We follow the internal variable formulation adopted by and , in which viscoelastic constitutive equations are expressed in integral form and reformulated using so-called internal variables. Conceptually, this approach consists of a set of elements with different shear relaxation timescales, arranged in parallel.

For a linear, compressible viscoelastic material, the non-dimensional constitutive equation takes the form 10σ1L=κ∇⋅u(t)I+2μ0d(t)-2∑iμimi(t), where κ is the non-dimensional bulk modulus and μ0 is the non-dimensional effective shear modulus given by 11μ0=∑iμi where μi are the non-dimensional shear moduli associated with each internal variable, mi. The deviatoric strain tensor is given as 12d=e-13Tr(e)I, where Tr(⋅) is the trace operator and the strain tensor, e, is 13e=12∇u+∇uT. We note that all non-dimensional moduli are obtained by division of terms by the characteristic shear modulus μ¯ (Table ). Each internal variable, mi, is defined by 14mi=1αi∫t0te-(t-t′)αid(t′)dt′, where αi is the non-dimensional Maxwell time for each element.

Equivalently, each internal variable evolves according to 15∂tmi+1αimi-d=0,mi(t0)=0.

This formulation provides a compact, flexible and convenient means to incorporate transient rheology into viscoelastic deformation models: using a single internal variable is equivalent to a simple Maxwell material; two correspond to a Burgers model with two characteristic relaxation frequencies; and using a series of internal variables permits approximation of a continuous range of relaxation timescales for more complicated transient rheologies (e.g., Extended Burgers). This formulation is also readily extensible to non-linear constitutive equations e.g.,. For example, we can represent a composite rheology that combines the effects of linear diffusion creep at low stresses with non-Newtonian power-law rheology (e.g., dislocation creep) at high stresses by modifying the viscosity to 16η=η01+||τ||τTn-1, where η0 is the same spatially varying Newtonian (i.e., stress-independent) viscosity as before, ||τ|| is the second invariant of the deviatoric stress tensor, τT is a transition stress beyond which power-law deformation starts to dominate, and n is the associated power-law exponent.

To complete the problem specification, we must also define the boundary conditions. At the top of the domain (i.e., Earth's surface), normal stress is balanced by the applied surface load such that 17n^⋅σ1L=-Bμρloadghloade^k, where ρload and hload are the non-dimensional density and vertical thickness of the surface load, and n^ is the outward unit normal vector to the boundary. At the base of the domain (i.e., core–mantle boundary), we impose a no-normal-displacement condition 18u⋅n^=0, which is consistent with the approximation of a rigid core following . Note that this choice departs from models that treat the core as an inviscid fluid and instead impose a free-surface condition e.g.,. After self-gravity has been introduced into G-ADOPT, switching to a free-surface should be a trivial extension.

Finally, we assume continuity of both displacement and traction across all internal boundaries, such that 19[u]-+=0,20[n^⋅σL1]-+=0, where [⋅]-+ denotes the jump in the associated property across an interface.

In summary, the governing equations for viscoelastic loading consist of the conservation of momentum (Eq. ) driven by the evolving surface load (Eq. ) along with the conservation of mass (Eq. ) and a viscoelastic constitutive equation (Eqs.  and ). Note that this linearised formulation is strictly relevant for small displacements with respect to the depth of the mantle and that the background stress field is assumed to be in hydrostatic equilibrium, implying no lateral variations in density . For simplicity, we have also neglected rotational and self-gravitational effects in this study. Nevertheless, they are particularly important when coupling the evolving ice and ocean loads through the sea-level equation and will be incorporated in future work.

To derive the finite-element discretisation of these governing equations, we first translate them into their weak form. By selecting appropriate function spaces that contain both solution fields and test functions, the weak form can be obtained by multiplying the equations by their test functions and integrating over the domain, Ω. For conservation of momentum, we use the (vector) test function, ϕ, to give 210=∫Ωϕ⋅∇⋅σ1L-Bμ∇ρ0u⋅ge^k-Bμρ1ge^kdx, which we then simplify by multiplying both sides by -1 and splitting into three components 220=S+H+D, where 23S=∫Ω-ϕ⋅∇⋅σ1Ldx, 24H=∫Ωϕ⋅Bμ∇ρ0u⋅ge^kdx, and 25D=∫Ωϕ⋅Bμρ1ge^kdx.

