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Geoscientific Model Development An interactive open-access journal of the European Geosciences Union
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Discussion papers
https://doi.org/10.5194/gmd-2019-86
© Author(s) 2019. This work is distributed under
the Creative Commons Attribution 4.0 License.
https://doi.org/10.5194/gmd-2019-86
© Author(s) 2019. This work is distributed under
the Creative Commons Attribution 4.0 License.

Submitted as: development and technical paper 26 Apr 2019

Submitted as: development and technical paper | 26 Apr 2019

Review status
This discussion paper is a preprint. A revision of the manuscript for further review has not been submitted.

Slate: extending Firedrake's domain-specific abstraction to hybridized solvers for geoscience and beyond

Thomas H. Gibson1, Lawrence Mitchell2, David A. Ham1, and Colin J. Cotter1 Thomas H. Gibson et al.
  • 1Department of Mathematics, Imperial College London, London, SW7 2AZ, UK
  • 2Department of Computer Science, Durham University, Durham, DH1 3LE, UK

Abstract. Within the finite element community, discontinuous Galerkin (DG) and mixed finite element methods have become increasingly popular in simulating geophysical flows. However, robust and efficient solvers for the resulting saddle-point and elliptic systems arising from these discretizations continue to be an on-going challenge. One possible approach for addressing this issue is to employ a method known as hybridization, where the discrete equations are transformed such that classic static condensation and local post-processing methods can be employed. However, it is challenging to implement hybridization as performant parallel code within complex models, whilst maintaining separation of concerns between applications scientists and software experts. In this paper, we introduce a domain-specific abstraction within the Firedrake finite element library that permits the rapid execution of these hybridization techniques within a code-generating framework. The resulting framework composes naturally with Firedrake's solver environment, allowing for the implementation of hybridization and static condensation as runtime-configurable preconditioners via the Python interface to PETSc, petsc4py. We provide examples derived from second order elliptic problems and geophysical fluid dynamics. In addition, we demonstrate that hybridization shows great promise for improving the performance of solvers for mixed finite element discretizations of equations related to large-scale geophysical flows.

Thomas H. Gibson et al.
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Status: closed (peer review stopped)
Status: closed (peer review stopped)
AC: Author comment | RC: Referee comment | SC: Short comment | EC: Editor comment
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Interactive discussion
Status: closed (peer review stopped)
Status: closed (peer review stopped)
AC: Author comment | RC: Referee comment | SC: Short comment | EC: Editor comment
Printer-friendly Version - Printer-friendly version Supplement - Supplement
Thomas H. Gibson et al.
Model code and software

Software used in 'Slate: extending Firedrake's domain-specific abstraction to hybridized solvers for geoscience and beyond' Zenodo/Firedrake https://doi.org/10.5281/zenodo.2587072

Tabula Rasa: experimentation framework for hybridization and static condensation T. H. Gibson https://doi.org/10.5281/zenodo.2616031

Thomas H. Gibson et al.
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Latest update: 18 Oct 2019
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Short summary
Galerkin finite element discretizations for atmospheric modeling often require the solution of ill-conditioned, saddle-point equations which can be efficiently solved using a hybridized method. By extending Firedrake's domain-specific abstraction, we provide a mechanism for the rapid implementation of hybridization methods for a wide class of methods. In this paper, we show that hybridization is an effective alternative to traditional block solvers for simulating geophysical flows.
Galerkin finite element discretizations for atmospheric modeling often require the solution of...
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