For the past several decades, the geodynamics community's workhorse for numerical simulations of the incompressible Stokes equations has been the use of (continuous) piecewise bilinear and/or trilinear velocity and piecewise constant (discontinuous) pressure finite elements, often in combination with the penalty method for the solution of the resulting linear systems e.g.,. This velocity–pressure pair is often referred to as the Q1×P0 Stokes element and sometimes as the Q1×Q0 element . It is used, for example, in the ConMan , SOPALE , SLIM3D , CitcomCU , CitcomS , Ellipsis , UnderWorld , DOUAR , and FANTOM codes and has therefore been used in hundreds of publications.

The popularity of this element can be explained by its very small memory footprint and ease of implementation and use. On the other hand, it has a rather low convergence order that makes it difficult to achieve high accuracy; maybe more importantly, the element is known not to satisfy the so-called Ladyzhenskaya–Babuška–Brezzi (LBB) condition condition e.g., and is therefore unstable. This instability noticeably manifests itself through oscillatory pressure modes (e.g., Fig. 18 of or Fig. 36 of ) and makes it not suited for large-scale three-dimensional simulations coupled to iterative solvers . The unreliability of the pressure also makes this element a dubious choice for models in which some of the parameters – e.g., the density or the viscosity – depend on the pressure.

The more modern alternative to this choice is the Taylor–Hood element that uses (continuous) polynomials of degree k for the velocity and of degree k-1 for the pressure, where k≥2.

Strictly speaking, only introduced the Q2×Q1 element on quadrilaterals. However, finite-element practitioners use the term “Taylor–Hood” for both the 2D and 3D cases, for the case of both simplex and quadrilateral–hexahedral meshes, and for all cases with k≥2. See also p. 98.

For the purpose of this paper, we are concerned with the accurate numerical solution of the incompressible Stokes equations: 1-∇⋅2ηε(u)+∇p=ρginΩ,2-∇⋅u=0inΩ, where η is the viscosity, ρ the density, g the gravity vector, ε(⋅) denotes the symmetric gradient operator defined by ε(u)=12(∇u+∇uT), and Ω⊂Rd,d=2 or 3 is the domain of interest. Both the viscosity η and the density ρ will, in general, be spatially variable; in applications, this is often through nonlinear dependencies on the strain rate ε(u) or the pressure, but the exact reasons for the spatial variability are not of importance to us here: what matters is that these coefficients may vary strongly and on short length scales.

In applications, the equations above will be augmented by appropriate boundary conditions and will be coupled to additional and often time-dependent equations, such as ones that describe the evolution of the temperature field or of the composition of rocks (see, for example, ). This coupling is also not of interest to us here.

Before delving into the details of numerical experiments, let us consider one other theoretical aspect. An interesting complication of geodynamics simulations compared to many other applications of the Stokes equations is that the hydrostatic component of the pressure is often vastly larger than the dynamic pressure, even though only the dynamic component is responsible for driving the flow. As we will discuss in the following, this has no importance when using the Q1×P0 or the Taylor–Hood elements, but it turns out to be rather inconvenient when using a stabilized formulation that contains an artificial compressibility term. This issue is also mentioned in the quote from reproduced in the Introduction and in .

To illustrate the issue, consider the force balance equation (Eq. ). We can split the pressure into hydrostatic and dynamic components, p=ps+pd, where we define the hydrostatic pressure via the relationship 11∂∂zps=ρref(z)gz(z), coupled with the normalization that ps=0 at the top of the domain. In defining ps this way, we have made the assumption that the vertical component gz of the gravity vector dominates its other components. Furthermore, we have introduced a reference density ρref that somehow reflects a depth-dependent profile. As we will discuss below, there is really no unique or accepted way to define this profile, though one should generally think of it as capturing the bulk of the three-dimensional variation in the density via a one-dimensional function.

