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

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.

Multigrid preconditioners for the mixed finite element dynamical core of the LFRic atmospheric model

“…Finally, an obvious downside of the approximate Schur-complement approach pursued here is the need for an additional outer solve of the mixed problem. As has been shown recently in Gibson et al (2020), this can be avoided by using a hybridised mixed finite element discretisation. In fact, for a gravity-wave test case a solver based on this hybridised approach has already been implemented in LFRic.…”

“…As has been shown recently in Gibson et al . (2020), this can be avoided by using a hybridised mixed finite element discretisation. In fact, for a gravity‐wave test case a solver based on this hybridised approach has already been implemented in LFRic.…”

A Hybridized Mixed Approach for Efficient Stress Prediction in a Layerwise Plate Model

“…If choosing such spaces is feasible when u is a scalar (e.g., in the case of antiplane elasticity), this is much harder for the general vectorial case of 2D/3D elasticity [45][46][47]. Hybridized mixed methods therefore consist of relaxing the a priori traction continuity requirement and including it in the variational formulation [48]. They are therefore easier to formulate, especially regarding the choice of the stress interpolation space, and offer computational advantages as it will be seen in the following section.…”

A compatible finite‐element discretisation for the moist compressible Euler equations

“…Further information on hybridisation techniques and their implementation can be found in Gibson et al . (2019; 2020). More on this particular linear solver and its performance will be detailed in future work.…”