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 Q1×P0 element, a
stabilized version of the equal-order Q1×Q1 element, or
more recently the stable Taylor–Hood element with continuous
(Q2×Q1) or discontinuous (Q2×P-1)
pressure. However, it is not clear which of these choices is
actually the best at accurately simulating “typical” geodynamic
situations.
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
Q1×Q1 element has great difficulty producing accurate
solutions for buoyancy-driven flows – the dominant forcing for
mantle convection flow – and that the Q1×P0 element is
too unstable and inaccurate in practice. As a consequence, we
believe that the Q2×Q1 and Q2×P-1 elements
provide the most robust and reliable choice for geodynamical simulations,
despite the greater complexity in their implementation and the
substantially higher computational cost when solving linear
systems.
Introduction
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.
This element is not only LBB-stable, but owing to its higher polynomial degree is
also convergent of higher order. It is therefore widely used in commercial flow solvers and
is also the default element for the Aspect code in geodynamics . This element
is obviously more difficult to implement, and building efficient solvers and preconditioners
is also more complicated .
However, these drawbacks can be mitigated by building
on one of the widely
available finite-element libraries that have appeared over the past 20 years; for example,
Aspect inherits all of its finite-element functionality from the deal.II library (see
). We will note that one can also use a number of variations of the underlying idea of the
Taylor–Hood element, for example on quadrilaterals and hexahedra by using Qk×P-(k-1)
(see, for instance, , , and )
in which the pressure is discontinuous and of (total) polynomial
degree k-1, but missing the part of the finite-element space
on every cell that distinguishes the space Qk on quadrilaterals and hexahedra
from the space Pk that is typically used on
triangles and tetrahedra.
The discontinuous space P-(k-1) for the
pressure can be interpreted in two incompatible ways: first, one can
map the corresponding space from the reference cell to each of the
cells of the mesh, as one also does for the velocity; or, one can
define shape functions directly in the
global coordinate system, without mapping from the reference cell. The
two agree on cells that are parallelograms but not on more general
meshes. Since our experiments are all on meshes where all cells are
rectangles, the distinction does not matter for the current paper,
but we point out that the error
estimates (Eq. ) stated in
Sect. only hold for the latter
definition. See , , and
Sect. 3.6.4 for more information.
Another variation is to enrich the pressure space by a constant shape
function on each cell (see, for example, , and the
references therein). All of these alternatives are stable for k≥2, and in keeping with common usage of the term, we will also refer
to all of these variations as Taylor–Hood or Taylor–Hood-like elements
even though they are strictly speaking not what Taylor and Hood
proposed in .
A third option is the use of Q1×Q1 elements with both
velocity and pressure using bilinear or trilinear shape functions. This
combination of elements is not LBB-stable by default, but
numerous stabilization techniques – typically adding a pressure-dependent term to the mass
conservation equation – have been proposed in the
literature see, e.g.,.
Herein, we will discuss in particular the variation by
that is simple to implement and does not involve any tunable parameter.
This approach is used in the Rhea code in conjunction with adaptive mesh
refinement (AMR), allowing for the numerical solution of whole Earth models at high resolutions
. Another example of the use of this method is the work of , also
using AMR, to study thermochemical mantle convection.
Both the ELEFANT code with an application to the 3D thermal state of
curved subduction zones and the GALE code , with application
to the 3D shapes of metamorphic core complexes or oceanic
plateau subduction , use the stabilized Q1×Q1 method.
Finally the ADELI code was coupled to a stabilized Q1×Q1 flow solver in
the context of lithosphere–asthenosphere interaction
studies .
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 not be used?
On the face of
it, this seems like a simple question: the consensus in the computational science community is
that using moderately high-degree elements (say, k=3 or k=4) yields the best accuracy for
a given computational effort (measured in CPU cycles) unless one wants to change the solver
technology to use matrix-free methods whereby even higher polynomial degrees become
more efficient. This conclusion is based on the higher convergence order
of higher-degree methods but balanced by the rapidly growing cost of matrix assembly and
linear solver effort for higher-degree methods. On the other hand, the recommendation to use
higher-degree methods is predicated on the assumption that the solution is smooth enough –
say, the velocity is in the Sobolev space Hk+1 of functions that
have, loosely speaking, at least k+1 derivatives – that one can actually achieve a convergence
rate of O(hk) in the energy norm and O(hk+1) in
the L2 norm, where h is the mesh size. This
assumption generally requires that all coefficients, such as density and viscosity, are sufficiently
smooth on length scales resolvable by the mesh. This may not be the case in realistic
geodynamics problems given that density and viscosity often depend discontinuously on the
solution variables (velocity or strain rate, pressure, temperature, and compositional variables);
indeed, in many models, the viscosity may vary by orders of magnitude on short length scales.
