Since the advent of plate tectonic theory, there has been a long and successful history of research software development within the geodynamics community. The earliest modelling tools provided fundamental new insight into the process of mantle convection, its sensitivity to variations in viscosity, and its role in controlling Earth's surface plate motions and heat transport (e.g. ). Although transformative at the time, computational and algorithmic limitations dictated that these tools were restricted to a simplified approximation of the underlying physics and, excluding some notable exceptions (e.g. ), to 2-D Cartesian geometries. They were specifically designed to address targeted scientific questions. As such, they offered limited flexibility, were not easily extensible, and were not portable across different platforms. Furthermore, since they were often developed for use by one or two expert practitioners, they were poorly documented: details of the implementation could only be determined by analysing the underlying code, which was often a non-trivial and specialised task.

Growing computational resources and significant theoretical and algorithmic advances have since underpinned the development of more advanced research software, which incorporates, for example, better approximations to the fundamental physical principles, including compressibility (e.g. ), mineralogical-phase transformations (e.g. ), multi-phase flow (e.g. ), variable and non-linear rheologies (e.g. ), and feedbacks between chemical heterogeneity and buoyancy (e.g. ). In addition, these more recent tools can often be applied in more representative 2-D cylindrical and/or 3-D spherical shell geometries (e.g. ). The user base of these tools has rapidly increased, with software development teams emerging to enhance their applicability and ensure their ongoing functionality. These teams have done so by adopting best practices in modern software development, including version control, unit and regression testing across a range of platforms and validation of model predictions against a suite of analytical and benchmark solutions (e.g. ).

Nonetheless, given rapid and ongoing improvements in algorithmic design and software engineering alongside the development of robust and flexible scientific computing libraries that provide access to much of the low-level numerical functionality required by geodynamical models, a next generation of open-source and community-driven geodynamical research software has emerged, exploiting developments from the forefront of computational engineering. This includes ASPECT (e.g. ), built on the deal.II , p4est and Trilinos libraries, Fluidity (e.g. ), which is underpinned by the PETSc and Spud libraries, Underworld2 (e.g. ), core aspects of which are built on the St Germain and PETSc libraries, and TerraFERMA , which has foundations in the FEniCS , PETSc and Spud libraries. By building on existing computational libraries that are highly efficient, extensively tested and validated, modern geodynamical research software is becoming increasingly reliable and reproducible. Its modular design also facilitates the addition of new features and provides a degree of confidence about the validity of previous developments, as evidenced by growth in the use and applicability of ASPECT over recent years.

However, even with these modern research software frameworks, some fundamental development decisions, such as the core physical equations, numerical approximations and general solution strategy, have been integrated into the basic building blocks of the code. Whilst there remains some flexibility within the context of a single problem, modifications to include different physical approximations or components, which can affect non-linear coupling and associated solution strategies, often require extensive and time-consuming development and testing, using either separate code forks or increasingly complex option systems. This makes reproducibility of a given simulation difficult, resulting in a lack of transparency – even with detailed documentation, specific details of the implementation are sometimes only available by reading the code itself, which, as noted previously, is non-trivial, particularly across different forks or with increasing code complexity . This makes scientific studies into the influence of different physical or geometrical scenarios, using a consistent code base, extremely challenging. Those software frameworks that try to maintain some degree of flexibility often do so at the expense of performance: the flexibility to configure different equations, numerical discretisations and solver strategies, in different dimensions and geometries, requires implementation compromises in the choice of optimal algorithms and specific low-level optimisations for all possible configurations.

A challenge that remains central to research software development in geodynamics, therefore, is the need to provide accurate, efficient, flexible, easily extensible, scalable, transparent and reproducible research software that can be applied to simulating a wide range of scenarios, including problems in different geometries and those incorporating different approximations of the underlying physics e.g.. However, this requires a large time commitment and knowledge that spans several academic disciplines. Arriving at a physical description of a complex system, such as global mantle convection, demands expertise in geology, geophysics, geochemistry, fluid mechanics and rheology. Discretising the governing partial differential equations (PDEs) to produce a suitable numerical scheme requires proficiency in mathematical analysis, whilst its translation into efficient code for massively parallel systems demands advanced knowledge in low-level code optimisation and computer architectures e.g.. The consequence of this is that the development of research software for geodynamics has now become a multi-disciplinary effort, and its design must enable scientists across several disciplines to collaborate effectively, without requiring each of them to comprehend all aspects of the system.

