat diffusion problems. The proposed error estimate is independent of the heuristically selected enrichment functions and provides a useful and reliable upper bound for the discretization errors of the PUFEM solutions. Numerical results show that the presented error estimate efficiently captures the effect of h‐refinement and q‐refinement on the performance of PUFEM solutions. It also efficiently reflects the effect of ill‐conditioning of the stiffness matrix that is typically experienced in the partition of unity based finite element methods. For a problem with a known exact solution, the error estimate is shown to capture the same solution trends as obtained by the classical L2 norm error. For problems with no known analytical solutions, the proposed estimate is shown to be used as a reliable and efficient tool to predict the numerical errors in the PUFEM solutions of 3D transient heat diffusion problems.