The staggered-grid finite-difference is a popular approach to discretizing the Stokes equation originally proposed by . This scheme is constructed using conservative second-order finite difference and a staggered arrangement of both velocity and pressure (Fig. a, b). The strong form of the Stokes equations (Eq. ) is discretized using central finite differences: 4rmom=∇^⋅τ-∇^P-b,5rcont=-1KΔ^PΔ^t-∇^⋅v,6τ=2ηD^-13(∇^⋅v)I, where the hat symbol denotes the discrete representation of the differential operators. The primitive solution fields v and p are obtained by minimizing the momentum and continuity residuals (rmom and rcont). The staggered FD scheme represents, to our knowledge, the most computationally efficient stencil that satisfies the Inf-Sup stability condition in regular meshes . For flows with smooth viscosity variations, FD achieves second-order accuracy in the L1 norm, while in the presence of sharp material heterogeneities, its accuracy reduces to first-order . Geometrically, FD has been successfully implemented in Cartesian and polar coordinates e.g., spherical configurations e.g., and on adaptively refined meshes e.g.. However, its application in non-orthogonal meshes remains rare e.g. due to the need for extended stencils and the associated increase in algorithmic complexity. Consequently, the treatment of free-surface boundary conditions remains a notable challenge e.g..

The Face-Centered Finite-Volume (FCFV) method is an emerging discretization scheme rooted in the principles of the hybridizable discontinuous Galerkin (HDG) method . This approach has been successfully employed to achieve full-field solutions for the Stokes equations in various types of viscosity distribution, including constant , discontinuous , and smoothly varying fields . The FCFV scheme satisfies the Inf-Sup stability condition, making it particularly suitable for solving the Stokes equations. The discrete system of equations is derived from the weak formulation of the Stokes problem, using a constant degree of approximation for both velocity and pressure fields: 7rmom=Γτ+PIn+2s{{v}}-v^-χTN,8rcont=-∑i=1nfacΓi1KΔ^PΔ^t+ni⋅v^i,9τ=2η∑i=1nfacΓi2Ωni⊗v^i+v^i⊗ni-13ni⋅v^iI,10v=1abΩ+∑i=1nfacΓisv^i, The primitive variables, namely the hybrid velocity vector (v^) and pressure (P), are obtained by minimizing the momentum and continuity residuals (rmom, rcont). The hybrid velocity is an independent variable defined on element faces, different from the element-wise velocity. It serves as a Lagrange multiplier–type trace variable that enforces weak continuity of the velocity field across elements. The symbol nfac corresponds to the number of faces of an element. ⋅ is the jump operator defined as ⊙=⊙R-⊙L, where ⊙ denotes quantities defined on elements located to the right (R) and left (L) of an interface. {{⋅}} is the average operator defined as {{⊙}}=12⊙R+⊙L. The discretization requires the outward-pointing normal to each face (n), the face area (Γ), the volume of the element (Ω), the stabilization parameter (s), the factor a (a=∑i=1nfacΓis), the traction vector (TN), and the function χ that indicates the presence of applied traction (0 or 1). The FCFV scheme requires the storage of the hybrid velocity vector and momentum residual at the midpoint of each element face, while the pressure, the deviatoric stress tensor and the continuity residual are stored in the centroids of the elements (Fig. c, d). The discretization considered here yields a first-order approximation of velocity and pressure fields , nevertheless second-order solutions can be constructed without altering the sparsity of the discretization . In this study, we implement the FCFV approach using structured quadrangular meshes, although the method is versatile and supports arbitrary element geometries, exhibiting low sensitivity to mesh distortions . Additionally, FCFV can naturally handle internal boundaries and traction boundary conditions, such as free surfaces. Due to these features, the FCFV method is becoming increasingly popular in the geodynamics community, as highlighted by .