Key to achieving this is to abstract, automate and compose the various processes involved in numerically solving the PDEs governing a specific problem (e.g. ) to enable a separation of concerns between developing a technique and using it. As such, software projects involving automatic code generation have become increasingly popular, as these help to separate different aspects of development. Such an approach facilitates collaboration between computational engineers with expertise in hardware and software, computer scientists and applied mathematicians with expertise in numerical algorithms, and domain-specific scientists, such as geodynamicists.

In this study, we introduce Firedrake to the geodynamical modelling community: a next-generation automated system for solving PDEs using the finite-element method (e.g. ). As we will show, the finite-element method is well-suited to automatic code-generation techniques: a weak formulation of the governing PDEs, together with a mesh, initial and boundary conditions, and appropriate discrete function spaces, is sufficient to fully represent the problem. The purpose of this paper is to demonstrate the applicability of Firedrake for geodynamical simulation whilst also highlighting its advantages over existing geodynamical research software. We do so via comparisons against a suite of analytical and benchmark cases of systematically increasing complexity.

The remainder of the paper is structured as follows. In Sect. , we provide a background to the Firedrake project and the various dependencies of its software stack. In Sect. , we introduce the equations governing mantle convection which will be central to the examples developed herein, followed, in Sect. , by a description of their discretisation via the finite-element method and the associated solution strategies. In Sect. , we introduce a series of benchmark cases in Cartesian and spherical shell geometries. These are commonly examined within the geodynamical modelling community, and we describe the steps involved with setting up these cases in Firedrake, allowing us to highlight its ease of use. Parallel performance is analysed in Sect. , with a representative example of global mantle convection described and analysed in Sect. . The latter case confirms Firedrake's suitability for addressing research problems at the frontiers of global mantle dynamics research. Other components of Firedrake, which have not been showcased in this paper but which may be beneficial to various future research endeavours, are discussed in Sect. .

The Firedrake project is an automated system for solving partial differential equations using the finite-element method e.g.. Using a high-level language that reflects the mathematical description of the governing equations e.g., the user specifies the finite-element problem symbolically. The high-performance implementation of assembly operations for the discrete operators is then generated “automatically” by a sequence of specialised compiler passes that apply symbolic mathematical transformations to the input equations to ultimately produce C (and C++) code . Firedrake compiles and executes this code to create linear or non-linear systems, which are solved by PETSc . As stated by , in comparison to conventional finite-element libraries, and even more so with handwritten code, Firedrake provides a higher-productivity mechanism for solving finite-element problems whilst simultaneously applying sophisticated performance optimisations that few users would have the resources to code by hand.

Firedrake builds on the concepts and some of the code of the FEniCS project e.g., particularly its representation of variational problems via the Unified Form Language (UFL) . We note that the applicability of FEniCS for geodynamical problems has already been demonstrated (e.g. ). Both frameworks have the goal of saving users from manually writing low-level code for assembling the systems of equations that discretise their model physics. An important architectural difference is that, while FEniCS has components written in C++ and Python, Firedrake is completely written in Python, including its run-time environment (it is only the automatically generated assembly code that is in C/C++, although it does leverage the PETSc library, written in C, to solve the assembled systems, albeit through its Python interface – petsc4py). This provides a highly flexible user interface with ease of introspection of data structures. We note that the Python environment also allows deployment of handwritten C kernels should the need arise to perform discrete mesh-based operations that cannot be expressed in the finite-element framework, such as sophisticated slope limiters or bespoke sub-grid physics.

Firedrake offers several highly desirable features, rendering it well-suited to problems in geophysical fluid dynamics. As will be illustrated through a series of examples below, of particular importance in the context of this paper is Firedrake's support for a range of different finite-element discretisations, including a highly efficient implementation of those based on extruded meshes, programmable non-linear solvers and composable operator-aware solver preconditioners. As the importance of reproducibility in the computational geosciences is increasingly recognised, we note that Firedrake integrates with Zenodo and GitHub to provide users with the ability to generate a set of DOIs corresponding to the exact set of Firedrake components used to conduct a particular simulation, in full compliance with FAIR (findable, accessible, interoperable, reusable) principles.