By splitting the pressure in this way, Eq. () can then be rewritten as follows: -∇⋅2ηϵ(u)+∇pd=ρg-ρrefgzezinΩ. Since this is the only equation in which the pressure appears, it is obvious that the velocity field so computed is the same whether or not one uses the original formulation solving for u and p or the one solving for u and pd. More concisely, the observation shows that the velocity field so computed does not depend on how one chooses the reference density ρref. The original formulation is recovered by using the simplest choice, ρref=0. As a consequence, many geodynamics codes use formulations that only compute the dynamic pressure pd using a reference density ρref(z). Importantly, however, there is no canonical way for this definition: one might choose a constant reference density, a depth-dependent adiabatic profile, or one computed at each time step by laterally averaging the current three-dimensional density field ρ(x,y,z,t); each of these options – and likely more – have been used in numerical simulations one can find in the literature. In any case, pressure-dependent coefficients such as the density or viscosity are then evaluated by using ps+pd, where pd is computed as part of the solution of the Stokes problem and ps is the hydrostatic pressure defined by Eq. () using the particular choice of reference density used by a code. On the other hand, the Aspect code notably always computes the full pressure instead of splitting it in hydrostatic and dynamic components (see the discussion in ) corresponding to the particular choice ρref=0.

The problem with the stabilized Q1×Q1 formulation – different from the use of the other element choices – is that the velocity field computed from the Stokes solution is not independent of the choice of the reference density. This is because the mass conservation equation is modified by the stabilization term and – in the simple case of a constant viscosity – reads 12-∇⋅u-1ηΠpd=0. Here, Π=(I-π0) is the operator that corresponds to the stabilization term in Eq. ().

To arrive at this form for the operator, one needs to rewrite Eq. () using (I-π0)qh,1η(I-π0)ph=qh,1η(I-π0)∗(I-π0)ph, where the asterisk denotes the adjoint operator. One then shows (I-π0)∗=(I-π0) and finally that Π=(I-π0)2=I-π0, which follows by recalling that projection operators are idempotent.

The point of these considerations is that different choices of ρref (including the choice ρref=0 that leads to the original formulation) do have an effect here because they lead to different pd=p-ps for which Πpd is different: that is, the amount of artificial compressibility depends on the splitting of the pressure into static and dynamic pressures. In other words, the discretization errors ‖u-uh‖L2 and ‖∇(u-uh)‖L2 discussed in the previous section will in general depend on the choice of the reference density profile, and the latter will need to be carefully defined in order to lead to acceptable error levels. As we will show in the benchmarking section, the specific choice of ρref in fact has a rather large effect. This is in line with the previously quoted comments in .

Let us end this section by commenting on two aspects of why this issue may not be as relevant in other contexts in which stabilized formulations have been used. First, in many important applications of the Stokes equations, the flow is not driven by buoyancy effects but by inflow and outflow boundary conditions e.g.,. Indeed, in those conditions both the density and the gravity vector are generally considered spatially constant, and the choice of reference density and hydrostatic pressure is then obvious and unambiguous. In these cases, computations are always performed with only the dynamic pressure because the hydrostatic pressure does not enter the problem at all except in the rare cases of fluids with pressure-dependent viscosities.

Second, while we have here considered the stabilization first introduced in , earlier stabilized formulations used a pressure Laplacian in place of the operator Π above. (See, for example, , or the variation in , as well as the analysis in .) That is, instead of Eq. () they used a formulation of the form 13-∇⋅u-ch2Δp=0, where c is a tuning parameter that also incorporates the viscosity. If one uses this formulation for cases in which the reference density is chosen as a function that is constant in depth – as was often done in earlier mantle convection codes considering the Boussinesq approximation – and if one computes in a Cartesian box with a constant gravity vector g=gez, then ps is a linear function, and consequently Δps=0. In other words, Δp=Δ(p-ps)=Δpd, which implies that the computed velocity field again did not depend on the exact choice of ρref as long as it was chosen constant. This property does not hold for the formulation of Dohrmann and Bochev because Πp≠Π(p-ps)=Πpd for linear pressures ps because Πps≠0: Π subtracts from ps the average value on each cell, leaving a piecewise linear discontinuous function.

Of course, whether one uses the Dohrmann–Bochev formulation (Eq. ) or the addition of a pressure Laplace as in Eq. (), the formulation is consistent. That is, as the mesh size h goes to zero, the added stabilization term also goes to zero. In the limit, the numerical solution therefore satisfies the original mass conservation equation. In other words, the limit is independent of the choice of ρref, even though the solutions on a finite mesh are not.