At this stage, it becomes necessary to define the finite element spaces that will be used for spatial discretisation. Generally, for each component of displacement, we use Q2 finite elements on hexahedral meshes (i.e., the piecewise continuous tri-quadratic tensor product of quadratic continuous polynomials in each direction). In Sect. , however, we instead use prismatic meshes that are unstructured in the horizontal but layered in the vertical (facilitated by Firedrake's inbuilt `extruded' mesh functionality, this space uses the piecewise continuous bi-quadratic tensor product of a quadratic polynomial in the horizontal and a quadratic polynomial in the vertical; ). For the deviatoric strain tensor (and hence the internal variable), since it is proportional to the gradient of displacement, we choose the discontinuous DG1 space (i.e., linear variations within each finite element cell and discontinuous jumps between cells) for each component. For purely radial variations in density, viscosity, and shear modulus, we choose the DG0 space (i.e., constant within a finite element cell but discontinuous between cells), while for laterally varying viscosity fields, we again select DG1 finite element functions.

Once these spaces have been chosen, we can integrate the divergence of the incremental Lagrangian stress term by parts within each element, Kn, to introduce weak boundary conditions and to move derivatives from the trial function to the test function according to 26S=∑n∫Kn∇ϕ:σ1Ldx-∫∂Knϕ⋅n^⋅σ1Lds, where n^ is the outward pointing normal vector to an element edge, represented by ∂Kn. Given that test functions are continuous across cell edges and we assume that there is no jump in stress across material discontinuities, Eq. () simplifies to 27S=∫Ω∇ϕ:κ∇⋅uI+2μ0d(u)-2∑iμimi(u)dx+∫∂Ωtopϕ⋅Bμρloadghloadn^ds, where we have substituted Eq. () for the constitutive relation and Eq. () for the boundary condition at the top of the domain, ∂Ωtop. For bottom (and side) boundaries where we specify no normal displacement (i.e., Eq. ), we utilise two different strategies depending on domain geometry. In Cartesian domains, we apply the boundary condition strongly by modifying the discrete test and trial spaces, such that the surface integral vanishes. For spherical domains, we implement that condition weakly using the Symmetric Interior Penalty Galerkin method . An example showing implementation of this approach for mantle convection problems in Firedrake can be seen in .

Following , we combine the hydrostatic pre-stress and buoyancy terms 28H+D=Bμ∑n∫Knϕ⋅∇ρ0u⋅ge^k-∇⋅ρ0uge^kdx, which can be written in an explicitly symmetric form 29H+D=12Bμ∫Ωρ0∇(u⋅ge^k)⋅ϕ+u⋅∇(ϕ⋅ge^k)dx-12Bμ∫Ωρ0∇⋅uge^k⋅ϕ+ge^k⋅u∇⋅ϕdx, by assuming that the background stress is in a state of hydrostatic equilibrium, such that surfaces of constant background density and gravitational potential coincide (e.g., Eqs. B22–B29 in Appendix B of ).

