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Tensor-Product Vertex Patch Smoothers for Biharmonic Problems

  • Julius Witte ORCID logo , Cu Cui ORCID logo , Francesca Bonizzoni ORCID logo and Guido Kanschat ORCID logo EMAIL logo
Published/Copyright: May 29, 2025

Abstract

We discuss vertex patch smoothers as overlapping domain decomposition methods for fourth order elliptic partial differential equations. We show that they are numerically very efficient and yield high convergence rates. Furthermore, we discuss low rank tensor approximations for their efficient implementation. Our experiments demonstrate that the inexact local solver yields a method which converges fast and uniformly with respect to mesh refinement and polynomial degree. The multiplicative smoother shows superior performance in terms of solution efficiency, requiring fewer iterations in both two- and three-dimensional cases. Additionally, the solver infrastructure supports a mixed-precision approach, executing the multigrid preconditioner in single precision while performing the outer iteration in double precision, thereby increasing throughput by up to 70 %.

Award Identifier / Grant number: 202106380059

Award Identifier / Grant number: EXC 2181/1 – 390900948

Award Identifier / Grant number: P2022N5ZNP

Award Identifier / Grant number: E53C23001670001

Award Identifier / Grant number: E53C24001950001

Funding statement: Cu Cui was supported by the China Scholarship Council (CSC) under grant no. 202106380059. Francesca Bonizzoni has received support from the project PRIN2022, MUR, Italy, 2023–2025, P2022N5ZNP “SIDDMs: shape-informed data-driven models for parametrized PDEs, with application to computational cardiology”. Francesca Bonizzoni is partially funded by “INdAM – GNCS Project”, codice CUP E53C23001670001. Francesca Bonizzoni is member of INdAM-GNCS. Guido Kanschat was supported by the Deutsche Forschungsgemeinschaft (DFG) under Germany’s Excellence Strategy EXC 2181/1 – 390900948 (the Heidelberg STRUCTURES Excellence Cluster).

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Received: 2024-11-30
Revised: 2025-04-29
Accepted: 2025-05-08
Published Online: 2025-05-29
Published in Print: 2025-07-01

© 2025 Institute of Mathematics of the National Academy of Science of Belarus, published by De Gruyter, Berlin/Boston

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