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 Aspect
so that it can use all of the element combinations above, and we will use these implementations
in the comparisons in this paper.
Goals of this paper. Having outlined the conflict between the expected superiority
of higher-degree elements for the Stokes equation on the one hand and the expected lack of
smoothness of solutions in realistic geodynamic cases, our goals in the paper are as follows.
Quantitatively compare the solution accuracy of the various options (Q1×P0,
Qk×Qk-1, Qk×P-(k-1) and stabilized Q1×Q1) using a variety of analytical
benchmarks for which the exact solution is known. As we will see
below, there is little point working with k>2 in geodynamics
applications, and so the only
cases we consider for Taylor–Hood-like elements are Q2×Q1 and Q2×P-1.
Extend these numerical comparisons to cases in which it is known that the
stabilized Q1×Q1 demonstrates problematic behavior that may make it
unusable in many practical situations. In particular, we will consider the case
of buoyancy-driven flows.
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 Q1×Q1 element for its inability to deal with large
lithostatic pressures and free surfaces absent special modifications of the formulation.
For example, comment on the need to modify the physical description of the problem
due to the stabilization (with references replaced by ones listed at the end of this paper).
All the models were run with the open source code Gale.
[…] Gale uses Q1–Q1 elements to describe the pressure and the velocity.
However, this formulation is unstable and a slight compressible term
is added in the divergence equation to stabilize it .
Ideally, this term should be applied on the dynamic
pressure and not on the full pressure. To fix this, a hydrostatic term
corresponding to the reference density and temperature profile, is
subtracted from the full pressure and the body force vector.
Few other negative comments concerning the Q1×Q1 element
appear on record in the published literature, although one can find the following quote in .
We do not consider the Q1×Q1/stab element , as stabilization of this element is achieved by introducing an artificial compressibility that dominates for flows mainly driven by buoyancy variations . In geophysical flow models this yields unphysical pressure artifacts for cases where both the free surface of the Earth and mantle flow are considered, because the driving density contrast between cold sinking plates and the warmer surrounding Earth's mantle is much smaller than the density difference between rocks and air . In our experience, this results in artificial “compaction” of the Earth’s mantle if Q1×Q1/stab element is used, which makes them unsuitable for these purposes.
Indeed, our numerical experiments will encounter a similar issue; see
Sect. .
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 Q1×P0 and Taylor–Hood elements, and consequently
believe that our discussions here are useful for the community.
The governing equations
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.
Discretization using finite-element methodsFormulation and basic error estimates
For the comparisons we intend to make in this paper,
Eqs. ()–() are discretized using the
finite-element method. A straightforward application of the Galerkin method yields
the following finite-dimensional variational problem:
find uh∈Uh,ph∈Ph
so that
ε(vh),2ηε(uh)-(∇⋅vh,ph)=(vh,ρg),-(qh,∇⋅uh)=0,
for all test functions vh∈Uh,qh∈Ph.
Here, (a,b)=∫Ωa(x)b(x)dx.
For simplicity, we have omitted terms introduced through the treatment
of boundary conditions. The finite-dimensional, piecewise polynomial spaces Uh and
Ph can be chosen in a variety of ways, as discussed in the Introduction. In
particular, if they are chosen as Uh=Qk and Ph=Qk-1 – i.e., the Taylor–Hood
element – then the discrete problem is known to satisfy the LBB condition and
the solution is stable .
Here, Qs is the space of continuous functions that are
obtained on each cell K of a mesh T by mapping polynomials of degree at
most s in each variable from the reference cell [0,1]d.
Likewise, the problem is stable if one chooses Uh=Qk and Ph=P-(k-1),
where now P-s is the space of discontinuous functions obtained by mapping
polynomials of total degree at most s from the reference cell. In both of
these cases, we expect from fundamental theorems of the finite-element
method (see, for example, ) that the convergence rates are
optimal, i.e., that the errors satisfy the relationships
‖∇(u-uh)‖L2=O(hk),‖u-uh‖L2=O(hk+1),‖p-ph‖L2=O(hk),
where h is the maximal diameter over all cells in the mesh T.