We now discretise in time using the implicit Backward Euler (BE) scheme. This choice allows us to take timesteps larger than the characteristic Maxwell time without compromising numerical stability. Such flexibility is particularly advantageous when the timescale of glacial loading is substantially slower than the Maxwell time – as is often the case in low-viscosity regions – thereby avoiding having to take prohibitively small timesteps in realistic simulations of glacial cycles cf.. Applying the BE scheme, the evolution of each internal variable in Eq. () becomes 30min+1=11+Δtαimin+Δtαid(un+1), where the superscript n refers to the previous timestep, n+1 is the next timestep, and Δt is the timestep duration. We then substitute this result into Eq. (), yielding 31S=∫Ω∇ϕ:κ∇⋅un+1I+2∑iηiαi+Δtd(un+1)-mindx+∫∂Ωtopϕ⋅Bμρloadghloadn^ds, where ηi is the non-dimensional viscosity of the internal variable mi. Recombining Eqs. () and (), the final system of equations is 32∫Ω∇ϕ:κ∇⋅un+1I+2∑iηiαi+Δtd(un+1)dx+12Bμ∫Ωρ0∇(un+1⋅ge^k)⋅ϕ+un+1⋅∇(ϕ⋅ge^k)dx-12Bμ∫Ωρ0∇⋅un+1ge^k⋅ϕ+ge^k⋅un+1∇⋅ϕdx=-∫∂Ωtopϕ⋅Bμρloadghloadn^ds+∫Ω∇ϕ:2∑iηiαi+Δtmindx, where implicit terms involving the unknown displacement at the next time step, un+1, have been collected on the left-hand side and explicit terms known from the current time step are on the right-hand side. Finding the new values of un+1 requires solving a linear system at each timestep, which is solved in Firedrake through PETSc's comprehensive linear algebra library . Since the system is symmetric, we employ the Conjugate Gradient method as the Krylov method. We also choose PETSc's algebraic multigrid package, GAMG, as a preconditioner since it has proven to be a robust and effective method for other large-scale problems in geodynamics e.g..

In testing, we have found that the ratio of bulk-to-deviatoric strain terms (which shows up in the first integral term of Eq. ) plays a crucial role in determining solver performance. In particular, when the ratio of bulk-to-shear modulus is high (as is the case when approximating incompressible materials), the resulting system can become poorly conditioned and leads to an increased number of iterations. This issue is exacerbated when timesteps greatly exceed the Maxwell time, as the relative contribution of the deviatoric strain term is reduced in that scenario. Since the ratio of bulk-to-shear modulus is typically modest (i.e., ∼2) for mantle rocks, we do not expect this behaviour to be problematic when using our software to run realistic GIA simulations. Nevertheless, when approximating incompressible materials by raising this ratio, there is an increase in computational cost. One way around this issue is to reformulate the system, such that the internal variable evolution equation is solved simultaneously with the momentum equation, yielding a coupled system whereby two equations are solved for two unknowns, un+1 and mn+1. Under this formulation, the momentum equation becomes 33∫Ω∇ϕ:κ∇⋅un+1I+2μ0d(un+1)dx+12Bμ∫Ωρ0∇(un+1⋅ge^k)⋅ϕ+un+1⋅∇(ϕ⋅ge^k)dx-12Bμ∫Ωρ0∇⋅un+1ge^k⋅ϕ+ge^k⋅un+1∇⋅ϕdx=-∫∂Ωtopϕ⋅Bμρloadghloadn^ds+∫Ω∇ϕ:2∑iμimn+1dx, and Eq. () is converted into its weak form by integrating over the domain and testing with ψ, a (tensor) DG1 finite element test function (i.e., numerical fields vary linearly within cells but can have discontinuities across cell boundaries), yielding 34∫Ωψ⋅min+1dx=∫Ωψ⋅11+Δtαimin+Δtαid(un+1)dx.

This formulation results in a system that is no longer symmetric due to the presence of coupling terms and we therefore use GMRES as the Krylov method. A key challenge is to find an efficient preconditioner for the coupled system. Within each iteration of GMRES, we employ a block Gauss-Seidel approach (facilitated by PETSc's FieldSplit functionality) whereby we independently solve Eq. () for un+1 and Eq. () for mn+1, using the previous value for the other unknown. The matrix for both of these individual inner solves is symmetric and, for the momentum Eq. (), we can use the same Conjugate Gradient and GAMG strategy as before, while the internal variable Eq. () is computationally simpler and can be solved using the Conjugate Gradient method preconditioned with successive over-relaxation.

Another benefit of the coupled formulation is that it provides a convenient means to model non-linear composite rheologies (e.g., the power-law formulation introduced in Eq. ). In these cases, the viscosity is a function of stress (i.e., depends on both the internal variable and displacement). Since the coupled formulation explicitly computes both of these variables at every solution step, it is therefore comparatively trivial to employ PETSc's SNES functionality to implement a non-linear solve strategy, for which we apply Newton's method with a line-search algorithm.

