Overcoming element quality dependence of finite elements with adaptive extended stencil FEM (AES‐FEM)

Conley, Rebecca; Delaney, Tristan J.; Jiao, Xiangmin

doi:10.1002/nme.5246

Title: Overcoming element quality dependence of finite elements with adaptive extended stencil FEM (AES‐FEM)

Journal Article · Wed Mar 23 00:00:00 EDT 2016 · International Journal for Numerical Methods in Engineering

DOI:https://doi.org/10.1002/nme.5246· OSTI ID:1401188

^[1]; Delaney, Tristan J. ^[1]; Jiao, Xiangmin ^[1]

Department of Applied Mathematics and Statistics Stony Brook University Stony Brook 11794 NY USA

Summary The finite element methods (FEMs) are important techniques in engineering for solving partial differential equations, but they depend heavily on element shape quality for stability and good performance. In this paper, we introduce the Adaptive Extended Stencil Finite Element Method (AES‐FEM) as a means for overcoming this dependence on element shape quality. Our method replaces the traditional basis functions with a set of generalized Lagrange polynomial basis functions , which we construct using local weighted least‐squares approximations. The method preserves the theoretical framework of FEM and allows imposing essential boundary conditions and integrating the stiffness matrix in the same way as the classical FEM. In addition, AES‐FEM can use higher‐degree polynomial basis functions than the classical FEM, while virtually preserving the sparsity pattern of the stiffness matrix. We describe the formulation and implementation of AES‐FEM and analyze its consistency and stability. We present numerical experiments in both 2D and 3D for the Poisson equation and a time‐independent convection–diffusion equation. The numerical results demonstrate that AES‐FEM is more accurate than linear FEM, is also more efficient than linear FEM in terms of error versus runtime, and enables much better stability and faster convergence of iterative solvers than linear FEM over poor‐quality meshes. Copyright © 2016 John Wiley & Sons, Ltd.

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Sponsoring Organization:: USDOE

OSTI ID:: 1401188

Journal Information:: International Journal for Numerical Methods in Engineering, Journal Name: International Journal for Numerical Methods in Engineering Vol. 108 Journal Issue: 9; ISSN 0029-5981

Publisher:: Wiley Blackwell (John Wiley & Sons)Copyright Statement

Country of Publication:: United Kingdom

Language:: English

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Cited by: 4 works

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