On the other hand, if one chooses Uh=Q1 and Ph=P0, i.e., the unstable
Q1×P0 element with piecewise linear continuous velocities
and piecewise constant discontinuous pressure, then the best
convergence rates one can hope for would satisfy the following
relationships based solely on interpolation error estimates:
‖∇(u-uh)‖L2=O(h),‖u-uh‖L2=O(h2),‖p-ph‖L2=O(h).
In practice, if the numerical solution shows pressure oscillations
see for instance, one will not even observe
the rates shown above but might in fact obtain a worse pressure
convergence rate, for example ‖p-ph‖L2=O(h1/2).
Finally, if one uses Uh=Q1 and Ph=Q1, then this unstable element
combination can be made stable if one replaces the discrete formulation () by the following stabilized version
due to :
ε(vh),2ηε(uh)-(∇⋅vh,ph)=(vh,ρg),(qh,∇⋅uh)-(I-π0)qh,1η(I-π0)ph=0.
Here, I is the identity operator and π0 is the projection onto piecewise constant
functions – i.e., π0f is the function that on
each cell is equal to the mean value of f on that cell. For this element, the rates one might hope
for are as follows (see again ):
‖∇(u-uh)‖L2=O(h),‖u-uh‖L2=O(h2),‖p-ph‖L2=O(h). report that for some test cases, one might in fact obtain
‖p-ph‖L2=O(ht) with t≈1.5, though it is not clear
whether this rate can be obtained for all possible applications.
We also observe
this improved rate in one of our benchmarks in Sect. .
We end this section by noting that in many of the setups we use in
Sect. , the boundary conditions we impose lead
to a problem in which the pressure is only determined up to an
additive constant. The same is then true for the linear system one
has to solve after discretization. As a consequence, we can only
meaningfully compute quantities such as ‖p-ph‖L2 if both the exact
and the numerical solution are normalized; a typical
normalization is to ensure that their mean values are zero. Aspect
enforces this normalization after solving the linear system.
A closer look at the error estimates
A comparison of Eq. () with Eqs. () and () would suggest that the Taylor–Hood element can obtain
substantially better rates of convergence if one only chooses the polynomial degree k
large enough.
However, this is an incomplete understanding because the O(hm) notation hides the fact that
the constants in this behavior depend on the solution. More specifically, a complete
description of the error behavior would replace Eq. () by the following
statement: there are constants C1,C2,C3<∞
so that
‖∇(u-uh)‖L2≤C1hk‖∇k+1u‖L2,‖u-uh‖L2≤C2hk+1‖∇k+1u‖L2,‖p-ph‖L2≤C3hk‖∇kp‖L2.
The validity of these statements clearly depends on the solution being regular
enough so that ∇k+1u and ∇kp actually exist and are square-integrable – in other words, that u∈Hk+1 and p∈Hk, where Hk represents the usual Sobolev function spaces.
For a concise definition of the Lebesgue space L2 and the
Sobolev spaces of functions Hk, see . Loosely
speaking, L2 is the set of all functions f for which the integral
of the square over the domain, ∫Ω|f(x)|2dx, is finite. We say that such
functions are “square-integrable”. Hk is the set of all
functions whose kth (weak) derivatives are square-integrable.
On the other
hand, all that is guaranteed by the existence theory for partial differential equations
is that u∈H1 and p∈L2=H0; any further smoothness should only
be expected if, for example, the domain Ω is convex and if viscosity η
and right-hand side ρg are also smooth. Indeed, this is
the case for many artificial benchmarks for which these functions are chosen a priori; on the
other hand, in “realistic” geodynamics applications, one might expect η and
ρ to be discontinuous at phase boundaries and potentially vary widely. In
such cases, one needs to accept that the solutions only satisfy
u∈Hq and p∈Hq-1 with q≥1 but possibly q<k+1. Numerical analysis predicts
that in such cases, the best-case rates in Eq. () will be
replaced by the following:
‖∇(u-uh)‖L2≤C1hmin{q-1,k}‖∇min{q,k+1}u‖L2,‖u-uh‖L2≤C2hmin{q,k+1}‖∇min{q,k+1}u‖L2,‖p-ph‖L2≤C3hmin{q-1,k}‖∇min{q-1,k}p‖L2.