The dynamic relaxation method is designed to iteratively solve both linear and nonlinear problems. The iterative procedure resembles an explicit time integration scheme, which is why it is also often referred to as a pseudo-transient integration method. As in classical iterative procedures, the DR method iteratively refines the unknown u until the magnitude of the residual of the partial differential operator r(u) drops below a given tolerance. Once this criterion is met, u is considered a solution satisfying r(u)=0. Note that the function r(u) may include physical transient terms; thus, this method is not limited to solving quasi-static problems e.g.. The algorithm is formulated by introducing the first and second-order derivative of the solution u in pseudo-time : 11Md2udt‾2+Mcdudt‾=r(u) where t‾ refers to pseudo-time, c is a damping parameter, and M is a preconditioner. Following previous work e.g., the second derivative is taken as the central derivative of the first derivative and the first derivative is taken as the average of the new and old values: 12d2udt‾2≈1Δt‾ΔuΔt‾new-ΔuΔt‾old13dudt‾≈12ΔuΔt‾new+ΔuΔt‾old. This leads to update rules for the rate of change of u, and u itself: 14ΔuΔt‾new=bΔuΔt‾old+aM-1r(u)15unew=uold+Δt‾ΔuΔt‾new where a=2Δt‾/(2+cΔt‾) and b=(2-cΔt‾)/(2+cΔt‾). Alternatively, the DR algorithm can be expressed in analogy to the conjugate gradient (CG) method , using the variable p=Δu/(βΔt‾): 16pnew=βpold+M-1r(u)17unew=uold+αpnew where p is analogous to the search direction, α=aΔt‾ and β=b. The main difference is that the CG method involves different expressions for the parameters α and β, which need to be reevaluated at each iteration, requiring computationally expensive reduction operations (e.g. vector norms).

In either case, two iteration parameters need to be determined as a function of the minimum (λmin) and maximum (λmax) eigenvalues of the discrete problem : 18Δt‾=cCFL2λmax19c=cdamp2λmin where cCFL and cdamp are scalar adjustment parameters. Whereas cCFL is strictly bounded (cCFL<1), no analogous upper bound applies to cdamp. In our experiments, values in the range ≈[0.5,1] were found to provide robust performance. For optimal convergence, at least in the case of linear problems, cCFL should be chosen as large as possible (e.g.  cCFL=0.999). In theory e.g., the optimal value of cdamp is 1.0. Nevertheless, we empirically observed that slightly lower values may further enhance convergence.

The maximum eigenvalue is found using the Gershgorin circle theorem: 20λmax⁡=max⁡i∂ri∂ui+∑j≠i∂ri∂uj. The minimum eigenvalue can only be approximated, using, for example, a Rayleigh quotient approach e.g.: 21λmin=ΔuTΔrΔuTMΔu, where Δu=unew-uold and Δr=r(unew)-r(uold). The evaluation of iterative parameters requires two reductions. Reduction operations should be avoided when possible given that reduction algorithms in shared memory architectures are not embarrassingly parallel, and global communication between different nodes is needed in distributed memory architectures. Nevertheless, this procedure can be performed every n iterations, together with the evaluation of the stopping criteria. This approach provides successful convergence while minimizing the cost of global communication. It should be noted that the estimate of λmin⁡ relies on differences in the residual. Before the iteration is initiated, this quantity is set to zero by default.

Lastly, the DR method requires the definition of a preconditioner, typically defined as the Jacobi or diagonal preconditioner: 22M=diag∂r∂u. While this choice allows for trivial inversion and parallelisation, it restricts the convergence property of the iteration scheme. In the context of matrix-free schemes, both the evaluation of the preconditioner and maximum eigenvalue estimates can be achieved by using automatic differentiation of the residual function.