In this section, let us present computational results for three analytical problems and a buoyancy-driven flow community benchmark. While the first of these (Sect. ) is simply used to establish the best convergence rates one can hope for in the case of smooth solutions, the remaining test cases were chosen because they illustrate aspects of what we think “typical” solutions of geodynamic applications look like in an abstracted, controlled way. In particular, the “SolCx” benchmark in Sect.  demonstrates features of solutions in which the mesh can be aligned with sharp features in the viscosity, and the “SolVi” benchmark in Sect.  does so in the more common case in which the mesh cannot be aligned. Finally, the “sinking block” case in Sect.  shows a buoyancy-driven situation in which all of the discussions of the previous section on the choice of a reference density will come into play. All of these cases are simple enough that we know (quantitative or qualitative features of) the solution to sufficient accuracy to investigate convergence rigorously.

While these benchmarks provide us with insight that allows us to conjecture which elements may or may not work in practical application, they still are just abstract benchmarks. As a consequence, we will consider an actual geodynamic application in Sect. .

All models are run with the Aspect code. We have limited ourselves to two-dimensional cases as we do not expect that three-dimensional models would shed any more light on the conclusions reached. Although Aspect is built for adaptive mesh refinement (AMR), we have chosen not to use this feature in order to reflect the fact that the majority of existing codes use structured meshes.

While the previous sections have built our intuition for which element may actually work in the context of geodynamics applications, they have only done so through abstract and idealized benchmarks. It is therefore interesting to investigate what one would find in more realistic setups, and consequently we have also investigated convergence for a situation still sufficiently simple that numerical simulations can reach reasonably high accuracy but that has more of the complexity one would generally find in “real” simulations. Given that the previous examples have highlighted the fact that the stabilized Q1×Q1 element has difficulties with the pressure approximation, we are specifically interested in a situation in which the material behavior is pressure-dependent.

To this end, we consider an example of continental extension here. The setup is similar to ones that can be found in , , , and , and we specifically use the one that can be found in the “continental extension” cookbook of the manual of the Aspect code . The situation we model here is characterized by the following building blocks: on a domain of size 400 km × 100 km, we impose an extensional horizontal velocity component of ±0.25 cm yr-1 on the sides and a vertical upward velocity of 0.125 cm yr-1 at the bottom. The tangential components are left free. At the top, we allow for a free boundary. More interestingly, we use a pressure- and temperature-dependent viscoplastic rheology of Drucker–Prager type with parameters for viscous deformation based on dislocation creep flow laws: 15ηdisl=A-1/nε˙-1+1/nexp⁡Q+pVnRT, where A is a material constant, n is an index typically between 3 and 4, Q is the activation energy, V is the activation volume, R the gas constant, T the temperature, and ε˙ is the effective strain rate (the square root of the second invariant of the corresponding tensor). Stresses are limited plastically at a yield stress σy=Ccos⁡(ϕ)+Psin⁡(ϕ) via a Drucker–Prager criterion where C is the cohesion and ϕ the angle of friction. We use distinct values for some of these parameters in the initially 20 km thick upper crust (wet quartzite), an initially 10 km thick lower crust (wet anorthite), and the mantle (dry olivine), which initially occupies the remaining 70 km in depth. Deformation is seeded by a weak area within the mantle lithosphere. We only carry out a single time step as obtained with a CFL number of 0.5.

A complete and concise description of this setup has more parameters than are worth spelling out in detail here. For a detailed description, see and the section of the Aspect manual along with the corresponding input files. For the purposes of this paper, the important part is that both the yield stress and the dislocation creep rheology depend on the pressure; as a consequence, we can anticipate that elements that result in poor pressure accuracy may not yield accurate simulations in general.

This setup produces localized shear zones that accommodate the majority of the deformation. Figure  illustrates the structure of the resulting solution. Each panel of the figure shows in its left half the solution produced by the stabilized Q1×Q1 element and its right half that produced by the Taylor–Hood Q2×Q1 element. Because the solution is symmetric, the two halves should be mirror images. It is, however, clear from several of the panels that this is not the case: the Q1×Q1 element produces large artifacts at depth where the pressure is large and the pressure dependence of the material strong.