Similar considerations apply for the Q1×P0 and the stabilized Q1×Q1
combinations; a closer examination yields the following rates that would
replace Eqs. () and ():
‖∇(u-uh)‖L2≤C1hmin{q-1,1}‖∇min{q,2}u‖L2,‖u-uh‖L2≤C2hmin{q,2}‖∇min{q,2}u‖L2,‖p-ph‖L2≤C3hmin{q-1,1}‖∇min{q-1,1}p‖L2.
In other words, we will only benefit from the added expense of the Taylor–Hood
element with k≥2 if the solution is sufficiently smooth, namely
if at least q>k≥2.
The question of whether q>2 indeed for a given situation is one of partial differential equation (PDE) theory and difficult to answer in general
without using particular knowledge of η, ρg, and Ω. On
the other hand, one can observe convergence rates experimentally for a number
of cases of interest, so in some sense, it would be legitimate
to ask the following question:
what is the regularity index q of typical solutions in geodynamics
applications? At the same time, this requires careful convergence
studies on problems that are already typically quite challenging to solve
on any reasonable mesh, let alone several further refined
ones. As a consequence, we cannot answer this question in the
generality stated above. Instead, we will
approach it below by considering a number of benchmarks that illustrate
typical features of geodynamic settings in an abstracted way (in Sect. ),
followed by a model application (in Sect. ). In
particular, the examples in Sect. and
will illustrate cases in which the exact solution is not
smooth enough to achieve the optimal convergence rate.
We end this section by noting that all of the estimates shown above
guarantee that the error on the left of an inequality
decreases at least at the rate shown on the right side, but
they do not state that on a given sequence of meshes, the rate might
not in fact be better. Indeed, this
often happens: for example, if one aligns meshes with a
discontinuity in coefficients (as we do for the SolCx benchmark
discussed in Sect. ), one often observes optimal
rates – or convergence rates between the minimal theoretically
guaranteed and the optimal ones –
for some elements even if the solution lacks
regularity. Actually observing the minimal theoretically guaranteed
convergence rate for solutions lacking regularity often
requires choosing randomly arranged meshes – a case we will not
consider herein.
Comments about the use of the Q1×Q1 element in geodynamics computations
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
∂∂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 densityρref 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
-∇⋅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
-∇⋅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.
Numerical results for artificial benchmarks
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.
The Donea and Huerta benchmark
Let us start our numerical experiments with the simple 2D benchmark
presented in . The exact definition involves lengthy
formulas not worth repeating here, but in short it consists of the
following ingredients: (i) the domain is a unit square, (ii) the
viscosity and density are set to 1, and (iii) velocity and
pressure fields are chosen to correspond to smooth polynomials describing
circular flow with no-slip boundary conditions. We then choose an
(unphysical) gravity vector field that produces these velocity and
pressure fields. This setup produces the smooth solution shown in
Fig. for which we would expect that the
higher-order Taylor–Hood element is highly accurate.
Donea and Huerta benchmark. Velocity (a) and pressure (b) fields
obtained on a 32×32 mesh with Q2×Q1 elements.
We verify this in Fig.
for the four element choices of interest in this work:
Q1×P0, stabilized Q1×Q1, Q2×Q1, and Q2×P-1.
Looking at the velocity error, we recover a cubic convergence rate (q=3) for the
Q2×Q1 and Q2×P-1 elements and a quadratic convergence rate (q=2) for those choices using the Q1 elements for the velocity. The pressure error is of linear rate for the Q1×P0 element and of quadratic rate for the Q2×Q1
and Q2×P-1 elements. All of these are as expected. For the stabilized Q1×Q1, we
obtain the better-than-expected rate of 1.5 already mentioned in ; see also
Sect. .
Figure shows the root mean square velocity as a function of the mesh
size as obtained with the four elements in question. Again, the
second-order elements are more accurate.
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
, which also has a smooth solution; we have obtained
identical error convergence rates.
Finally, we also investigate the cost associated with solving this
problem using the various elements.
Fig. shows the number of outer FGMRES iterations
iterations of the Stokes solver as a function of the mesh size.
The concrete number of iterations of course depends on the
preconditioner used – here the one described in . The
important point of the figure, however, is how the number of
iterations changes (or does not) with the mesh size h.
