Geodynamical simulations over the past decades have widely been
built on quadrilateral and hexahedral finite elements. For the
discretization of the key Stokes equation describing slow, viscous
flow, most codes use either the unstable

Herein, we provide a systematic comparison of all of these
elements for the first time. We use a series of benchmarks that illuminate different
aspects of the features we consider typical of mantle convection
and geodynamical simulations. We will show in particular that the stabilized

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

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

The more modern alternative to this choice is the Taylor–Hood element that uses
(continuous) polynomials of degree

Strictly speaking,

The discontinuous space

A third option is the use of

The availability of all of these options leads us to the main question of this paper: which
element should one use in geodynamics computations based on the Stokes
equations? Or, in the absence of clear-cut conclusions, which ones should

Such considerations put into question whether higher-order methods are really worth the effort
for actual geodynamics simulations. Given these divergent theoretical thoughts, the only way to resolve
the question is by way of numerical comparisons. We have consequently extended

Quantitatively compare the solution accuracy of the various options (

Extend these numerical comparisons to cases in which it is known that the
stabilized

Conclude our considerations by comparing the available options using a realistic geodynamical application. This will allow us to draw conclusions as to what element one might want to recommend for geodynamics applications.

While we have approached this study with an open mind and without a strong prior idea of which
element might be the best, let us end this Introduction by noting that members
of the crustal dynamics and mantle convection communities have occasionally expressed
a dislike of the stabilized

All the models were run with the open source code Gale.
[…] Gale uses

We do not consider the

We are not aware of any other significant publications in the
geodynamics literature that specifically discuss the relative
trade-offs between the elements we consider herein, specifically
between the

For the purpose of this paper, we are concerned with the accurate numerical solution of the
incompressible Stokes equations:

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,

For the comparisons we intend to make in this paper,
Eqs. (

On the other hand, if one chooses

Finally, if one uses

We end this section by noting that in many of the setups we use in
Sect.

A comparison of Eq. (

However, this is an incomplete understanding because the

For a concise definition of the Lebesgue space

Similar considerations apply for the

In other words, we will only benefit from the added expense of the Taylor–Hood
element with

We end this section by noting that all of the estimates shown above
guarantee that the error on the left of an inequality
decreases

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

To illustrate the issue, consider the force balance equation (Eq.

By splitting the pressure in this way, Eq. (

The problem with the stabilized

To arrive at this form for the operator, one needs to
rewrite Eq. (

The point of these considerations is that different choices of

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

Second, while we have here considered the stabilization first introduced in

Of course, whether one uses the Dohrmann–Bochev formulation
(Eq.

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.

While these benchmarks provide us with insight that allows us to

All models are run with the

Let us start our numerical experiments with the simple 2D benchmark
presented in

Donea and Huerta benchmark. Velocity

We verify this in Fig.

Figure

These results are not surprising: the solution
is smooth, and consequently one would expect to obtain optimal order convergence in all
cases. One can carry out similar experiments for the SolKz benchmark

Finally, we also investigate the cost associated with solving this
problem using the various elements.
Fig.

The concrete number of iterations of course depends on the
preconditioner used – here the one described in

Donea and Huerta benchmark. Error convergence as a function of the mesh size

Donea and Huerta benchmark.

The SolCx benchmark is a common benchmark found in many geodynamical
papers

We show the velocity and pressure fields in Fig.

As in the Donea and Huerta benchmark, we compute the velocity and pressure error convergence for
all four elements. Those are shown in Fig.

Figure

SolCx benchmark.
Velocity

SolCx benchmark. Error convergence as a function of the
mesh size

SolCx benchmark. Number of FGMRES solver iterations as a function of the mesh size

The SolCx benchmark in the previous section allows for aligning mesh interfaces with the discontinuity in the
viscosity. This is an artificial situation that will, in general, not
happen in actual large-scale
geodynamics applications for which the interfaces between materials may be at arbitrary
locations and orientations in the domain and may also move with
time. An example is the simulation
of a cold subducting slab (with correspondingly
large viscosity) surrounded by hot low-viscosity mantle material.
Consequently, it is worth considering a
situation in which it is impractical to align mesh and viscosity interfaces. This is
done by the SolVi inclusion benchmark, which solves
a problem with a viscosity that is discontinuous along a circle. This in turns leads to
a discontinuous pressure along the interface,
which is difficult to represent accurately. Using the regular meshes used by
a majority of codes, the discontinuity in the
viscosity and pressure then never aligns with cell boundaries. Even though

SolVi benchmark with inclusion of radius 0.2.
Velocity

A characteristic of the analytic solution is that the pressure is zero inside the inclusion, while outside it follows the relation

SolVi benchmark. Left to right:

Since harmonic averaging yields the lowest errors we select this averaging and now turn
to the pressure field for all elements as shown in Fig.

SolVi benchmark. Pressure field for the

SolVi benchmark. Pressure on the horizontal ray starting from the center of the inclusion at

As discussed in Sect.

It consists of a two-dimensional

In a geodynamical context, the block could be interpreted as a detached slab
(

In the following, we will denote as Method 1 the approach whereby we do calculations with the density
field as specified above.
Method 2 consists of a “reduced” density field from which the quantity

We have carried out measurements for all four elements with

When using the full density, we see that all elements, with the
exception of the stabilized

When reduced densities are used results are unchanged for the stable elements
(only

Sinking block benchmark.

In Fig.

These observations illustrate the unreliable nature of the results obtained with
stabilized

Sinking block benchmark with

Finally, in Fig.

Sinking block benchmark.
Normalized pressure

In addition to the slow convergence of the

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

To this end, we consider an example of continental extension here. The
setup is similar to ones that can be found in

A complete and concise description of this setup has more parameters than are
worth spelling out in detail here. For a detailed description,
see

This setup produces localized shear zones that accommodate the
majority of the deformation. Figure

Application example.

This effect is also demonstrated in a different way in
Fig.

Application example.

To investigate the origin of these convergence problems of the

Application example. Nonlinear residual as a function of nonlinear iteration step for all four elements and for different mesh resolutions.

Our interpretation of this experiment is that the inability of
the

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.

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.

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

We conclude from these thoughts that the stabilized

We believe that the

In summary, we think that the Taylor–Hood variations

The experiments we have shown do not provide clear guidance on whether
one should use the

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

Finally, there are other more exotic elements one could work
with. Examples include the Rannacher–Turek element

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.

CT conceived the study and ran all models. WB implemented
the stabilized

The contact author has declared that neither they nor their co-author has any competing interests.

Publisher's note: Copernicus Publications remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

The authors thank Dave May, Matthew Knepley, and an anonymous reviewer
for their comments that have helped improve the paper. We thank the Computational Infrastructure for Geodynamics
(

CIG is funded by the National Science Foundation under award EAR-1550901. Additional support was provided by the National Science Foundation through awards EAR-1550901 and OAC-1835673 as part of the Cyberinfrastructure for Sustained Scientific Innovation (CSSI) program, as well as EAR-1925595.

This paper was edited by Susanne Buiter and reviewed by Dave May, Matthew Knepley, and one anonymous referee.