This effect is also demonstrated in a different way in Fig.  where we show laterally averaged quantities for the different elements and different mesh resolutions. Even though it is clear from Fig.  that lateral averaging should result in a better approximation (than pointwise evaluations) of the correct quantities for a given depth, Fig.  shows that even the average is far from correct. On the other hand, the figure shows that with increasing mesh resolution, the solutions produced by the Q1×Q1 seem to converge to the solutions generated by the other elements – albeit very slowly and at what one might consider an unacceptable cost.

To investigate the origin of these convergence problems of the Q1×Q1 element, one should recall that the model is nonlinear. As a consequence, the artifacts may be related to the discretization or to a failure of the nonlinear iteration – and the two may be connected. All of the solutions we show were taken after 100 Picard iterations to resolve the nonlinearity of the model, with nonlinear convergence shown in Fig. . (One could accelerate convergence by using a Newton solver – – but this is not relevant for the work herein.) Looking at the evolution of the nonlinear residual during these iterations, we see that it decreases quickly and for most element choices then plateaus at about 10-5 relative to the starting residual. In contrast, for the stabilized Q1×Q1 element, increasing the mesh resolution yields lower nonlinear residuals – but even on the finest mesh, the nonlinear residuals are still substantially worse than for any of the other elements, with no apparent progress after about 20 iterations. Of course, we are not the first to observe that convergence is hard to come by for these sorts of problems (see, for example, ), and recent approaches to regularize visco(–elasto)–plastic deformation by and have been found to improve the convergence behavior of the nonlinear solvers.

Our interpretation of this experiment is that the inability of the Q1×Q1 element to generate accurate pressure fields leads to values for the pressure-dependent rheology that are so far away from their correct values – and, indeed, from the values on nearby cells – that they greatly increase the condition number of the linear systems that have to be solved in each nonlinear iteration. The resulting difficulty of solving these Picard steps accurately then affects the speed with which the nonlinear residual is reduced by the Picard iteration to the point at which the condition number is so large that convergence can no longer be achieved. Only mesh refinement, with the attendant increased accuracy of the pressure solution (and, consequently, a more accurate viscosity), helps to restore the ability to actually solve this problem to small nonlinear residuals.

In this contribution, we have provided a side-by-side comparison of the most widely used quadrilateral finite elements. As outlined in the Introduction, most finite-element solvers used in the geodynamics community rely on one or the other of these. At the same time, we are not aware of a comprehensive comparison of their relative strengths – or their weaknesses, as they may be.

Using the artificial linear benchmarks discussed in Sect. , we can infer that when the solution is smooth, the Taylor–Hood variations Q2×Q1 and Q2×P-1 provide far better accuracy than the lower-order elements Q1×P0 and the stabilized Q1×Q1. This advantage is largely lost when one considers problems in which the viscosity is discontinuous. Since we believe that the real Earth has relatively narrow phase transition zones where the viscosity may jump by large factors, benchmarks like the SolVi one in Sect.  are relevant and illuminate important aspects.

From these considerations, one may conclude that the Taylor–Hood variations are too expensive – in terms of their number of degrees of freedom and the attendant memory and CPU time cost. However, we believe that this is not so.

For buoyancy-driven flows such as the sinking block benchmark in Sect. , the stabilized Q1×Q1 element is largely unable to reproduce the correct solution and, furthermore, depends on using a formulation in which one subtracts a reference density from the actual density; this is equivalent to defining a hydrostatic pressure profile and only attempting to solve for the “dynamic” component of the pressure. Crucially, however, there are many ways of defining such a reference density, neither of which is canonical and “obviously right” in complex mantle convection simulations. Since the solution obtained with the stabilized Q1×Q1 element strongly depends on the specific choice of reference density, we conclude that the element cannot be made robust for the kinds of flows we encounter in real mantle convection situations. We have also verified this assertion using an application in which we consider continental extension (Sect. ) and in which the inability to produce accurate pressure solutions also greatly affects the convergence of the nonlinear solver to the point at which the computed solution must be considered unusable. We have shown that these errors can be reduced when choosing very fine meshes, but the attendant cost is unacceptable when compared with that of using other elements on far coarser meshes.