In addition to easy implementation of non-linear rheologies, the coupled system ensures that convergence of the momentum solve is no longer dependent on timestep size and therefore offers greater robustness in cases involving either long timesteps or large bulk-to-shear modulus ratios. Nevertheless, it comes at the computational expense of additional iterations of the outer GMRES algorithm to maintain coupling. We therefore generally prefer to use the substitution method in simulations that use compressible, linear rheologies.

We also note that an alternative, more common approach to solve the momentum equation in the incompressible limit is to reintroduce pressure as a Lagrange multiplier in order to enforce the conservation of volume constraint ∇⋅u=0 e.g.. The resulting saddle-point system can be effectively solved using a Schur complement factorisation strategy e.g., which we have implemented during code development but do not discuss further here since it is not applicable to real-Earth simulations with realistic levels of compressibility.

To summarise this section, we have outlined the viscoelastic loading equations and discretisation as currently implemented in G-ADOPT. In particular, by adopting the internal variable formulation of viscoelasticity , we are able to accommodate a broad class of complex rheologies with relative ease in comparison to many of the traditional, Maxwell-based approaches. Additionally, our implementation of an implicit timestepping scheme enables stable integration even when timesteps are significantly longer duration than the Maxwell time. This flexibility should prove particularly advantageous in longer glacial loading scenarios, where low-viscosity regions that have Maxwell times on the order of a year would otherwise impose prohibitively small timestep constraints.

In this section, we assess the accuracy of G-ADOPT in solving simple viscoelastic loading problems that have analytical solutions for both compressible and incompressible cases. Such benchmarks provide a stringent check as they also allow us to test whether, as the temporal and spatial resolution is refined, the error decreases at the expected theoretical rate.

For the first scenario, we consider the vertical surface displacement, ur, of a 2D Maxwell viscoelastic half-space that is subjected to emplacement of an instantaneous, sinusoidal surface load. By rearranging Eq. (2b) from , we can simulate a loading, rather than an unloading scenario. The analytical solution is 35ur=F01-11+feα/τ+1-e-t/(τ+feα)1+feα/τcos(kx), where F0 is the load thickness (where its density is equal to the mantle density), α=ημ is the Maxwell time, η is the viscosity, μ is the shear modulus, and τ is the viscous relaxation timescale given by 36τ=2kηρg, where k is the wavenumber, ρ is the density, and g is the gravitational acceleration. The compressibility parameter, fe, is defined as 37fe=λe+2μλe+μ, where λe is the (first) elastic Lamé parameter, given by 38λe=κ-2μ3, where κ is the bulk modulus. When the material is incompressible, then fe→1.

For consistency with the analytical solution, both compressible buoyancy and advection of hydrostatic pre-stress must be neglected. To ensure correct asymptotic behaviour in the viscous limit, we must include a surface integral term at Earth's surface, ∫∂Ωtopϕ⋅n^Bμρ0gukds, where uk is the component of non-dimensional displacement in the e^k direction. This term is equivalent to the surface term that would be obtained after integrating Eq. () by parts e.g., and is similar to the free-surface feedback encountered in mantle convection .

We adopt a mantle viscosity of η=1×1021 Pa s and shear modulus of μ=1×1011 Pa, yielding a Maxwell time of α≈320 years. For the compressible case, the bulk modulus is κ=2×1011 Pa, yielding fe=1.43. Density and gravitational acceleration are set to 4500 kg m⁻³ and 10 m s⁻², respectively. The applied load has a height of 1 km and wavelength of 375 km (equivalent to 18 of the domain depth), leading to a viscous relaxation time of τ≈24000 years.

Figure a shows the temporal evolution of peak surface displacement in the analytical solutions over a duration of 160 Maxwell times (i.e., approximately 2.2 viscous relaxation timescales). Both compressible and incompressible cases exhibit similar behaviour: an immediate elastic displacement is followed by a gradual approach towards isostatic equilibrium. Including compressibility leads to a slightly larger initial elastic deformation (∼19 m, as opposed to ∼13 m; Fig. b), but the viscous asymptote is the same for both scenarios (Fig. c).

