Solving the Stokes equation by an optimal domain decomposition method derived algebraically involves the use of nonstandard interface conditions whose discretisation is not trivial. For this reason the use of approximation methods such as hybrid discontinuous Galerkin appears as an appropriate strategy: on the one hand they provide the best compromise in terms of the number of degrees of freedom in between standard continuous and discontinuous Galerkin methods, and on the other hand the degrees of freedom used in the nonstandard interface conditions are naturally defined at the boundary between elements. In this paper, we introduce the coupling between a well chosen discretisation method (hybrid discontinuous Galerkin) and a novel and efficient domain decomposition method to solve the Stokes system. We present the detailed analysis of the hybrid discontinuous Galerkin method for the Stokes problem with nonstandard boundary conditions. The full stability and convergence analysis of the discretisation method is presented, and the results are corroborated by numerical experiments. In addition, the advantage of the new preconditioners over more classical choices is also supported by numerical experiments.
Discontinuous Galerkin (DG) methods have been first introduced in the early 1970s  and they have benefited of a wide interest from the scientific community. The main advantages of these methods are their generality and flexibility as they can be used for a variety of partial differential equations on unstructured meshes. Moreover, they can preserve local properties such as mass and momentum conservation while ensuring a high-order accuracy. However, the cost of these advantages is a larger amount of degrees of freedom in comparison to the continuous Galerkin methods  for the same approximation order.
A good compromise between the previous methods, while preserving the high order, are the hybridised versions of DG using divergence conforming spaces such as Raviart–Thomas (RT) and Brezzi–Douglas–Marini (BDM) . These methods are a subset of the hybrid discontinuous Galerkin (HDG) methods, introduced in  for second-order elliptic problems, and extended to three-dimensional Stokes equation in . The authors present there the mixed formulation of HDG methods defined locally on each element. They consider many types of boundary conditions that involve normal and tangential velocity, pressure, and tangential stress. The formulations of the methods are similar, the only difference lies in the choice of the numerical traces. A refined analysis of HDG methods for the Stokes equation with Dirichlet boundary conditions was presented in , where optimal convergence in the HDG norm is proven, and superconvergent postprocessings are introduced. An alternative strategy to approximate the Stokes problem was used in , where a combination of HDG and a symmetric interior penalty (somehow linked to the work ) is presented and analysed.
In a further paper  this approach is extended to Darcy, and coupled Darcy–Stokes flows. The new formulation includes different degrees of polynomials for finite element spaces associated with different variables.
In all HDG discretisations, one important component is the Lagrange multiplier defined on the inter element edges (faces in 3D). In the work  the authors approximate the Navier–Stokes equation using -conforming spaces (rather than the usual -conforming ones) together with an HDG approach. Now, in order to reduce the number of degrees of freedom needed for the Lagrange multiplier, the authors introduce an edge-wise projection into a finite element space of lower degree. This projection was also shown to be of paramount importance to establish the connection between the hybrid high-order  and the HDG methods presented in .
Even considering the decrease of the number of degrees of freedom provided by the HDG methods with projection, modern applications lead to linear systems whose size is too large to allow the use of direct solvers. Thus, parallel solvers are becoming increasingly important in scientific computing. A natural paradigm to take advantage of modern parallel architectures is the Domain Decomposition method, see e.g. [28, 24, 31, 12]. Domain decomposition methods are iterative solvers based on a decomposition of a global domain into subdomains. At each iteration, one (or two) boundary value problem(s) are solved in each subdomain and the continuity of the solution at the interfaces between subdomains is only satisfied at convergence of the iterative procedure. The partial differential equation is the one of the global problem.
For Additive Schwarz methods and Schur complement methods, the boundary conditions on the interfaces between subdomains, or, interface conditions (IC), are Dirichlet or Neumann boundary conditions. For scalar equations there is a consensus on this choice of IC. On the other hand, for systems of differential equations, such as the Stokes or incompressible elasticity equations, alternatives like normal velocity-tangential flux (NVTF) or tangential velocity-normal flux (TVNF) IC should be superior to the pure velocity (Dirichlet like) or pure stress (Neumann like) IC, see [12, Section 6.6] and the references therein. In , this choice was motivated by symmetry considerations, while in [14, 5, 4], they were obtained by an analysis of the systems of partial differential equations by symbolic techniques, mostly linked to the Smith factorization . Similar attempts to derive more intrinsic IC to the nature of the equation to solve were derived  for the Euler system.
Due to the difficulty of implementing these IC, previous numerical tests were restricted to decompositions where the boundaries of subdomains are rectilinear so that the normal to the interface is easy to define. The underlying domain decomposition method was a Schur complement method. That is mainly the reason we have considered and analysed a specific HDG method where this kind of degrees of freedom are naturally present.