An example of the application of the DR method to a variable coefficient Poisson problem is depicted in Fig. . The problem is defined as: 23r(u)=-∇⋅q-b,24q=-k∇u where k=f(x). The differential operators are discretized with both the FD and the FCFV over a 1D domain Ω=-12,12 and Dirichlet boundary conditions u-12=1, u12=2 (Fig. a). The coefficient k varies with x (Fig. b) and is expressed as f(x)=1+100(1+exp⁡(-200(x+0.1))-1-1+exp⁡(-200(x-0.1))-1). The forcing term is set to zero so that b(x)=0. The problem converges for different resolutions to a relative residual of 10-10, with evaluation of iterative parameters and convergence check every 100 iterations. The frequency with which the parameters are reevaluated can influence the convergence behavior. In the present study, a value of 100 was empirically determined to provide an appropriate compromise. The typical convergence behavior of the DR method is shown in Fig. c. As a consequence of diagonal preconditioning, the number of iterations needed to reach convergence depends linearly on the size of the problem (Fig. d). It is worth noting that for the 1D Poisson case, the FD and the FCFV deliver very similar results, both in terms of the numerical solution and convergence behavior.

The DR method is particularly well-suited to solving symmetric positive-definite problems. The solution of indefinite problems, such as those arising in saddle-point systems such as the Stokes equations, requires additional modifications.

In order to test the robustness of the proposed solution procedure and to evaluate its scaling behavior, we have designed a model configuration that combines known challenging elements for geodynamic flow solvers. These elements include the presence of large and sharp viscosity variations, as well as the enforcement of the incompressibility constraint. This type of flow problem is known to be pathological for geodynamic solvers that rely on iterative solution procedures .

Both viscosity variations and incompressibility contribute to a deterioration in the conditioning of the discrete system. While this does not pose a problem for direct solvers, it is problematic for iterative solvers, whose performance strongly depends on the spectral properties of the discrete system.

The selected model configuration consists of a unit volume cubic domain, centered at the origin. The viscosity within the inclusions varies according to the viscosity contrast (Δη=ηmaxηmin). The background viscosity ηref is defined as ηref=10log⁡10(ηmin)+log⁡10(ηmax) and is set to 1. A total of 50 spherical inclusions, with variable radii and positions, are distributed throughout the domain (see Appendix ). Weak inclusions have their viscosity set to the minimum viscosity value (ηweak=ηmin), while strong inclusions have their viscosity set to the maximum viscosity value (ηstrong=ηmax). The flow is driven by boundary conditions that prescribe pure shear deformation in the x–y plane, with extension along the x-axis. No flow occurs across the z boundaries, and free-slip conditions are applied to all faces of the domain.

An example of the model results is presented in Fig. , showcasing the model configuration and the resulting flow field structure for a viscosity contrast spanning four orders of magnitude. The computations were performed with a numerical resolution of 3203 cells (66 Mdof) for the FD (Fig. a) and 2563 cells (168 Mdof) for the FCFV (Fig. b). These resolutions correspond to the largest resolution that can be achieved on a single GPU with 32 GB of DRAM (NVIDIA Tesla V100-SXM2-32GB).

Overall, the two methods exhibit a consistent and satisfactory qualitative agreement. Small differences between the two methods can be observed, particularly in regions where inclusions are in close proximity. These discrepancies stem from the different spatial resolution used for the FD and the FCFV methods, their different order of accuracy, and their different treatments of interfaces. A more detailed comparison of the FD and FCFV results is provided in the Appendix . The stopping criteria of the PH/DR scheme was set to 10-6, the inner DR stropping criteria was set to 10-3, the penalty parameter was set to 15 times the mean of the viscosity field (γ=15〈η〉).

Although the previous examples were performed in the incompressible limit, the PH/DR scheme can adequately handle compressible deformation or flow. Here, we show examples of compressible flow performed in 2D, using FD. The model configuration consists of a domain Ω∈[0,1]×[0,1] with a circular inclusion of radius 0.2 (Fig. a). The background viscosity is set to 1 and the inclusion viscosity is 100. Pure shear background deformation is applied at the boundaries. Compressibility is accounted for by including a bulk viscosity term: 28∇⋅τ-∇P=0,29∇⋅vs+Pηb=0 The degree of compressibility varies with the value of bulk viscosity (ηb). To solve this problem, one can use PH/DR, setting the penalty factor independently of the bulk viscosity. Here, we express the penalty factor as the quasi-harmonic average of a reference numerical bulk viscosity (γnum=60〈η〉) and the physical bulk viscosity, γphys=ηb, so that γ=1/γphys+1/γnum-1. In this case, several Powell–Hestenes iterations will be needed to reach convergence. Alternatively, one can set the penalty factor to the bulk viscosity (γ=ηb). In that case, PH/DR becomes equivalent to a basic DR scheme and only one Powell–Hestenes iteration is needed to reach convergence. We observe that in the case of compressible problems (ηb=2.0, equivalent to a Poisson ratio of 0.286), the basic DR can be beneficial, leading to a lower number of iterations compared to PH/DR (Fig. b). The PH/DR solver exhibits a characteristic sawtooth convergence history that reflects the update of pressure at each Powell–Hestenes iteration (Fig. b). In contrast, in weakly compressible cases (ηb=2×103, equivalent to a Poisson ratio of 0.499), the PH/DR schemes outperform the standard DR iteration approach. In the context of geodynamic problems involving weakly compressible-to-incompressible materials, PH/DR is preferred.