Our focus here is on mantle convection, the slow creeping motion of Earth's mantle over geological timescales. The equations governing mantle convection are derived from the conservation laws of mass, momentum and energy. The simplest mathematical formulation assumes a single incompressible material and the Boussinesq approximation , under which the non-dimensional momentum and continuity equations are given by 1∇⋅σ=+Ra0Tk^=0,2∇⋅u=0, where σ= is the stress tensor, u is the velocity and T is the temperature. k^ is the unit vector in the direction opposite to gravity and Ra0 denotes the Rayleigh number, a dimensionless number that quantifies the vigour of convection: 3Ra0=ρ0αΔTgd3μ0κ. Here, ρ0 denotes the reference density, α the thermal expansion coefficient, ΔT the characteristic temperature change across the domain, g the gravitational acceleration, d the characteristic length, μ0 the reference dynamic viscosity and κ the thermal diffusivity. Note that the above non-dimensional equations are obtained through the following characteristic scales: length d, time d2/κ and temperature ΔT.

When simulating incompressible flow, the full stress tensor, σ=, is decomposed into deviatoric and volumetric components: 4σ==τ=-pI, where τ= is the deviatoric stress tensor, p is dynamic pressure and I is the identity matrix. Substituting Eq. () into Eq. () and utilizing the constitutive relation 5τ==2μϵ˙=2μsym(∇u)=μ∇u+∇uT, which relates the deviatoric stress tensor, τ=, to the strain-rate tensor, ϵ˙=sym(∇u), yields 6∇⋅μ∇u+∇uT-∇p+Ra0Tk^=0. The viscous flow problem can thus be posed in terms of pressure, p, velocity, u, and temperature, T. The evolution of the thermal field is controlled by an advection–diffusion equation: 7∂T∂t+u⋅∇T-∇⋅κ∇T=0. These governing equations are sufficient to solve for the three unknowns together with adequate boundary and initial conditions.

For the derivation of the finite-element discretisation of Eqs. (), () and (), we start by writing these in their weak form. We select appropriate function spaces V, W and Q that contain respectively the solution fields for velocity u, pressure p and temperature T and also contain the test functions v,w and q. The weak form is then obtained by multiplying these equations by the test functions and integrating over the domain Ω: ∫Ω∇v:μ∇u+∇uTdx-∫Ω∇⋅vpdx8-∫ΩRa0Tv⋅k^dx=0 for all v∈V,9∫Ωw∇⋅udx=0 for all w∈W,∫Ωq∂T∂tdx+∫Ωqu⋅∇Tdx+∫Ω∇q⋅κ∇Tdx10=0 for all q∈Q. Note that we have integrated by parts the viscosity and pressure gradient terms in Eq. () and the diffusion term in Eq. () but have omitted the corresponding boundary terms, which will be considered in the following section.

Equations ()–() are a more general representation of the continuous PDEs in strong form (Eqs. , and ), provided suitable function spaces with sufficient regularity are chosen (see for example ). Finite-element discretisation proceeds by restricting these function spaces to finite-dimensional subspaces. These are typically constructed by dividing the domain into cells or elements and restricting it to piecewise polynomial subspaces with various continuity requirements between cells. Firedrake offers a very wide range of such finite-element function spaces seefor an overview. It should be noted however that, in practice, this choice is guided by numerical stability considerations in relation to the specific equations that are being solved. In particular, the choice of velocity and pressure function spaces used in the Stokes system is restricted by the Ladyzhenskaya–Babuška–Brezzi (LBB) condition (see , for an overview of common choices for geodynamical flow). In this paper, we focus on the use of the familiar Q2Q1 element pair for velocity and pressure, which employs piecewise continuous bi-quadratic and bilinear polynomials on quadrilaterals or hexahedra for velocity and pressure respectively. In addition, to showcase Firedrake's flexibility, we use the less familiar Q2P1DG pair in a number of cases, in which pressure is discontinuous and piecewise linear (but not bilinear). For temperature, we primarily use a Q2 discretisation but also show some results using a Q1 discretisation.