To explore numerical accuracy as a function of timestep size, we fix the mesh resolution and perform a suite of simulations over a range of timestep durations. We use a 2D domain of depth 1 and width 0.5 in non-dimensional terms that generally contains 320×320 cells, except in the incompressible, long-duration case where this resolution was doubled to 640×640 cells to ensure that the time discretisation error dominates over any spatial discretisation errors. Since the Backward Euler timestep method is unconditionally stable and first-order accurate, these tests should ideally achieve a first-order convergence rate.

At short timescales where elastic deformation is expected to dominate, we undertake a single timestep with durations of Δt=0.1α down to 0.0015625α. To approximate incompressibility within our compressible viscoelastic formulation, we set the bulk modulus to a high value of κ=1×1015 Pa (equivalent to fe=1.0001). For both compressible and incompressible cases, as Δt reduces, we can see that our viscoelastic response gets systematically closer to the instantaneous elastic response from the analytical solutions at time t=0 (Fig. b) and that the surface L2-errors of this difference reduce with first-order accuracy (Fig. d).

At long timescales where viscous deformation dominates, we run numerical simulations with a timestep duration that systematically varies from 16α down to 0.5α. Since the relative contribution of elastic compressibility to total displacement is much lower over long timescales, we use a less extreme bulk modulus of κ=1×1013 Pa for the incompressible case (i.e., fe=1.01). Once again, we observe that, as timestep size reduces, numerical simulations approach the two analytical solutions (Fig. c) and that their time-integrated global L2-error reduces with first-order convergence (Fig. e).

Despite the physical simplicity necessitated by these analytical solutions, the results demonstrate that G-ADOPT accurately reproduces the analytical response for both compressible and incompressible Maxwell halfspaces and that its implicit time-stepping scheme achieves the expected first-order accuracy in time. In the following benchmarks, we next evaluate G-ADOPT against other finite element and semi-analytical codes using more realistic loading scenarios in a 3D domain.

have recently presented a valuable suite of test simulations created using three published GIA models: two finite element codes – Abaqus and Aspect – and one semi-analytical code that assumes an axisymmetric spherical Earth, Taboo . A schematic of their model setup is shown in Fig. , for which the domain is a 3D Cartesian box measuring 1500×1500 km laterally and 2891 km in depth (and therefore can only be approximately represented in Taboo). Two loading scenarios were considered: (i) a short-duration event that has similar spatiotemporal scales to modern ice changes, whereby a 100 m-thick ice sheet linearly grows for 100 years before being held constant for another 100 years; and (ii) a long-duration scenario that is more similar to a glacial cycle, whereby 1 km of ice linearly accumulates over 90 kyr and then fully melts by 100 kyr, followed by a further 10 kyr of evolution in the absence of further loading changes. In both cases, the ice sheet is represented by a disc with a radius of 100 km.

The default Earth structure follows the 1D layered model of , composed of a lithosphere, upper mantle, transition zone, and lower mantle (Table ). To assess sensitivity to lateral viscosity variations, a second 3D-Earth structure also incorporates a cylindrical anomaly with a radius of 100 km and a reduced viscosity of 1×1019 Pa s directly beneath the ice sheet, extending from 70–170 km depth. Since Taboo assumes spherical symmetry, it is therefore limited to modelling 1D cases only.