We further applied the PH/DR scheme to the solution of a coupled multi-physics problem. The equations governing deformation and fluid flow in a Darcy poro-viscous medium can be expressed following : 30∇⋅τ‾-∇P‾=0,31∇⋅vs+P‾-Pf(1-ϕ)ηϕ=0,32∇⋅qD-P‾-Pf(1-ϕ)ηϕ=0. Here, the Darcy flux is expressed as qD=-kf/ηf∇Pf. The overbar denotes phase-averaged quantities, ϕ is porosity, and the superscripts s and f indicate solid and fluid properties, respectively. k and η refer to permeability and viscosity. An elliptic equation for fluid pressure can be obtained by substituting the Darcy flux into Eq. ().

The PH/DR scheme is extended by including fluid pressure updates within the inner DR iterations. The update parameters for fluid pressure are obtained from the automatic DR procedure described in Sect. . The pseudo-time steps and damping coefficients for velocity and fluid pressure are determined independently, with each variable assigned its own value. As in previous examples, the total pressure is updated at each Powell–Hestenes iteration.

The test case we consider uses a configuration similar to that in the previous example (Sect. ). The shear viscosity of the inclusion is assigned a value 100 times greater than that of the matrix, while the bulk viscosity is taken as twice the shear viscosity. In addition, the background porosity is set to 1×10-3, and the fluid permeability-viscosity ratio kf/ηf is set to 1×10-3. The penalty factor is defined as γ=1γphys+1γnum-1, where γphys=(1-ϕ)ηϕ and γnum=5〈ηs〉.

Figure  shows examples of solution fields and the scaling properties of the PH/DR scheme. We compute the two-phase flow problem with a resolution varying from 312 to 9922 cells and apply a tolerance of 10-7. For the chosen parameters, pronounced differences are observed between the total and fluid pressure fields (Fig. a, b). The total pressure field is sharply defined and zero inside the stiff inclusion, in contrast to the fluid pressure field, which is smoother and diffuses within the inclusion. We observe the characteristic sawtooth convergence history for both momentum and fluid continuity (Fig. c).

We find that low-resolution models require a relatively high number of iterations (Fig. d). Nevertheless, the number of DR iterations per Nx drops to a stable low value (approximately 20) for resolutions greater than 1002 cells. This indicates a linear dependence of the number of iterations on the problem size, which is consistent with the behavior documented in the previous examples.

In this example, we show how nonlinear material properties can be incorporated into the PH/DR solution strategy, considering the case of non-associated plasticity. With such nonlinear problems, parameters such as the effective viscosity vary throughout the nonlinear solution procedure, which directly affects the eigenvalues of the discrete operator. It is thus important to reevaluate the maximum pseudo-time step (Eq. ) during the iterations to ensure stability of the procedure. Here, we reevaluate the pseudo-time step every 100 iterations by applying the Gershgorin circle theorem (Eq. ).

We adopt a single-phase formulation and apply a visco-elasto-viscoplastic material model . An additive decomposition of the deviatoric strain rate and the divergence rate is assumed: 33ε˙=ε˙v+ε˙e+ε˙p34∇⋅v=∇⋅ve+∇⋅vp where the superscripts v, e, and p correspond to the viscous, elastic, and plastic components, respectively.