In this paper, we want to combine appropriate HDG discretisation and the associated domain decomposition methods mentioned above using nonstandard IC. The combination of the two is meant to provide a competitive solving strategy for this kind of partial differential equation system. A different, but somewhat related, approach can be found in  where a DG type discretisation is coupled to a discrete Helmholtz decomposition to propose some preconditioners.
Concerning the HDG discretisation, in approach similar to the one presented in this work, but using completely discontinuous spaces, is given in . Our analysis is related to the one in that paper, but the method presented herein uses -conforming spaces, which implies in turn that the Lagrange multipliers are scalar valued. The combination of these two facts reduces the number of degrees of freedom significantly. In addition, the use of nonstandard boundary conditions (motivated by the newly defined domain decomposition preconditioners) makes the analysis somehow more involved.
The rest of the paper is organised as follows. To start with, we introduce the problem and notation in Section 2. In Section 3, we present the hybridisation of a symmetric interior penalty Galerkin method that allows us to impose the TVNF and NVTF boundary conditions in quite a natural way. The formulation is similar to the one from  with Dirichlet boundary conditions. In addition to different kinds of boundary conditions, we include a local edge-based projection in order to reduce the dimension of the space of Lagrange multipliers. Our analysis follows the one from  (see also  for a more detailed version). Thanks to the HDG discretisation, we can consider domain decomposition methods with arbitrary shape of the interfaces and Schwarz type methods. In Section 4, the Additive Schwarz methods are defined at the algebraic level. Section 5 contains the numerical results, including the convergence validation of the HDG method and a comparison of the domain decomposition preconditioners. Finally, some conclusions are drawn.
2 Notation and Preliminary Results
Let Ω be an open polygonal domain in with Lipschitz boundary . We use boldface font for tensor or vector variables, e.g. is a velocity vector field. The scalar variables will be italic, e.g. p denotes a pressure scalar value. We define the stress tensor and the flux . In addition, we denote normal and tangential components as follows: , , , , where is the outward unit normal vector to the boundary Γ and is a vector tangential to Γ such that .
For , we use the standard space with the following norm:
Let us define, for , the following Sobolev spaces:
where, for , , and
In addition, we will use following standard semi-norm and norm for the Sobolev space :
In this work, we consider the two-dimensional Stokes problem
where is the unknown velocity field, the pressure, the viscosity, which is considered to be constant, and is a given function. For we consider two types of boundary conditions:
tangential-velocity and normal-flux (TVNF)(2.2)
normal-velocity and tangential-flux (NVTF)(2.3)
which together with (2.1) define two boundary value problems. We will detail the analysis for the TVNF boundary value problem
since considering the NVTF boundary conditions (2.3) instead is very similar. We will just add a remark when necessary to stress the differences between them. The restriction to homogeneous Dirichlet conditions on is made only to simplify the presentation.
Let be a regular family of triangulations of made of triangles. For each triangulation , denotes the set of its edges. In addition, we set for each , and . We define the following Sobolev spaces on the triangulation and the set of all edges in :
with the corresponding broken norms.
The following results will be very useful in what follows.
Lemma 2.1 (Inverse and Trace Inequalities).
There exist , independent of , such that for every and every polynomial function v in K the following inequalities hold:
Moreover, there exists , independent of , such that for any , the following local trace inequality holds:
We now introduce the finite element spaces used for discretisation. First, we present them for the particular case in which we consider the TVNF boundary condition. For this case for , we discretise the velocity using the Brezzi–Douglas–Marini space (see [2, Section 2.3.1]) given by
The choice of the above finite element spaces is motivated by two requirements. On the one hand, we have imposed that the finite element space must be conforming in , while on the other hand, it has to satisfy the inf-sup condition with respect to a broken -norm of the velocity. The BDM space appears then as a natural alternative satisfying both these requirements. In fact, other popular alternatives, such as the Raviart–Thomas space, while conforming in do not satisfy the inf-sup condition with respect to the norm used in this work.
In addition, for we denote by the BDM projection defined in [2, Section 2.5]. The HDG formulation includes a Lagrange multiplier over the internal edges. In order to propose a discretisation with fewer degrees of freedom, we discretise the Lagrange multiplier using the spaces
Furthermore, we introduce for all the -projection defined as follows. For every , is the unique element of satisfying
and we define by for all .
Let us denote . The pressure is discretised using the following space:
In addition, we define the local -projection for each defined as follows. For every , is the unique element of satisfying
We will also use the following results.
Lemma 2.2 (Approximation Results).