This
number is nearly constant with increasing resolution for the stable or stabilized elements,
while it becomes exceedingly large for the unstable Q1×P0
element, reflecting the fact that lack of LBB stability corresponds to
the smallest eigenvalue of the
system matrix tending to zero – and thereby driving the condition
number to infinity. Indeed, our linear solver does not converge in the
1000 iterations we chose as a limit for the smallest mesh sizes.
Donea and Huerta benchmark. Error convergence as a function of the mesh size h.
(a) Velocity error ||u-uh||L2. (b) Pressure error
||p-ph||L2. The two leftmost points are missing for Q1×P0
since the solver failed to converge; the data points for Q2×Q1 and Q2×P-1 are on top of each other.
Donea and Huerta benchmark. (a) Root mean square velocity as a function of the
mesh size h. The dotted line is the analytical value. (b) Number of FGMRES solver iterations as a function of the mesh size h.
The SolCx benchmark
The SolCx benchmark is a common benchmark found in many geodynamical
papers e.g.,.
It uses a discontinuous viscosity profile with a large jump in the viscosity value along
the middle of the domain, resulting in a discontinuous pressure field.
The domain is a unit square, boundary conditions are free-slip on all edges,
and the gravity vector points downwards with |g|=1.
The density for SolCx is given by ρ(x,y)=sin(πy)cos(πx) and the viscosity field is
such that
η(x,y)=1,if0≤x≤0.5106if0.5<x≤1.
We show the velocity and pressure fields in Fig. . The discontinuous jump of
the viscosity field by a factor of 106 results in separate convective cells on the left
and right sides of the domain, though with vastly different strengths. The pressure also reflects
this disjoint behavior.
As in the Donea and Huerta benchmark, we compute the velocity and pressure error convergence for
all four elements. Those are shown in Fig. .
As documented in , the second-order element with discontinuous pressure Q2×P-1
performs better (pressure error convergence is O(h2)) than its continuous pressure
counterpart Q2×Q1 (convergence is only O(h1/2), but the
better convergence order with the discontinuous pressure can only be obtained if the
discontinuity in the viscosity is aligned with cell boundaries – which is the case here.
Also of interest here is the fact that the Q1×P0 outperforms the Q1×Q1
element for both velocity and pressure. All of these observations are readily explained by
the fact that a discontinuous pressure can only be approximated well when using discontinuous
pressure elements with cell interfaces aligned with the discontinuity in the viscosity.
Figure shows the number of outer FGMRES iterations of the Stokes solver as a function of the mesh size.
We find this time that this number is nearly constant with increasing resolution for all four elements. Unsurprisingly the
Q1×P0 element requires more iterations than all the others but by less than a factor of 2. The quadratic elements
require the same number of iterations, while the stabilized Q1×Q1 requires only half their number:
this is surprising, but the conclusions from the previous paragraph remain about it being the least accurate of all
four elements here.
SolCx benchmark.
Velocity (a) and pressure (b) fields
obtained on a mesh with a resolution of 32×32
grid with the Q2×Q1 element.
SolCx benchmark. Error convergence as a function of the
mesh size h. (a) Velocity error; (b) pressure error.
SolCx benchmark. Number of FGMRES solver iterations as a function of the mesh size h.
The SolVi (circular inclusion) benchmark
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 Aspect can use
arbitrary unstructured meshes (and can also use curved cell edges),
we will honor the setup of this benchmark by only considering
regular meshes.
derived a simple analytical solution for the pressure and velocity fields for such a circular
inclusion under pure shear, and this benchmark is showcased in many
publications .
The velocity and pressure fields are shown in Fig. .
SolVi benchmark with inclusion of radius 0.2.
Velocity (a) and pressure (b) fields
obtained on a 256×256 mesh using Q2×Q1 elements.
A characteristic of the analytic solution is that the pressure is zero inside the inclusion, while outside it follows the relation
p=4ϵ˙ηm(ηi-ηm)ηi+ηmri2r2cos(2θ),
where ηi=103 is the viscosity of the inclusion, ηm=1
is the viscosity of the background medium, r=x2+y2,
θ=arctan(y/x), and ϵ˙=1 is the applied strain
rate if one were to extend the domain to infinity. The formula above
makes it clear that the pressure is discontinuous along the perimeter
of the disk, with the jump largest at
θ=0,±π2,π.
thoroughly investigated this problem with various
numerical methods (FEM, FDM), with and without tracers, and
conclusively showed how various schemes of averaging the density and viscosity
lead to different results.
also come to this conclusion and also considered how
averaging the coefficient on each cell affects the number of
iterations necessary to solve the linear systems.