All that is required for the implementation of these choices is that a basis can be found for the function space such that each solution can be written as a linear combination of basis functions. For example, if we have a basis ϕi of the finite-dimensional function space Qh of temperature solutions, then we can write each temperature solution as 11T(x)=∑iTiϕi(x), where Ti represents the coefficients that we can collect into a discrete solution vector T¯. Using a Lagrangian polynomial basis, the coefficients Ti correspond to values at the nodes, where each node i is associated with one basis function ϕi, but this is not generally true for other choices of finite-element bases.

In curved domains, boundaries can be approximated with a finite number of triangles, tetrahedrals, quadrilaterals or hexahedrals. This can be seen as a piecewise linear (or bilinear/trilinear) approximation where the domain is approximated by straight lines (edges) between vertices. A more accurate representation of the domain is obtained by allowing higher-order polynomials that describe the physical embedding of the element within the domain. A typical choice is to use a so-called iso-parametric representation in which the polynomial order of the embedding is the same as that of the discretised functions that are solved for.

Finally, we note that it is common to use a subscript h for the discrete, finite-dimensional function subspaces and Ωh for the discretised approximation by the mesh of the domain Ω, but since the remainder of this paper focusses on the details and implementation of this discretisation, we simply drop the h subscripts from here on.

Firedrake provides a complete framework for solving finite-element problems, highlighted in this section through a series of examples. We start in Sect.  with the most basic problem – isoviscous, incompressible convection, in an enclosed 2-D Cartesian box – and systematically build complexity, initially moving into more realistic physical approximations (Sect. ) and, subsequently, geometries that are more representative of Earth's mantle (Sect. ). The cases examined and the challenges associated with each are summarised in Table .

We assess parallel scalability using a 3-D spherical shell case similar to that presented in Sect. , albeit incorporating a temperature-dependent viscosity, following the relation 42μ=exp[E(0.5-T)], where E is a parameter that controls the temperature dependence of viscosity. In the example considered – Case A4 from – we set E=ln⁡(100), leading to thermally induced viscosity contrasts of 102 across the computational domain. For completeness, our steady-state results, highlighting the consistency of our results for this case with the predictions of , are displayed in Fig. , although for the purposes of parallel scaling analyses, we run simulations for 20 time steps only.

We focus on weak scaling, where the problem size and the number of processing cores are simultaneously increased. Cases are examined on 24, 192, 1536 and 12 288 cores, maintaining 4096 elements per core and ensuring a constant element aspect ratio across all resolutions examined. Simulations were examined on the Gadi supercomputer at the National Computational Infrastructure (NCI) in Australia, using compute nodes with 2×24 core Intel Xeon Platinum 8274 (Cascade Lake) 3.2 GHz CPUs and 192 GB RAM per node. Linking the nodes is the latest-generation HDR InfiniBand technology in a Dragonfly+ topology, capable of transferring data at up to 200 GB s-1.

The most challenging aspect of weak parallel scaling is solver performance as the problem size increases. Whilst the amount of computation in equation assembly typically scales linearly with the number of DOFs – before taking parallel aspects such as communication into account – solver scaling is generally worse. In the case of iterative solvers, this is due to a deterioration in the conditioning of the matrix, driving an increase in the number of iterations required for convergence. As a result, even if the cost per iteration scales linearly, the overall cost will not. This implies that, for weak scaling, the amount of work per core may increase rapidly despite the number of DOFs per core remaining consistent.

The deterioration in conditioning is intimately related to the fact that an increase in resolution increases the ratio between the smallest and largest resolvable length scales. For elliptic operators, like the viscosity matrix K, the condition number scales with the square of that ratio e.g.. Multigrid approaches, which separate smaller and larger length scales on a hierarchy of fine to coarse meshes, are commonly used to address this problem, which motivates the choice of the algebraic multigrid preconditioner, GAMG, used here. Such approaches aim to maintain a constant or only slowly increasing number of iterations and, thus, a near-linear scaling of the overall cost as the problem size increases. This can be a challenge however, as, for instance, an increase in resolution will require more multigrid levels, which will lead to an increased set-up time and cost per iteration. In practice, when configuring the multigrid method, a compromise needs to be found between the effectiveness of a multigrid in limiting the number of iterations and not allowing the set-up and costs per iteration to grow too rapidly. The two options, gamg_threshold and gamg_square_graph, specified in our solver set-up ensure a balance between multigrid effectiveness and coarse grid complexity.

