# Solver composition across the PDE/linear algebra barrier

## TL;DR

This paper introduces enhancements in the Firedrake library that enable the development of flexible, composable preconditioners for PDE systems, decoupling solver strategies from specific model discretizations and allowing runtime configuration.

## Contribution

It extends PETSc's algebraic solver composability in Firedrake, enabling auxiliary operators and independent, runtime-configurable solver development for PDE discretizations.

## Key findings

- Enhanced solver composability with auxiliary operators
- Decoupling solver design from specific PDE models
- Runtime configuration of preconditioners

## Abstract

The efficient solution of discretisations of coupled systems of partial differential equations (PDEs) is at the core of much of numerical simulation. Significant effort has been expended on scalable algorithms to precondition Krylov iterations for the linear systems that arise. With few exceptions, the reported numerical implementation of such solution strategies is specific to a particular model setup, and intimately ties the solver strategy to the discretisation and PDE, especially when the preconditioner requires auxiliary operators. In this paper, we present recent improvements in the Firedrake finite element library that allow for straightforward development of the building blocks of extensible, composable preconditioners that decouple the solver from the model formulation. Our implementation extends the algebraic composability of linear solvers offered by the PETSc library by augmenting operators, and hence preconditioners, with the ability to provide any necessary auxiliary operators. Rather than specifying up front the full solver configuration, tied to the model, solvers can be developed independently of model formulation and configured at runtime. We illustrate with examples from incompressible fluids and temperature-driven convection.