We repeat these experiments here but with our larger set of different
elements. Specifically,
results obtained with no averaging inside the element (“No”), arithmetic averaging (“Arith”),
geometric averaging (“Geom”), and harmonic averaging (“Harm”) are shown in Fig. .
We see that
(i) all four elements show the same rate of convergence: O(h) for velocity errors
and O(h0.5) for pressure errors;
(ii) harmonic averaging always yields lower errors, validating the findings of ;
(iii) the number of iterations in the Stokes solver is the lowest for the stabilized Q1×Q1 element;
and (iv) this number is not strongly affected by the method of averaging (with the exception of the Q2×P-1 element).
The observation that none of the elements reach their optimal
convergence rate also supports our decision, briefly mentioned in the
“Goals of this paper” part of the Introduction, to not further
investigate higher-order Taylor–Hood elements Qk×Qk-1 or
Qk×P-(k-1) with k>2: we know from experiments such as
the current one that these elements will not yield better convergence
orders despite their additional cost.
SolVi benchmark. Left to right: Q1×P0, stabilized Q1×Q1,
Q2×Q1, and Q2×P-1. Top to bottom: velocity error,
pressure error, and number of FGMRES iterations for the
Stokes solve. The individual lines in each graph correspond to different ways
of averaging coefficients on each cell: dotted lines use the correct
unaveraged values of coefficients at each quadrature point;
dash-dotted lines compute the arithmetic average of the values at
the quadrature points on a cell and use the average for all quadrature
points; dashed lines use the geometric average; solid lines use the
harmonic average. The gray dotted line in the first two rows indicates O(h) convergence for velocity and
O(h0.5) for pressure.
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. .
We find that the recovered pressures on the line y=1 follow the analytical solution
outside the inclusion but are less accurate inside the inclusion where it should be identically zero
(Fig. ).
SolVi benchmark. Pressure field for the Q1×P0, stabilized Q1×Q1,
Q2×Q1, and Q2×P-1 elements from left to right and top to bottom
at resolution 128×128 with no averaging. Note the different color
scales, illustrating the differing size of overshoots and undershoots for
the different discretizations.
SolVi benchmark. Pressure on the horizontal ray starting from the center of the inclusion at x=1.
The sinking block
As discussed in Sect. , the stabilized Q1×Q1 element is sensitive
to the choice of a reference density profile as not only the computed
pressure, but also the computed velocity field, depends on
this choice. This is only relevant for buoyancy-driven flows, but because none of the
benchmarks shown previously are driven by buoyancy effects in the
presence of a background lithostatic pressure to any significant degree,
let us next consider a setup in which this is the dominant effect. To this end, we
perform an experiment based on a benchmark similar or identical to the ones presented in , , ,
and .
It consists of a two-dimensional 512×512 km domain filled with a fluid (the “mantle”)
of density ρ1=3200 kg m-3 and viscosity η1=1021 Pa s.
A square block of size 128×128 km is placed in the domain and
is centered at location (xc, yc) = (256, 384 km) so as to ensure that its sides
align with cell boundaries at all resolutions, avoiding
cases in which the quadrature within one element corresponds to different
density or viscosity values. It is filled with a fluid
of density ρ2=ρ1+δρ and viscosity η2.
The gravity vector points downwards with |g|=10 m s-2. Boundary conditions
are free-slip on all sides.
The pressure null space is removed by enforcing ∫ΩpdV=0, and only one time step is carried out.
The benchmark then solves for the instantaneous
pressure and velocity field for this setup.
In a geodynamical context, the block could be interpreted as a detached slab
(δρ>0) or a plume head (δρ<0). As such its viscosity and density can vary (a cold slab has a
higher effective viscosity than the surrounding mantle, while it is the other way
around for a plume head). The block density difference δρ can then vary from a
few to several hundred kilograms per cubic meter (kg m-3) to represent a wide array of scenarios.