A breakdown of key parallel scaling results is presented in Fig. . Panel (a) displays the average number of iterations per solve over the 20 time steps. We find that the number of pressure (the Schur complement solve: fieldsplit_1) and energy solve iterations remains flat (12 and ∼10.5 respectively), whilst the number of velocity solve iterations (inversion of the matrix K, using the GAMG preconditioner: fieldsplit_0) increases only slowly, from ∼41 to ∼51, over a greater than 3 order of magnitude increase in problem size and number of processor cores. This demonstrates algorithmic scalability on up to 12 288 cores and ∼50×106 elements (which corresponds to ∼1.26×109 velocity and pressure degrees of freedom).

Parallel scalability can also be assessed by analysing the growth in CPU time of the dominant components of our problem: assembly of finite-element systems (Fig. b), set-up of the algebraic multigrid (GAMG) preconditioner (Fig. c), and time spent solving the Schur complement system (Fig. d). We find that the assembly time is a negligible fraction of this problem. The set-up time for our GAMG preconditioner grows from ∼240 s on 24 cores to ∼470 s on 12 288 cores. This is understandable given the high communication costs associated with setting up various multigrid levels, particularly for problems incorporating null spaces and near-null spaces, as is the case here. We note, however, that this is not a concern: as a fraction of the entire solution time for the Schur complement solve (Fig. d), GAMG set-up remains small. We do observe an increase in time required for solution of the Schur complement (Stokes solve) from ∼6500 s on 24 cores to ∼12100 s on 12 288 cores. This results primarily from the minor increase in the number of velocity solve iterations and the increased cost per iteration (Fig. e), which rises from 155 s on 24 cores to 225 s on 12 288 cores, reflecting costs associated with increasing the number of multigrid levels for higher-resolution problems. The total time spent in running this problem mirrors the time spent in solving the Schur complement system (Fig. f), indicating where future optimisation efforts should be directed.

In this section, we demonstrate application of Firedrake to a time-dependent simulation of global mantle convection in a 3-D spherical shell geometry, at a realistic Rayleigh number. We assume a compressible mantle, under the ALA, and a temperature-, depth-, and strain-rate-dependent rheology, in line with the viscoplastic case analysed in Sect. . Viscosity increases below the mantle transition zone and we include a brittle-failure-type yield-stress law, ensuring that yielding concentrates at shallow depths. As with the examples provided above, calculations are performed using a hexahedral Q2Q1 element pair for velocity and pressure. We use a Q2 discretisation for temperature and, given the increased importance of advection at higher Rayleigh numbers, incorporate stabilisation through a streamline upwinding scheme, following . We employ a cubed sphere mesh with 98 304 elements on each spherical surface and extrude it radially through 64 layers, with spacing reduced adjacent to the top and bottom boundaries of the domain. This results in a problem with ∼1.26×109 velocity and pressure degrees of freedom and ∼5.0×107 temperature degrees of freedom. Our solution strategy for the Stokes equations is similar to the spherical shell examples presented above, albeit exploiting PETSc's SNES functionality using a set-up based on Newton's method to handle the non-linearity in the system. The solution strategy for the energy equation is identical to the previous example.

We achieve a (basally heated) Rayleigh number of 7.5×107 in the asthenosphere, which is comparable to estimates of Earth's mantle e.g. and also includes internal heating at a non-dimensional heating rate of 10. The simulation is spun up with free-slip and isothermal boundaries at both surfaces. After the model reaches a quasi-steady state (i.e. when the surface and basal Nusselt numbers both change by less than 0.1 % over 10 consecutive time steps), surface velocities are assimilated through a kinematic boundary condition, according to 230 Myr of plate motion histories , using the Python interface to GPlates e.g.. Our simulation then runs forward towards the present day. This case is therefore similar to the simulations examined when addressing questions from the very frontiers of geodynamical research, albeit incorporating a more representative treatment of mantle and lithosphere rheology (e.g. ). The simulation was executed across 1344 CPUs, on the same architecture as outlined above, and took ∼92 h.

