Monolithic Overlapping Schwarz Domain Decomposition Methods with GDSW Coarse Spaces for Incompressible Fluid Flow Problems

Heinlein, Alexander; Hochmuth, Christian; Klawonn, Axel

doi:10.1137/18m1184047

Cited by 18 publications

(35 citation statements)

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“…As the systems become very ill‐conditioned for small h , we need a scalable preconditioner to guarantee fast convergence of the iterative method. Therefore, we will apply monolithic overlapping Schwarz preconditioners for saddle point problems (cf, References , and ). In particular, we will improve the performance of the monolithic preconditioners with GDSW‐type coarse spaces introduced in Reference .…”

Section: Two‐level Overlapping Schwarz Preconditioners For Saddle Poimentioning

confidence: 99%

“…We consider monolithic two‐level preconditioners with generalized Dryja‐Smith‐Widlund (GDSW) coarse spaces for incompressible fluid flow problems introduced in Reference . GDSW coarse spaces were originally introduced in References and for linear second‐ and fourth‐order elliptic partial differential equations.…”

Section: Introductionmentioning

confidence: 99%

“…For these elliptic problems, the GDSW coarse basis functions are energy minimal extensions representing the null space of the elliptic operator. In Reference , this concept was extended to coarse basis functions that are saddle point extensions of the null spaces of velocity and pressure of the indefinite saddle point operator. One significant advantage of GDSW coarse spaces is that they can be applied to arbitrary geometries and domain decompositions, whereas the use of classical Lagrangian coarse basis functions requires a coarse triangulation.…”

Section: Introductionmentioning

confidence: 99%

“…Moreover, we will consider restricted and scaled Schwarz operators, introduced by Cai and Sarkis, in the first level of our monolithic preconditioners. Furthermore, we employ two alternative strategies to improve the additive and sequential coupling of the two levels, which was used in Reference : multiplicative but sequential coupling of the levels and additive coupling combined with the concurrent computation of the levels. Finally, for nonlinear or time‐dependent problems, we will employ different recycling strategies ranging from the reuse of symbolic factorizations (SFs) of the local overlapping and nonoverlapping matrices to the complete reuse of the coarse basis and matrix.…”

Section: Introductionmentioning

confidence: 99%

“…The parallel implementation employed in our numerical simulations is based on our parallel implementation of monolithic two‐level preconditioners described in References , and . The implementation is available in the FROSch framework as a part of the ShyLU package in Trilinos.…”

Section: Introductionmentioning

confidence: 99%

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Reduced dimension GDSW coarse spaces for monolithic Schwarz domain decomposition methods for incompressible fluid flow problems

Heinlein

Hochmuth

Klawonn

2019

Numerical Meth Engineering

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Summary Monolithic preconditioners for incompressible fluid flow problems can significantly improve the convergence speed compared with preconditioners based on incomplete block factorizations. However, the computational costs for the setup and the application of monolithic preconditioners are typically higher. In this article, several techniques are applied to monolithic two‐level generalized Dryja‐Smith‐Widlund (GDSW) preconditioners to further improve the convergence speed and the computing time. In particular, reduced dimension GDSW coarse spaces, restricted and scaled versions of the first level, hybrid, and parallel coupling of the levels, and recycling strategies are investigated. Using a combination of all these improvements, for a small time‐dependent Navier‐Stokes problem on 240 message passing interface (MPI) ranks, a reduction of 86% of the time‐to‐solution can be obtained. Even without applying recycling strategies, the time‐to‐solution can be reduced by more than 50% for a larger steady Stokes problem on 4608 MPI ranks. For the largest problems with 11 979 MPI ranks, the scalability deteriorates drastically for the monolithic GDSW coarse space. On the other hand, using the reduced dimension coarse spaces, good scalability up to 11 979 MPI ranks, which corresponds to the largest problem configuration fitting on the employed supercomputer, could be achieved.

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Section: Two‐level Overlapping Schwarz Preconditioners For Saddle Poimentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Reduced dimension GDSW coarse spaces for monolithic Schwarz domain decomposition methods for incompressible fluid flow problems

Heinlein

Hochmuth

Klawonn

2019

Numerical Meth Engineering

Self Cite

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An aneurysm‐specific preconditioning technique for the acceleration of Newton‐Krylov method with application in the simulation of blood flows

Liu,

Qi,

Cai

2023

Numer Methods Biomed Eng

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In this paper, we develop an algorithm to simulate blood flows in aneurysmal arteries and focus on the construction of robust and efficient multilevel preconditioners to speed up the convergence of both linear and nonlinear solvers. The work is motivated by the observation that in the local aneurysmal region, the flow is often quite complicated with one or more vortices, but in the healthy section of the artery, the principal component of blood flows along the centerline of the artery. Based on this observation, we introduce a novel two‐level additive Schwarz method with a mixed‐dimensional coarse preconditioner. The key components of the preconditioner include (1) a three‐dimensional coarse preconditioner covering the aneurysm; (2) a one‐dimensional coarse preconditioner covering the central line of the healthy section of the artery; (3) a collection of three‐dimensional overlapping subdomain preconditioners covering the fine meshes of the entire artery; (4) extension/restriction operators constructed by radial basis functions. The blood flow is modeled by the unsteady incompressible Navier–Stokes equations with resistance outflow boundary conditions discretized by a stabilized finite element method on fully unstructured meshes and the second‐order backward differentiation formula in time. The resulting large nonlinear algebraic systems are solved by a Newton‐Krylov algorithm accelerated by the new preconditioner in two ways: (1) the initial guess of Newton is obtained by solving a linear system defined by the coarse preconditioner; (2) the Krylov solver of the Jacobian system is preconditioned by the new preconditioner. Numerical experiments indicate that the proposed preconditioner is highly effective and robust for complex flows in a patient‐specific artery with aneurysm, and it significantly reduces the numbers of linear and nonlinear iterations.

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Hybrid analysis and modeling, eclecticism, and multifidelity computing toward digital twin revolution

2021

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Most modeling approaches lie in either of the two categories: physics‐based or data‐driven. Recently, a third approach which is a combination of these deterministic and statistical models is emerging for scientific applications. To leverage these developments, our aim in this perspective paper is centered around exploring numerous principle concepts to address the challenges of (i) trustworthiness and generalizability in developing data‐driven models to shed light on understanding the fundamental trade‐offs in their accuracy and efficiency and (ii) seamless integration of interface learning and multifidelity coupling approaches that transfer and represent information between different entities, particularly when different scales are governed by different physics, each operating on a different level of abstraction. Addressing these challenges could enable the revolution of digital twin technologies for scientific and engineering applications.

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Monolithic Overlapping Schwarz Domain Decomposition Methods with GDSW Coarse Spaces for Incompressible Fluid Flow Problems

Cited by 18 publications

References 50 publications

Reduced dimension GDSW coarse spaces for monolithic Schwarz domain decomposition methods for incompressible fluid flow problems

Reduced dimension GDSW coarse spaces for monolithic Schwarz domain decomposition methods for incompressible fluid flow problems

An aneurysm‐specific preconditioning technique for the acceleration of Newton‐Krylov method with application in the simulation of blood flows

Hybrid analysis and modeling, eclecticism, and multifidelity computing toward digital twin revolution

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