The configuration in G-ADOPT is set up to closely imitate Aspect's implementation of the benchmark, which includes: neglecting self-gravity; using a constant gravitational acceleration of 9.815 m s⁻²; setting viscosity in the elastic lithosphere to 1×1040 Pa s (thereby ensuring negligible viscous deformation of this layer over the course of the simulation); and exploiting axisymmetry to model only one quarter of the domain with no-normal-flow boundary conditions imposed on its side walls and base. Ice loading is implemented using a smooth hyperbolic tangent profile with a 1 km “rollover” length-scale and the density of ice is assumed to be 931 kg m⁻³. We use Gmsh to construct a horizontally unstructured mesh, refined to 5 km under the ice and coarsened to 200 km in the far field . Vertically, each rheological layer is discretised with a fixed number of DG0 elements (e.g., 10 per layer in the default setup), providing constant values within elements but permitting sharp boundaries between layers. Timesteps are 10 and 1000 years for the short- and long-loading scenarios, respectively. For reference, Abaqus uses a similar meshing approach to G-ADOPT, with horizontal resolution of 5–200 km and 8 vertical cells per rheological layer; Aspect uses ∼6 km isotropic resolution in the top 100 km of the domain, transitioning to 50 km elsewhere; and Taboo uses a maximum spectral degree of 4096, equivalent to approximately 5 km horizontal resolution, and is semi-analytical in the vertical direction . Since the benchmarks all assume incompressibility, we again approximate incompressibility by setting the bulk modulus to 1×1014 Pa (i.e., 1000 times larger than the reference shear modulus). Note that, to ensure robustness in these incompressible simulations, we solve the coupled system (i.e., Eqs.  and ) for reasons outlined in Sect. .

Figure  compares maximum vertical displacements beneath the ice load across all four codes in each of the four benchmark combinations of Earth model and loading scenario. The upper row shows results for 1D Earth structure, while the lower row includes the low-viscosity cylindrical anomaly. Agreement is generally excellent in all cases where cross-code comparisons are possible (Fig. a–c), affirming consistency with G-ADOPT's implementation. In the final panel (Fig. d), we present results from G-ADOPT for the long-loading scenario with 3D viscosity – a configuration not illustrated in . In comparison to the same loading scenario on a 1D Earth, the presence of a low-viscosity patch accelerates the surface response consistent with expectations, but only modestly: the maximum vertical displacement differs by just ∼2 % relative to the 1D case at 90 kyr, compared to a ∼60 % difference in the short-duration loading scenario. This result is consistent with the fact that, even for the 1D Earth model, the ∼100 kyr loading timescale is substantially longer than the ∼320 year Maxwell time of the upper mantle (N.B., the equivalent Maxwell time for the low-viscosity patch is only ∼3 years).

To further assess the robustness of G-ADOPT, we conduct a resolution and parameter sensitivity analysis for the scenario involving long-duration loading of a 1D Earth model. Horizontal mesh sensitivity tests demonstrate that even as few as five elements beneath the ice-covered region (i.e., 20 km horizontal resolution) are sufficient to accurately capture the pattern of deformation (Fig. a). Changing the vertical resolution has a bigger effect (Fig. b). Halving the default values to five elements per layer (i.e., 20 vertical elements in total) results in practically indistinguishable results, but reducing to two cells per layer produces peak differences of ∼1 %, which increases to ∼6 % when only using one cell per layer. Regarding the impact of timestep size, these tests neatly demonstrate the effectiveness of our implicit time-stepping scheme. Even when the timestep is increased to 10 kyr, which is more than 30 times greater than the Maxwell time of the upper mantle in these models, the solution remains stable and accurate throughout the loading phase and produces only minor errors during the unloading phase (Fig. c). For this case, the computational cost per timestep is doubled because the number of outer iterations per timestep increases, but this still leads to an effective speed up of a factor of five in comparison to the 1 kyr simulation. By way of comparison, Aspect used significantly shorter timesteps (e.g., 50 years) to achieve comparable results in this benchmark .

We can also use this test scenario to demonstrate the impact of realistic levels of compressibility by changing the ratio of bulk-to-shear modulus (Fig. d). A value of 1000 closely approximates incompressibility and turns out to be indistinguishable from a value of 100. By the time it is lowered to 1.94, there is a ∼20 % increase in maximum vertical displacement, reflecting the larger elastic deformation caused by loading compressible materials. This result further underscores the importance of including compressibility in GIA simulations . However, it also requires taking greater care during model setup to ensure that a density inversion (with respect to depth) does not occur at any point during the simulation. Such a scenario leads to convective instabilities that can grow, depending on the timescales, and have been widely documented in GIA literature .

