Quasi-optimal mesh sequence construction through Smoothed Adaptive Finite Element Methods

Mulita, Ornela; Giani, Stefano; Heltai, L.

Abstract

We propose a new algorithm for Adaptive Finite Element Methods (AFEMs) based on smoothing iterations (S-AFEM), for linear, second-order, elliptic partial differential equations (PDEs). The algorithm is inspired by the ascending phase of the V-cycle multigrid method: we replace accurate algebraic solutions in intermediate cycles of the classical AFEM with the application of a prolongation step, followed by a fixed number of few smoothing steps. Even though these intermediate solutions are far from the exact algebraic solutions, their a-posteriori error estimation produces a refinement pattern that is substantially equivalent to the one that would be generated by classical AFEM, at a considerable fraction of the computational cost. We quantify rigorously how the error propagates throughout the algorithm, and we provide a connection with classical a posteriori error analysis. A series of numerical experiments highlights the efficiency and the computational speedup of S-AFEM.

Citation

Mulita, O., Giani, S., & Heltai, L. (2021). Quasi-optimal mesh sequence construction through Smoothed Adaptive Finite Element Methods. SIAM Journal on Scientific Computing, 43(3), A2211-A2241. https://doi.org/10.1137/19m1262097

• Journal Article Type: Article
• Acceptance Date: Mar 30, 2021
• Online Publication Date: Jun 17, 2021
• Publication Date: 2021
• Deposit Date: Mar 31, 2021
• Publicly Available Date: Mar 31, 2021
• Journal: SIAM Journal on Scientific Computing
• Print ISSN: 1064-8275
• Electronic ISSN: 1095-7197
• Publisher: Society for Industrial and Applied Mathematics
• Peer Reviewed: Peer Reviewed
• Volume: 43
• Issue: 3
• Pages: A2211-A2241
• DOI: https://doi.org/10.1137/19m1262097
• Public URL: https://durham-repository.worktribe.com/output/1250507