As shown in Appendix A.2 of , one can independently vary η1, ρ2, and η2
and measure |vz| for each combination: the quantity ν=|vz|η1/δρ is then
found to be a simple function of the ratio η⋆=η2/η1. At high enough mesh
resolution all data points collapse onto a single line.
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 ρ1 has been uniformly removed
so that the block has a density δρ, while the surrounding fluid has
zero density. As discussed above, the two choices will result in
different pressure but the same velocity fields.
We have carried out measurements for all four elements with
η⋆∈[10-4:106]
and δρ/ρ1∈{0.25%,1%,4%}
corresponding to δρ∈{8,32,128} kg m-3.
Results for ν=f(η⋆) for all elements, the three block density values, and five different
mesh resolutions are shown in Fig. for the two methods.
When using the full density, we see that all elements, with the
exception of the stabilized Q1×Q1 element, yield results which
align on a single curve on the plots once sufficient resolution is reached.
We find that measurements pertaining to a given resolution but different δρ
are always collapsed onto a single line. It is worth noticing
that the Q2×P-1 element results seem to be the least resolution-dependent.
On the other hand, the stabilized Q1×Q1 element yields very anomalous results
which are orders of magnitude off at all resolutions, especially for
η1/η2≫1. In addition, we find that for this element, the value of δρ strongly affects
the measurements, as expected based on the discussions in
Sect. ; as a result, the curves for the same mesh
resolution but different δρ2 no longer coincide
(see Fig. b).
When reduced densities are used results are unchanged for the stable elements
(only Q2×Q1 results are shown in Fig. e),
and the results for the stabilized Q1×Q1 results are substantially improved. For values η1/η2<1
we see that all results align on the expected curve, but this is far from true
for η1/η2≫1 even at high resolution.
Sinking block benchmark. (a–d)ν=|vz|η1/δρ as a function of η⋆=η2/η1
as obtained with the four elements with full density;
(e, f) same with reduced density for only two element types.
Legend:
•16×16 resolution,
□32×32 resolution,
▪64×64 resolution,
△128×128 resolution,
▴256×256 resolution.
Colors represent the element used. For each mesh resolution, we show
separate curves for δρ/ρ1∈{0.25%,1%,4%};
for all but the stabilized Q1×Qq element, these curves coincide.
Note the different y axis used
for the stabilized Q1×Q1 element in (b) and (f).
In Fig. we show the velocity field in
the case η⋆=10-4 (i.e., the viscosity of the block is
10 000 times smaller
than the surrounding mantle) and δρ=8 kg m-3.
When the Q2×Q1 element is employed in conjunction with Method 1 we
see in Fig. a that the velocity field is strongest inside the block
with a maximum value of about 5 mm yr-1 in its center.
We see that the Q2×Q1 and Q2×P-1 elements
yield nearly identical results (Fig. b), so
we consider this to be the correct solution of the physical experiment.
The same setup with the stabilized Q1×Q1 (left half of Fig. c)
yields a velocity field that is also maximal in the middle of the block but nearly
1000 times larger in amplitude.
If we now switch to Method 2 (right half of Fig. c) the amplitude of the velocity
is reduced by 2 orders of magnitude, but it is still much too large compared to the
true solution.
These observations illustrate the unreliable nature of the results obtained with
stabilized Q1×Q1 elements in the context of buoyancy-driven flows.
Looking at Fig. f we see that increasing the resolution to 512×512
or 1024×1024 would probably yield the expected curve, but such resolutions are
intractable in three dimensions
and better results can be obtained at much lower resolutions with other elements.
Sinking block benchmark with δρ/ρ=0.25 % and η⋆=104
on a 256 × 256 element mesh.
(a) Viscosity and velocity field.
(b) Velocity field obtained with the Q2×Q1 element (left of vertical white line)
and Q2×P-1 element (right of vertical line), both using full density;
(c) velocity field obtained with stabilized Q1×Q1 with full density (left) and
stabilized Q1×Q1 with reduced density (right).
Finally, in Fig. we plot the normalized pressure
p⋆=p/(δρgLb) at the center of the block
(where
Lb is the size of the block) as a function of the viscosity ratio η⋆ in
the case in which a reduced density field is used. For the Q2×Q1 and
the stabilized Q1×Q1 elements, the pressure at this point is
uniquely defined since the elements have continuous pressures.