Our results are illustrated in Fig. . We find that the present-day upper-mantle convective planform is dominated by strong downwellings in regions of plate convergence. In the middle mantle, cold downwellings are prominent beneath North America and South-East Asia, whilst remnants of older subduction are visible above the core–mantle boundary. The location of hot upwelling material is strongly modulated by these downwellings, with upwelling plumes concentrating in two clusters beneath the African continent and the central Pacific Ocean (i.e. away from regions that have experienced subduction over the past 150 Myr or so). The cluster of plumes in the Pacific is reasonably circular, whilst those beneath Africa extend in a NW–SE-trending structure, which to the north curves eastward under Europe and to the south extends into the Indian Ocean.

Further analysis of this proof-of-concept simulation is beyond the scope of this study. However, when combined with the benchmark and parallel scaling analyses presented above, our model predictions, which are consistent with those from a number of previous studies (e.g. ), confirm Firedrake's applicability for time-dependent global mantle dynamics simulations of this nature.

Firedrake is a next-generation automated system for solving variational problems using the finite-element method (e.g. ). It has a number of features that are ideally suited to simulating geophysical fluid dynamics problems, as exemplified by its use in application areas such as coastal ocean modelling , numerical weather prediction and glacier flow modelling . The focus of this paper has been to demonstrate Firedrake's applicability for geodynamical simulation, with an emphasis on global mantle dynamics. To do so, we have presented, analysed and validated Firedrake against a number of benchmark and analytical cases of systematically increasing complexity, building towards a time-dependent global simulation at realistic convective vigour.

To introduce its core components and illustrate the elegance of setting up and validating a geodynamical model in Firedrake, we started with a simple, incompressible, isoviscous case in an enclosed 2-D Cartesian box. Setting up this problem was straightforward, requiring only a weak formulation of the governing equations for specification in the UFL, together with a mesh, initial and boundary conditions, and appropriate discrete function spaces. We utilised Firedrake's built-in meshing functionality and default direct solver options and demonstrated the framework's accuracy for simulations of this nature: in only 56 lines of Python (excluding comments and blank lines), we reproduced results from the well-established benchmark study of .

Representative simulations of mantle and lithosphere dynamics, however, incorporate more complicated physics. To demonstrate Firedrake's applicability in such scenarios, we next set up 2-D simulations that accounted for compressibility through the anelastic liquid approximation and a non-linear viscosity that depends upon temperature, depth and strain rate. Our results were validated through comparison to the benchmark studies of and respectively. For compressible cases, despite the governing equations differing appreciably from their incompressible counterparts, the modifications required to our set-up were minimal, with the most notable change being the UFL describing the relevant PDEs. For the viscoplastic rheology case, where viscosity varied by several orders of magnitude across the domain, an appropriate solution strategy was required to deal with non-linear coupling between strain rate and viscosity: Firedrake's fully programmable solver interface and seamless coupling to PETSc facilitated the straightforward use of PETSc's SNES . Taken together, these examples highlight one of Firedrake's key benefits: by leveraging the UFL , associated strategies for automatic assembly of finite-element systems, and PETSc , the framework is easily extensible, allowing for straightforward application to problems involving different physical approximations, even when they require distinct solution strategies.

This is further highlighted with the transition from 2-D to 3-D. With modifications to only a few lines of Python, the basic 2-D Cartesian case described above was easily extended to 3-D, allowing for comparison and validation against the well-established benchmark results of . However, the direct solvers used for our 2-D cases quickly become computationally intractable in 3-D, necessitating the use of an iterative approach. Firedrake's programmable solver interface facilitates the straightforward inclusion of Python dictionaries that define iterative solver parameters for the Stokes and energy systems. A number of different schemes have been advocated by the geodynamical modelling community e.g., but in all 3-D simulations examined herein, the Schur complement approach was utilised for solution of our Stokes system, exploiting the fieldsplit preconditioner type to apply preconditioners, including an algebraic multigrid, to different blocks of the system. A Crank–Nicolson scheme was utilised for temporal discretisation of the energy equation, with a standard GMRES Krylov method with SOR preconditioning used for solution. We have demonstrated that such solution strategies are effective and scalable, with algorithmic scalability confirmed on up to 12 288 cores.