To test the scalability of G-ADOPT, we have undertaken a weak scaling analysis on Gadi (Australia's highest performance CPU-based supercomputer) using the long-duration loading scenario on a 1D Earth and a structured mesh. We use a number of cores ranging from 104 up to 6656 (i.e., from 1 to 64 Intel Xeon Sapphire Rapids nodes) and aim to keep the number of degrees of freedom per core constant at ∼30000. The results show that, with a bulk-to-shear modulus ratio of 100, increasing the problem size by a factor of 64 causes the solve time per timestep to increase by ∼30 % (Table ), which is consistent with previous scaling results for mantle convection modelling within Firedrake . As discussed in Sect. , for the case with a smaller bulk-to-shear modulus ratio of 1.94, conditioning of the system improves and the number of iteration counts is significantly reduced. However, the overall scaling performance is slightly worse with the solve time per timestep effectively doubling as the problem size increases by a factor of 64.

In summary, for a 3D Cartesian box with potential presence of lateral viscosity variations, G-ADOPT accurately reproduces benchmark results from published codes. It also exhibits desirable performance characteristics, including efficient timestepping and stability across a range of resolutions. These results validate the core solver and lay the groundwork for investigating more complex loading scenarios, including those involving compressibility and more complex rheologies.

In this section, we demonstrate the flexibility of the internal variable formulation to model both power-law and transient rheologies by extending the short- and long-duration loading 1D-Earth benchmarks of Sect. . Compressibility is implemented by adopting a bulk-to-shear modulus ratio of 1.94 (corresponding to a Poisson’s ratio of 0.28), as that value is standard in Abaqus implementations . As discussed before, the impact of compressibility alone is to increase the magnitude of the peak displacement through time in comparison to the incompressible Maxwell case, consistent with expectations (Fig. ).

For composite power-law rheology, we use the same reference viscosity as before and adopt a transition stress of τT=0.2 MPa and exponent of n=3 (consistent with case 1 of ). All other aspects of the setup including grid resolution, timestep size, boundary conditions, and loading configuration are identical to those in Sect. . In the short-duration loading scenario, the maximum vertical displacement increases with respect to the compressible Maxwell model, becoming more obvious after the load stops growing at 100 years and reaching ∼13 % more by 200 years (Fig. a). In the long-duration loading scenario, the same situation is observed during loading, although the peak vertical displacement is only ∼1.7 % larger at the time of maximum loading at 90 kyr. During unloading, however, the composite power-law rheology recovers substantially faster than its Maxwell equivalent, reaching 20 % of its peak displacement by 103 kyr in comparison to 109 kyr for the Maxwell case. We also verified that setting the transition stress to a high value of τT=10 MPa (i.e., ensuring that the power-law component is not activated) gave equivalent results to the Maxwell case.

A Burgers rheology can be conceptually considered as a standard Maxwell body (composed of a spring and dashpot connected in series) connected in series to a single Kelvin-Voigt element (a spring and dashpot connected in parallel). It requires using two internal variables to represent the two characteristic relaxation timescales of the system . To maintain the same elastic response as in the compressible Maxwell case, we halve the shear modulus of each internal variable element, thereby preserving the total effective modulus. Similarly, we halve the viscosity of the long-term viscous element, so that the overall relaxation timescale remains consistent with the Maxwell cases presented earlier. We then explore two different cases in which the viscosity of the element controlling short-term relaxation is set to either one-half or one-tenth that of the long-term element.

In both cases, adding in a transient element with lower viscosity amplifies the response in comparison to the compressible Maxwell case (as was found for the power-law rheology), particularly where rapid changes in the load interact strongly with the shorter relaxation timescale (Fig. ). This behaviour is most obvious in the short-duration loading scenario, but it is also visible in the unloading phase of the long-duration scenario. We also verified that setting the same viscosity for both internal variables (i.e., reducing the Burgers model to a single timescale) reproduces the original Maxwell response, confirming the internal consistency of our implementation.

It is noteworthy that, for short-duration loading, displacement curves for the compressible Maxwell case, and the compressible power-law and Burgers simulations (Fig. a), exhibit comparable features to those produced for the incompressible Maxwell model with a low-viscosity patch (Fig. c). Although the underlying deformation mechanisms differ, this similarity illustrates the difficulty of disentangling the effects of local viscosity variations, compressibility, and non-Maxwell rheologies e.g.,. Nevertheless, it also highlights the flexibility of the G-ADOPT framework and reinforces its potential future value to these debates.