There are other considerations to believing that the procedure of trying to subtract a reference density (or a hydrostatic pressure) cannot be a successful strategy. For example, simulations of free or deformable surfaces (at the Earth's surface as well as at the core–mantle boundary) require accurate knowledge of the total pressure. This is true for coupled formulations of flow and surface deformation as well as approaches such as the “sticky air” method . But similar considerations also apply to nonlinear material laws in which the pressure enters the viscosity or, more commonly, phase computations that determine the density and other thermodynamic material properties from the pressure and the temperature. Indeed, one could conjecture that the stabilized Q1×Q1 element would also fail for compressible Stokes simulations, though we have not verified this here.

We conclude from these thoughts that the stabilized Q1×Q1 element is not a viable choice for mantle convection simulations. It is important to point out that the cases we consider to be crucial here – buoyancy-driven flows, large hydrostatic pressures, and pressure-dependent rheologies – are uncommon in most of the engineering applications for which the Q1×Q1 was originally developed; as a consequence, it is not surprising that what we find here contradicts substantial parts of the engineering literature wherein the element remains widely used.

In summary, we think that the Taylor–Hood variations Q2×Q1 and Q2×P-1 present the best compromise for robust mantle convection and crustal dynamics simulations based on the finite-element method. This is not because these elements are “obviously better” than the others but due more to a “last man standing” argument: the other choices simply disqualified themselves by failing to provide adequate accuracy in one situation or another. At the same time, the lack of regularity one expects of typical scenarios also implies that we should not expect higher-order Taylor–Hood elements Qk+1×Qk or Qk+1×P-k with k>2 to provide substantially better accuracy compared to their much higher computational cost. Although we have only shown results for two-dimensional simulations, experience – including the experience with the Aspect code used here that solves two- and three-dimensional problems within the same framework – suggests that all of these considerations would also apply to the three-dimensional (hexahedral) analogs of the ones we have used.

The experiments we have shown do not provide clear guidance on whether one should use the Q2×Q1 or Q2×P-1 element. But other considerations can provide such guidance. Most notably, the elements with discontinuous pressure elements (namely, the Q2×P-1 but also the Q1×P0 elements) have the “local conservation” property for which the velocity satisfies ∫K∇⋅uh=∫∂Kn⋅uh=0 on every cell K of the mesh, a property also satisfied by the exact solution. Local conservation is useful when considering that the velocity computed in geodynamics models is often used in a second step to advect both the temperature field and chemical compositions (see, for example, ). A comprehensive investigation of the interplay of local conservation and transport can be found in .

Of course, the choices we have considered here are not the only ones. One could, for example, consider “simplicial” (triangular and tetrahedral) elements instead of the quadrilateral and hexahedral ones we have used here. Indeed, some existing mantle convection codes use this strategy. One successful example is the TERRA-NEO code that uses equal-order linear tetrahedra stabilized by means of a pressure-stabilization approach based on the addition of linear least-squares terms (the “PSPG” approach, see ); other examples include Fluidity , MILAMIN , and LaCoDe . While we have not evaluated simplicial elements, one might conjecture that many of the same conclusions would also hold: the unstable P1×P0 provides low accuracy and is unstable, the stabilized P1×P1 has difficulties with buoyancy-driven flows and large hydrostatic pressures, and the Taylor–Hood element P2×P1 is expensive but robust.

Finally, there are other more exotic elements one could work with. Examples include the Rannacher–Turek element , the Crouzeix–Raviart element , or the DSSY element . We have not investigated these kinds of choices for four reasons: (i) the paper at hand is long enough as it stands; (ii) these elements are not widely used, both within and outside our community; (iii) many of these elements are difficult to implement in one regard or another, including complications with boundary conditions and with dealing with unstructured and possibly curvilinear cells; and finally, (iv) the elements mentioned above are not as widely available or completely implemented in common software frameworks, and their use thus requires substantial additional implementation work.

While we have not investigated these two possible directions for alternatives to the elements we have considered, we think that such studies would be interesting. We hope that our careful choice of test cases might also be useful to such studies.