Cartesian simulations offer a means of better understanding the physical mechanisms controlling mantle convection, but a 3-D spherical shell geometry is required to simulate global mantle dynamics. We have demonstrated how Firedrake's built-in meshing and extrusion functionality facilitates the effortless transition to such geometries (in addition to comparable 2-D cylindrical shell geometries), whilst its Python user interface allows for the simple inclusion of a radial gravity direction and boundary conditions that are not aligned with Cartesian directions. The convergence properties and accuracy of our simulations in a 3-D spherical shell geometry have been demonstrated through comparison to the extensive set of analytical solutions introduced by and a series of low Rayleigh number isoviscous and temperature-dependent viscosity simulations from . We observed super-convergence for the Q2Q1 element pair at fourth and second order for velocity and pressure respectively.

Having validated Firedrake against this broad suite of cases, we finally applied the framework to a simulation of global mantle convection at realistic convective vigour. We assumed a compressible mantle and a non-linear temperature-, depth- and strain-rate-dependent viscosity, assimilating 230 Myr of plate motion histories through a kinematic surface boundary condition. These prescribed plate velocities modulate underlying mantle flow, such that the predicted present-day convective planform is dominated by cold downwellings in regions of plate convergence, with upwellings concentrating elsewhere, particularly beneath the African continent and the Pacific Ocean. Our model predictions, which are consistent with those from a number of previous studies (e.g. ), reproduce first-order characteristics of the structure of Earth's mantle imaged through seismology (e.g. ) and the geographical distribution of mantle plumes (e.g. ). They serve as a proof of concept, confirming Firedrake's applicability for time-dependent, global simulations of this nature and, accordingly, its suitability for addressing research problems from the very frontiers of global geodynamical research.

Despite this, several components of Firedrake have not been fully examined in this paper. Many of these will likely be useful for geodynamical simulation and, accordingly, will be examined in the future. These include the following.

A range of finite elements: in most examples considered herein, we utilised a continuous Q2Q1 element pair for velocity and pressure with a Q2 discretisation for temperature. In addition, in some cases we instead employ the Q2P1DG finite-element pair for the Stokes system or a Q1 discretisation for temperature. Despite this, we have not fully demonstrated Firedrake's support for a wide array of finite elements, including continuous, discontinuous, H(div) and H(curl) discretisations and elements with continuous derivatives such as the Argyris and Bell elements seefor an overview. Some of these could offer major advantages for geodynamical simulation.

To support these statements and further demonstrate the potential of the framework in exploring challenging non-linear problems, we briefly consider the non-linear benchmark case of , with a strain-rate- and pressure-dependent Drucker–Prager rheology. In , a number of solution strategies are explored for this case (amongst others), with the study advocating the use of two modifications to the Jacobian: (i) adding an additional term to Eq. () that is the transpose of the second term, thus restoring the symmetry of K, and (ii) scaling those terms associated with ∂η/∂u by a spatially varying αSPD, calculated at the Gauss points according to 43αSPD=1 if 1-a:b‖a‖‖b‖2<csafety2η(ϵ˙(u)),csafety2η(ϵ˙(u))1-a:b‖a‖‖b‖2otherwise, where a=ϵ˙ and b=∂η∂ϵ˙. This rescaling acts as a stabilisation, ensuring that K remains positive definite. It should be noted that the pressure dependence of the Drucker–Prager rheology also leads to additional terms in the top-right block of the Stokes Jacobian matrix, in addition to G in Eq. (), making the overall system asymmetric regardless.

In traditional codes, the implementation of such additional terms in the Jacobian and the proposed modifications (stabilisation) require significant development. Analytical expressions for ∂η/∂u and ∂η/∂p must be derived for each specific rheological relationship analysed (as is done in the appendices of ), and the assembly of any additional terms may require a significant overhaul of existing code and data structures as, for example, sparsity structures may change. In Firedrake, the full Jacobian is derived symbolically and the code for its assembly generated automatically, making the entire process automatic, even for highly complex rheologies. We were able to implement the Jacobian modifications proposed in in only seven lines of Python code (the full Python script for this case is available in the repository accompanying this paper) and, as illustrated in Fig. , we obtain similar results. As indicated in , the convergence of the problem gets more challenging with increased resolution, and although a reasonably converged result can be obtained for the case shown in Fig.  at a resolution of 1024×512, this is insufficient to resolve the details of the unstructured mesh domain used in , who reported non-convergence for this case. Firedrake's ability to choose from a large variety of discretisation types, including unstructured meshes, and its flexibility to adapt and experiment with the solution strategy open up numerous avenues to further investigate the challenges in this and other highly non-linear problems.