In summary, we have demonstrated G-ADOPT's ability to model compressible power-law and transient viscoelastic rheologies using the internal variable formulation and the important impact of these phenomena on GIA predictions, particularly during rapid loading changes. In the next section, we demonstrate extension of these simulations to domains with spherical geometry.

A significant strength of G-ADOPT is that it is relatively trivial to change the mesh to suite a range of different domain geometries. We demonstrate this aspect by extending our previous simulations from the 3D Cartesian box to a 3D spherical Earth. We adopt a model domain from the benchmark of , which modelled an incompressible Maxwell rheology with purely radial viscosity variations. While we cannot repeat this benchmark as we have not yet implemented self-gravity in G-ADOPT, we have extended it to a more complex rheology by using a compressible, Burgers (i.e., transient) rheology that also contains lateral viscosity variations.

To generate the mesh, we first use Firedrake's inbuilt functionality to create a cubed-sphere surface mesh with 24 576 horizontal elements (equivalent to 6 refinements). We then “extrude” the mesh in the vertical direction to create 40 vertical layers (10 per rheological layer), as in Sect. . A couple of additional changes with respect to the Cartesian setup are that we provide a rotational null space to the Firedrake solver options to improve iterative solver performance and provide a weak implementation of the no-normal-flow boundary condition on the core-mantle boundary. We retain the same radial profiles of density, shear modulus, and long-term (i.e., steady-state) viscosity as in , but these viscosities are further modulated by a (ℓ=4,m=1) spherical harmonic function that introduces lateral viscosity variations spanning two orders of magnitude (Fig. ). For the short-term viscosity in the Burgers model, we assign a value that is a factor of 10 times smaller, while compressibility is set using a bulk-to-shear modulus ratio of 1.94, corresponding to a Poisson's ratio of approximately 0.28. A 1 km-thick ice disc with a radius of 10° latitude is placed on the North Pole and applied instantaneously at the start of the simulation via a Heaviside step function. Ice density is 931 kg m⁻³ and gravity is constant: g=9.815 m s⁻². A timestep of 50 years is used to resolve rapid deformation following the instantaneous ice loading event.

Figure  shows slices through the spherical domain at times of 1, 2, 5 and 10 kyr after the load is applied. Early on in the simulation, deformation velocities are large and concentrated in the upper mantle. Despite the axisymmetric nature of the load, there is a bias in their direction, with greater movement towards the neighbouring region of low viscosity. There is also a higher amplitude, shorter wavelength peripheral bulge developing on this side of the ice sheet in comparison to the direction of the highest viscosity. After 5 kyr, the remaining deformation is now concentrated on the higher viscosity side of the domain and has much lower velocity, consistent with expectations that the higher viscosity region has a longer Maxwell time and therefore takes longer to reach isostatic equilibrium. By 10 kyr, the majority of deformation has finished and the initially highly asymmetric surface displacement has gradually restored to a more symmetric pattern. At equilibrium, the surface displacement should match the symmetry of the applied load (since we are still using radially symmetric elastic structure).

This simulation is designed to showcase the relative ease in which G-ADOPT can be switched between different model domain geometries. It also simultaneously demonstrates its ability to model more complex, Earth-like problems that include compressibility, lateral viscosity variations, and transient rheologies. We believe that these aspects should render it a valuable future tool for modelling viscoelastic surface loading problems in future work, such as those routinely encountered in GIA. In the next section, we illustrate another key strength of G-ADOPT – namely its ability to compute automatic adjoints.

As emphasised in the introduction, a major motivation for developing a new viscoelastic deformation code in G-ADOPT is the availability of an automatically-derived discrete adjoint that can be used to invert any observational constraints for unknown Earth structure and/or loading histories. Here, we use a synthetic 2-D annulus domain loaded with ice at its surface and demonstrate this adjoint capability by performing gradient-based optimisations for ice thickness and mantle viscosity. While this example is intentionally kept simple, our goal is to establish a clear and controlled foundation for tackling more complex and realistic inverse problems in future work.