Abstract
We approximate the solution of the Stokes equations by a new quasi-optimal and pressure robust discontinuous Galerkin discretization of arbitrary order. This means quasi-optimality of the velocity error independent of the pressure. Moreover, the discretization is well-defined for any load which is admissible for the continuous problem and it also provides classical quasi-optimal estimates for the sum of velocity and pressure errors. The key design principle is a careful discretization of the load involving a linear operator, which maps discontinuous Galerkin test functions onto conforming ones thereby preserving the discrete divergence and certain moment conditions on faces and elements.
1 Introduction
This paper is a new contribution to the research programme initiated in [17, 27], which aims at designing quasi-optimal and pressure robust discretizations of the Stokes equations
for the largest possible class of inf-sup stable pairs of finite element spaces.
To illustrate our results, let
Analogously, we say that the given discretization is quasi-optimal and pressure robust when there is a constant
Any discretization fulfilling the above error estimates
is defined for any admissible load f in the weak formulation of (1.1),
inherits the approximation properties of the underlying spaces V and Q, irrespective of the regularity of
Whereas the first two properties are desirable in the discretization of any equation, the third one is specific to the (Navier–)Stokes equations. Its importance has been pointed out in [19] and further investigated in various other references, see, e.g., [16].
Most Stokes discretizations based on nonconforming pairs fail to
fulfill (1.2). Analogously, most discretizations with other pairs
than divergence-free ones fail to fulfill (1.3). Both claims
follow from the abstract results in
[17, 25]. Indeed, the combination
of (1.2) and (1.3) has been available for a long time only
for discretizations based on conforming and divergence-free pairs,
like the one of Scott and Vogelius [22].
The importance of pressure robustness was observed in [19],
where pressure robust schemes are proposed using
The discretization proposed in [27] uses the first-order nonconforming Crouzeix-Raviart pair and can be written as follows: find
where the forms a and b are as in the original discretization described in [9]. The operator E maps into continuous piecewise polynomials and preserves the discrete divergence and the averages on the mesh faces. This idea has been generalized in [17] to a wide class of pairs, under the same conditions on E. The only difference is that the form a needs to be augmented with additional terms. In this paper we propose a different approach, which does not require any augmentation of a, at the price of a more involved construction of E.
More precisely, we propose a class of discontinuous Galerkin
discretizations of arbitrary order
The remaining part of this paper is organized as follows. In Section 2 we propose the new discretization and motivate the above-mentioned conditions on E. Section 3 is devoted to the construction of E and to the derivation of the error estimates. Finally, in Section 4 we investigate numerically the proposed discretization in the lowest-order case. We indicate Lebesgue and Sobolev spaces and their norms as usual, see, e.g., [5].
2 Discontinuous Galerkin Discretization of the Stokes Equations
2.1 Stokes Equations
Let
Here
where c only depends on the geometry of Ω, see, e.g., [4, Theorem 8.2.1]. Moreover, introducing the kernel of the divergence operator
we infer
2.2 Meshes and Polynomials
Let
where
for all
We denote by
for all
where
where
The spaces
The approximation of the pressure space involved in the Stokes equations (2.1) motivates the use of the one-codimensional subspace
We shall repeatedly make use of the following integration by parts formula:
where
2.3 Discontinuous Galerkin Discretization
We consider a discontinuous Galerkin (dG) discretization of order
and
where
Motivated by the abstract results in [25], we let
Introducing the discrete divergence
for all
This shows that
i.e. the discrete velocity
Remark 1 (Alternative Definition of
div
dG
).
Note that we could equivalently define the discrete divergence as the linear operator
Indeed, testing with
for all
To assess the quality of the discretization (2.6), we introduce the scalar product
where the penalty parameter η is the same as in (2.5). We measure the velocity error
Remark 2 (Notation for
dG
Discretization).
The label “
Remark 3 (Comparison with Conforming and Nonconforming Discretizations).
One main advantage of dG discretizations of the Stokes equations is that they are inf-sup stable for all polynomial degrees
In what follows, we write C for a positive nondecreasing function of the shape constant
2.4 Stability
The so-called inverse trace inequality [10, Lemma 1.46] implies that there is a
constant
Hence, simple algebraic manipulations reveal that the form
(2.10)
for all
for all
The following discrete counterpart of (2.2) follows from (2.10) and the theory of saddle point problems. The discrete stability constant involves, in particular, the operator norm of
Lemma 4 (Discrete Well-Posedness and Stability).
Let
Proof.
Since
where
A discrete counterpart of (2.3) additionally holds, under the assumption that
Lemma 5 (Stability of the Discrete Velocity).
Under the assumptions of Lemma 4,
for any load
if and only if
Proof.