For the other two elements the pressure is discontinuous across element edges, and
it is therefore not uniquely defined at our measurement point. We have then
chosen to measure it at four locations corresponding to (xc±δx,yc±δy),
where δx=δy=0.1 m, and show the normalized
pressures at all four of these locations in the figure. For the
Q2×P-1 element, the difference between these values is
negligible but not so for the Q1×P0 for which the pressure
is a stairstep function with very different values depending on which
step an evaluation point is on. The distance between the two lines
for the Q1×P0 element decreases with mesh refinement
(indicating convergence of the pressure to the true value), but only
slowly and, matching the observation in Sect. ,
at the cost of not only a fine mesh but also very large numbers of
linear solver iterations.
Sinking block benchmark.
Normalized pressure p/(δρgLb) in the center of the block as a function of the
viscosity ratio η⋆. These computations use a 256 × 256
mesh and the reduced
density. For the Q1×P0 and Q2×P-1 elements with
their discontinuous pressure spaces, we show the normalized pressures
at several slightly displaced points (xc±δx,yc±δy). For the Q2×P-1 element, the difference is not
visible, but for the Q1×P0 this yields the two very
different red curves; this is due to the fact that the pressure for
this element forms a stairstep function for which two of the
evaluation points are on a lower and two on a higher step.
In addition to the slow convergence of the Q1×P0 element,
the most striking conclusion of this benchmark is that for
buoyancy-driven flows, the solution obtained using the stabilized
Q1×Q1 element on typical meshes not only strongly
depends on the choice of the otherwise arbitrary reference density, but
is also almost entirely unreliable even on meshes that are already
quite fine.
Numerical results for a model application
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:
ηdisl=A-1/nε˙-1+1/nexpQ+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.
Application example. (a) Vertical component of the velocity field; (b) pressure field;
(c) effective viscosity field; (d) effective strain rate field.
In all panels, the left half (left of the vertical line) shows data
obtained with the stabilized Q1×Q1 element, whereas the
right half shows results obtained with the Q2×Q1
element. Note the large deviations between the two towards the bottom
of the domain. All results were obtained on an 800×200 mesh with a cell size of
0.5 km.
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.
Application example.
(a) Laterally averaged effective viscosity;
(b) laterally averaged velocity magnitude. The line styles chosen
become increasingly assertive (dotted to solid lines) as mesh
resolution is increased.
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.
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 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.
Conclusions
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.
We believe that the Q1×P0 element is also not a viable
choice. As shown by several of the analytical benchmarks, the errors
that result from using this element can be orders of magnitude larger
than the corresponding errors that result from the Taylor–Hood-type
elements. This is no longer the case once we consider discontinuous
viscosity profiles (see Sect. ), but this element is
also unable to accurately solve the buoyancy-driven case discussed in
Sect. . Furthermore, as pointed out before, this
element is not LBB-stable, which, despite considerable efforts
in the past decades, has limited its use in combination with iterative methods:
because of the corresponding condition number increase,
the number of iterations is found to grow in a somewhat unpredictable manner
with an increase in resolution. This may explain why, despite the Citcom codes' success over 2 decades
with studies based on models counting up to ∼ 100 million elements on
several hundred processors e.g.,, the current generation
of massively parallel codes relies on either
stable or stabilized elements ,
or they use the finite-difference method .
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.
Code availability
Aspect is an open-source code licensed under the GNU
Public License (GPL) version 2 or later. It is available at https://aspect.geodynamics.org/.
Version 2.2.0-pre (master, commit hash 6b134ad4c) was used for this work, and the corresponding
input files for the benchmarks are already available in the current
distribution and discussed in the software's manual.
Author contributions
CT conceived the study and ran all models. WB implemented
the stabilized Q1×Q1 element in Aspect.
Both authors discussed the results and jointly wrote the paper.
Competing interests
The contact author has declared that neither they nor their co-author has any competing interests.
Disclaimer
Publisher's note: Copernicus Publications remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Acknowledgements
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
(http://geodynamics.org, last access: 21 June 2021) for their support of the Aspect code. Cedric Thieulot also wishes to thank
Riad Hassani for his help at the very early stages of this work. Wolfgang Bangerth gratefully acknowledges support by the National Science
Foundation.
Financial support
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.
Review statement
This paper was edited by Susanne Buiter and reviewed by Dave May, Matthew Knepley, and one anonymous referee.
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