The viscoplastic Drucker–Prager yield and potential functions are defined as: 35f=τII-Psin⁡Φ-Ccos⁡Φ-λ˙ηvp36q=τII-Psin⁡Ψ where τII, C, Φ, Ψ, and ηvp denote the second deviatoric stress invariant, the cohesion, the friction angle, the dilatancy angle, and the viscoplastic viscosity, respectively.

The model domain is Ω∈[0,2]×[0,1] km2 and contains a circular inclusion of radius 1×10-1 km (Fig. a). A pure shear loading rate of -1×10-15 s⁻¹ is applied, inducing horizontal extension. The inclusion has a low viscosity (1×1010 Pa s), while the surrounding matrix is effectively elasto-plastic (1×1023 Pas). Both shear and bulk moduli are set to 3×1010 Pa. The cohesion is set to 50 MPa, the friction and dilatancy angles are set to 35 and 5°, respectively, and the viscoplastic viscosity is 2×1020 Pas. The simulation proceeds over 50 time steps, each of size Δt=4×1010 s.

The temporal evolution of the second invariant of the deviatoric strain rate is shown in Fig. a, b, and c, while the evolution of the mean stress is shown in Fig. d. The model undergoes an initial phase of elastic loading (Fig. a), after which shear banding initiates, propagates (Fig. b), and reflects off the domain boundaries (Fig. c). As a result, stress is limited after 30 kyr and undergoes minor fluctuations witnessing underlying structural softening (Fig. d). With a relative tolerance of 1×10-6, the number of iterations per Nx varies between 5 and 25 (Fig. e). Low iteration counts correspond to linear-elastic steps, while higher counts occur during nonlinear steps involving significant changes in the solution pattern (e.g. reflections). The convergence history for the most challenging step is shown in Fig. f, where the typical sawtooth convergence pattern of the PH/DR is also observed.

Overall, the model converges reliably without the need for manual tuning of numerical parameters throughout the simulation.

In this section, we investigate the benefits of the proposed PH/DR method for solving two practical geodynamic problems. The main benefit of the PH/DR method over the APT approach is its ability to automatically adapt to transient changes in solution patterns, caused by variation of the model geometry or by the emergence of internal features. The PH/DR approach can therefore determine parameters that ensure convergence while also significantly reducing the overall wall time. The results presented below are intended to be indicative. In principle, the APT method could deliver results that are at least as good as those of the PH/DR method. However, this would require manually tuning the iteration parameters at every time step, which is not a desirable task. Several simulations were performed using the open-source code JustRelax.jl , which features both the PH/DR solver and the Accelerated Pseudo-Transient (APT) method .

The first model is a simplified subduction setup based on the configuration of . Simulations were performed over a model duration of 25 Myr, after which the slab tip reaches a depth of approximately 500 km (Fig. ). We observe that the use of the PH/DR scheme significantly reduces cumulative wall time compared to the APT method (Fig. a). The number of iterations per grid point in one dimension is also well behaved, fluctuating around 20 iterations per grid point. For the same tolerance, the APT scheme requires about 200 iterations per grid point and exhibits larger fluctuations (Fig. b). Overall, both approaches converge at each iteration and yield similar results (Fig. c, d).

The second model is a crustal-scale shear banding setup, which features a visco-elasto-viscoplastic rheology and temperature-dependent creep laws. The reader is referred to for the detailed model configuration and material parameters. The simulations are run for 100 time steps, in which the solution pattern evolves significantly in response to shear banding and frictional viscoplasticity. The results presented in Fig.  show the cumulative wall time and the number of iterations over 100 time steps for various model resolutions. Models performed with the PH/DR scheme consistently result in 2 to 12 times shorter wall times (Fig. a). Unlike APT models, simulations using the PH/DR method provide a reliable and well-defined range of iterations per time step. Nevertheless, the overall shear banding patterns are similar (Fig. c, d). The minor differences may be attributed to the fact that APT models could not reach convergence at every time step.