Assume first that (2.12) holds. Testing the first equation of (2.6) with the elements of
In view of (2.8), we are allowed to set
Then the inclusion
We derive the claimed a priori estimate in view of the boundedness of
Conversely, assume (2.11) holds and
there is
This contradiction confirms that
Remark 6 (Pressure Robustness).
The a priori estimate (2.3) reveals that the velocity u in the Stokes equations (2.1) solely depends on
2.5 Quasi-Optimality
We now look for conditions ensuring that the discretization (2.6) enjoys (1.2). To this end, we first investigate the approximation of the velocity field u in (2.1) by
Lemma 7 (Approximation by
Z
dG
).
Let
Moreover, it holds
Proof.
Let
cf. [5, Chapter 12.5]. Then the Banach–Nec̆as theorem (see, e.g., [11, Theorem 2.6]) ensures the existence of a unique solution
together with the estimate
Recall that u is in
where we have used the inverse trace inequality (2.9) for the term involving
We conclude taking the infimum over all
Remark 8 (Size of
δ
dG
).
The bound of
After this preparation, we observe that the dG discretization (2.6) fits into the abstract framework of [17, Section 2]. Hence, applying [17, Lemma 2.6], we derive that the following consistency conditions are necessary for quasi-optimality (1.2)
(2.13)
Differently from [17], we deal with these
conditions assuming that
Lemma 9 (Consistency by Moment-Preserving Operators).
Assume that the operator
(2.14)
for all
Proof.
Let
In view of (2.14), we can replace w by
This entails that (2.13b) holds. Arguing similarly, we infer that
for all
Remark 10 (Alternative Approach to Consistency).
The implication (2.14)
The next theorem states that (2.14) is indeed a sufficient condition for quasi-optimality. The essence of this result and a partial proof can be found also in [2, Section 6].
Theorem 11 (Quasi-Optimality).
Assume that the operator
and
Proof.
We first estimate the velocity error. For this purpose, let
Setting
We bound
We bound
for all
where we have used
Next, in order to estimate the pressure error, let
Let
We bound
We bound
We insert the estimates of
We derive the claimed estimate of the pressure error by means of the previous estimate of the velocity error. ∎
2.6 Quasi-Optimality and Pressure Robustness
The assumptions in Theorem 11 do not guarantee that the discretization (2.6) is pressure robust in the sense of (1.3). We illustrate this by a numerical experiment in Section 4.2. Similarly as in [17], we achieve pressure robustness by the additional assumption that the operator
Theorem 12 (Quasi-Optimality and Pressure Robustness).
Assume that the operator
for all
and
3 A Moment- and Divergence-Preserving Operator
Motivated by the error estimates in Theorem 12, we now aim at designing a linear operator
which satisfies the following conditions:
(3.1)
We restrict ourselves to the case
3.1 Outline of the Construction
We first outline the strategy underlying our construction before we enter into the technical details. We shall obtain
Our construction has a recursive structure in the sense that the definition of
The first operator
The third operator
Finally, the fourth operator
For
3.2 Preliminary Observations
The main difficulty in the construction of
Lemma 13 (
∇
P
ℓ
-
1
(
K
)
-Moments).
Let
Proof.
Let
where the second identity follows from the assumption that E satisfies (3.1b) and (3.1c). The equivalent definition of the discrete divergence in Remark 1 entails that
We conclude invoking once again the element-wise integration by parts formula and recalling that q vanishes in
The above lemma suggests that we should enforce (3.1d) only on some complement of
where w and
for all
Recall the convention
Lemma 14 (Decomposition of Vector-Valued Polynomials).
For all
The operator
Proof.
Assume
As mentioned before, the construction of the operators
The contravariant and the covariant Piola’s transformations, respectively, map functions
Remarkably, we have
Moreover, the contravariant Piola’s transformation is such that
and
3.3 Construction of
E
dG
We now construct an operator
Definition of
E
1
.
Each polynomial in
where
Definition of
E
2
.
We define
Then, for
Note that each summand involves an extension from F to
Definition of
E
3
.
We define
and introduce a linear operator
This constrained quadratic minimization problem is uniquely solvable as a consequence of [13, Theorem 3.1]. Note that we can equivalently rewrite (3.13) as a discrete Stokes-like problem, with velocity space
where each summand vanishes on
Definition of
E
4
.
Denote by
Lemma 14 ensures that this problem is uniquely solvable. Then, for
where each summand vanishes on
Reference triangle
3.4 Preservation Properties of
E
dG
In this subsection we prove that the operator
Lemma 15.
Let
We are now prepared to prove the claimed properties of
Theorem 16 (Preservation Properties of
E
dG
).
The operator
Proof.