It is important to point out that some common components of geodynamical models have not been showcased herein and, to our knowledge, have not yet been explored within the Firedrake framework. These include, for example, a free-surface boundary condition and the ability to model multiple-material flows, often implemented in geodynamical models using the particle-in-cell technique. Our goal for this paper is to provide solid foundations for future work in Firedrake that we, and others in the geodynamical modelling community, can build upon. Nonetheless, we see no fundamental reason why any component of other geodynamical modelling tools cannot be incorporated within Firedrake. For example, the TerraFERMA framework of , which is built on FEniCS, has been able to match the free-surface benchmarks of – a similar implementation would be straightforward in Firedrake. For multi-material flows, solving an advection equation, for example with a discontinuous Galerkin scheme and appropriate limiters e.g., would be straightforward. In addition, particle-in-cell schemes have been successfully developed and tested with FEniCS , and we see no fundamental reason that such functionality cannot be incorporated within Firedrake. Finally, Firedrake's flexibility would make exploring different advection schemes straightforward, rendering it very well-suited to level-set approaches e.g..

We note that the automated approach underpinning Firedrake has the potential to revolutionise the use of adjoints and other inverse schemes in geodynamics. Adjoint models have made an enormous impact in fields such as meteorology and oceanography. However, despite significant progress (e.g. ), their use in other scientific fields, including geodynamics, has been hampered by the practical difficulty of their derivation and implementation. In contrast to developing a model directly in Fortran or C++, high-level systems, such as Firedrake, allow the developer to express the variational problems to be solved in near-mathematical notation through the UFL. As such, these systems have a key advantage: since the mathematical structure of the problem is preserved, they are more amenable to automated analysis and manipulation, which can be exploited to automate the derivation of adjoints e.g. and the generation of the low-level code for the derived models. Exploring the use of such an approach in geodynamics will be an important avenue for future research.

Finally, given the importance of reproducibility in the computational geosciences, we note that Firedrake integrates with Zenodo and GitHub to provide users with the ability to generate a set of DOIs corresponding to the exact set of Firedrake components used to conduct a particular set of simulations. In providing our input scripts and a DOI for the version of Firedrake used herein, we ensure traceable provenance of model data, in full compliance with FAIR principles.

Firedrake is a next-generation system for solving variational problems using the finite-element method (e.g. ). It treats finite-element problems as a composition of several abstract processes, using separate and open-source software components for each. Firedrake's overarching goal is to save users from manually writing low-level code for assembling the systems of equations that discretise their model physics. It is written completely in Python and exploits automatic code-generation techniques to apply sophisticated performance optimisations. Firedrake creates a separation of concerns between employing the finite-element method and implementing it: this is a game changer, as it opens up these problems to a new class of user and developer.

In this paper, we have confirmed Firedrake's applicability for geodynamical simulation by configuring and validating model predictions against a series of benchmark and analytical cases of systematically increasing complexity. In all cases, Firedrake has been shown to be accurate and efficient, and we have also demonstrated that it is flexible and easily extensible: by leveraging the UFL and PETSc, it can be effortlessly applied to problems involving different physical approximations (e.g. incompressible and compressible flow, isoviscous and more complex non-linear rheologies), even if they require distinct solution strategies. We have illustrated how Firedrake's built-in mesh generation utilities and extrusion functionality provide a straightforward mechanism for examining problems in different geometries (2-D and 3-D Cartesian, 2-D cylindrical and 3-D spherical shells) and how its fully programmable solver dictionary and customisable preconditioner interface, both of which are seamlessly coupled to PETSc, facilitate straightforward configuration of different solution approaches. Parallel scalability has been demonstrated on up to 12 288 compute cores. Finally, using a more representative simulation of global mantle dynamics, where the distribution of heterogeneity is governed by imposed plate motion histories , we have confirmed Firedrake's suitability for tackling challenges at the very forefront of geodynamical research. We note that all simulation data presented herein have traceable provenance: in providing our input scripts and a DOI for the exact set of Firedrake components employed, Firedrake facilitates transparency and reproducibility, in full compliance with FAIR principles.