Let
Proof of (3.1b). By construction, each summand in the definitions (3.14) and (3.16) of
Rearranging terms, we infer that
Proof of (3.1c). Each summand in the definition (3.16) of
Rearranging terms, we infer that
Proof of (3.1d). For all
is an equivalent formulation of (3.1d). Moreover, according to the decomposition stated in Lemma 14, we can split (3.17) into two independent conditions with test functions in
Next, recall the definitions of
We combine this identity with the previous one and we rearrange terms according to the definition (3.2) of
Rearranging terms, we infer that (3.17) holds for all
3.5 Stability of
E
dG
In this subsection we prove that the operator
Lemma 17 (Local
L
2
-Estimate of
E
1
).
For all
Next, we prove that
Proposition 18 (Local
L
2
-Estimate of
E
dG
).
The operator
Proof.
First of all, we recall that
According to Lemma 17, we only need to bound the last three summands in the right-hand side. We estimate these terms one by one.
Estimate of
Recalling also the definition of
for all
on each face F involved in the above summation. This fact and an inverse trace inequality entail that
Then Lemma 17 yields
Estimate of
Since
The inclusion
where we have made use also of the identity
Then Lemma 17 and inequality (3.18) yield
Estimate of
Since
Then Lemma 17 and inequalities (3.18) and (3.19) yield
The local estimate in Proposition 18 ensures that
Theorem 19 (Stability of
E
dG
).
The operator
Proof.
Let
We square both sides in this inequality and sum over all
We conclude by recalling the definition of the norm
3.6 Main Results
We are now able to derive the main result of this paper. For this purpose, we invoke [24, Corollary 1] and derive the following upper bound of the velocity best error
for all
Theorem 20 (Quasi-Optimality and Pressure Robustness by
E
dG
).
Let
and
The above design of
According to [26, Proposition 3.4], the resulting operator is moment-preserving and stable, in the sense that it satisfies conditions (3.1a), (3.1b) and (3.1d). Then the following weaker counterpart of Theorem 20 readily follows from Theorem 11.
Theorem 21 (Quasi-Optimality by
E
~
dG
).
Let
and
3.7 Construction of
E
dG
for
d
=
3
We end this section with some comments concerning the extension of the previous results to the discretization of the Stokes equations in dimension
for all
This can be verified by noticing that the operator
The construction of
With these ingredients, the statements and the proofs of the results in Sections 3.4–3.6 can easily be adapted to the case
4 Numerical Experiments
We now discuss the results obtained when approximating the solution of the Stokes equations (2.1) with
We test the discretization (2.6) of the Stokes equations with
The first option differs from the others, in that
We consider only the first-order discretization in our experiments, i.e. we set
in (2.6). The numerical results are obtained with the help of ALBERTA 3.0 [15, 21].
4.1 Smooth Exact Solution
We first consider a test case with smooth exact solution, namely
We discretize the domain Ω by the following family of structured meshes. Given
For
where
The crisscross mesh
Velocity errors of the standard (
| N |
|
EOC |
|
EOC |
|
EOC |
| 4 | 8.2516e
|
8.3795e
|
8.5337e
|
|||
| 5 | 3.8937e
|
0.54 | 3.9344e
|
0.55 | 4.1273e
|
0.52 |
| 6 | 1.8797e
|
0.53 | 1.8910e
|
0.53 | 2.0231e
|
0.51 |
| 7 | 9.2180e
|
0.51 | 9.2477e
|
0.52 | 1.0007e
|
0.51 |
| 8 | 4.5621e
|
0.51 | 4.5698e
|
0.51 | 4.9756e
|
0.50 |
Pressure errors of the standard (
| N |
|
EOC |
|
EOC |
|
EOC |
| 4 | 4.4477e
|
4.4862e
|
4.3843e
|
|||
| 5 | 2.2248e
|
0.50 | 2.2377e
|
0.50 | 2.2109e
|
0.49 |
| 6 | 1.1142e
|
0.50 | 1.1178e
|
0.50 | 1.1109e
|
0.50 |
| 7 | 5.5781e
|
0.50 | 5.5878e
|
0.50 | 5.5692e
|
0.50 |
| 8 | 2.7912e
|
0.50 | 2.7937e
|
0.50 | 2.7884e
|
0.50 |
4.2 Jumping Pressure
In order to investigate the pressure robustness of the three discretizations, we consider a test case with smooth exact velocity and rough exact pressure, namely
We use unstructured meshes to make sure that the discontinuity of p along the line
The data displayed in Figure 3 (right) show that the velocity error of the quasi-optimal and pressure robust discretization fully exploits the regularity of u and converges to zero at the maximum decay rate
Initial mesh
Funding source: Deutsche Forschungsgemeinschaft
Award Identifier / Grant number: KR 3984/5-1
Funding statement: Christian Kreuzer gratefully acknowledges partial support by the DFG research grant “Convergence Analysis for Adaptive Discontinuous Galerkin Methods” (KR 3984/5-1). The research of Pietro Zanotti was supported by the GNCS-INdAM through the program “Finanziamento giovani ricercatori 2019-2020”.
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