There exists , independent of , such that for all and , , the following interpolation estimates hold:
local Brezzi–Douglas–Marini approximation(2.8)
trace -projection approximation(2.9)
local -projection approximation(2.10)
3 Hybrid Discontinuous Galerkin Method
In this section, we introduce the HDG method proposed in this work, study its well-posedness, and analyse its error.
3.1 The Discrete Problem
From now on we will use to denote the element-wise gradient. First, we multiply the first equation from (2.1) by a test function and integrate by parts. This gives
Since the normal and tangential vectors are perpendicular (), we can split (3.1) as
In addition, since is continuous across all interior edges, we have
and we can add this to (3.3) to get
where is a stabilisation parameter. Hence, we define the velocity bilinear form as
where and is a stabilisation parameter, and as
With these definitions we propose the HDG method for the TVNF boundary value problem (2.4):
Find such that for all ,
The use of -conforming spaces not only decreases the number of degrees of freedom in comparison to , but leads as well to a simpler bilinear form b, as the jump terms appearing in that reference are no longer needed for stability or consistency.
3.2 Well-Posedness of the Discrete Problem
Let us consider the following semi-norm:
The semi-norm defined by (3.7) is a norm on .
Since is a semi-norm, we only need to show that
Let us suppose and . Then in all , and thus for all . Now, since in every K, we have
Since is single valued on all the edges in , we have that is continuous in Ω. Moreover, belongs to , so is also continuous in Ω. Then, is continuous in Ω, and thus in Ω. Finally,
which finishes the proof since on every edge. ∎
There exists such that, for all and , we have
Let us start with (3.8). Using the Cauchy–Schwarz inequality, we get
Therefore, using the triangle inequality and the trace -projection approximation (2.9), we get
Thus, using the Cauchy–Schwarz inequality gives
To show the well-posedness of (3.6) we need the ellipticity of the bilinear form a and an inf-sup condition for the bilinear form b. We start by showing that a is elliptic with respect to .
There exists such that for all ,
First, since for all , we have
To bound the middle term in terms of the other two, we consider two cases.
Case 1: . Then (3.11) reduces to
which proves (3.10) with .
Case 2: . Then (3.11) becomes
Using the Cauchy–Schwarz inequality, we deduce
Since is a piecewise polynomial, we can apply the discrete trace inequality (2.6) to the second term, followed by Young’s inequality to arrive at
Finally, if we suppose and set
The next step towards stability is proving the inf-sup condition for .
There exists independent of such that
According to the Fortin criterion, see [17, Lemma 4.19], we need to prove that there exists a Fortin operator such that for every the following conditions hold:
Let and let us consider the operator . It is well known, see [2, Section 2.5], that satisfies (3.14). To prove (3.15) we denote . Then using the discrete trace inequality (2.6) and the fact that the projection is a bounded operator, we get
Applying the triangle inequality for the last term of (3.16), we arrive at
Using the stability of leads to
Finally, using the trace -projection approximation (2.9) for the third term, we get
which finishes the proof. ∎
Using the last two results and the standard theory of variational problems with constraints [2, Section 4.2], we deduce there exists a unique solution of (3.6). In addition, thanks to the derivation carried out in Section 3.1, method (3.6) is consistent as the following result shows.
3.3 Error Analysis
In this section, we present the error estimates for the method. These estimates are proved using the norm
The first step is the following version of Cea’s lemma.
Let us denote
Now Lemma 3.3 yields the following continuity for B: there exists such that
Thus, we get (3.20) with . ∎
We prove the following error estimate using standard interpolation estimates.
Lemma 3.8 (HDG Error).
Let us consider the Fortin operator defined in the proof of Lemma 3.5. If , then by using the triangle inequality and boundedness of the projection , we get
Next we use the local trace inequality (2.7) to get
Let be the usual Lagrange interpolant of degree k of (see [17, Example 1.31]). Using the triangle inequality followed by the local inverse inequality (2.5), the local Lagrange approximation [17, Example 1.106] and (2.8), we see that (3.26) becomes
Finally, the local -projection approximation (2.10) gives
3.4 NVTF Boundary Conditions
As we mentioned before, the analysis in case of NVTF boundary conditions (2.3) is similar. Thus, we just highlight the main differences. If we consider NVTF boundary conditions (2.3), then to discretise the velocity we use the BDM space
For the Lagrange multiplier we use the polynomial space , while the pressure is discretised using
The product space then becomes and we pose the following discrete problem:
Find such that for all ,
Concerning the analysis, the proofs of all the results presented in the last sections remain essentially unchanged.
4 The Domain Decomposition Preconditioner
Let us assume that we have to solve the linear system
where is the matrix arising from discretisation of the Stokes equations on the domain Ω, is the vector of unknowns and is the right-hand side. To accelerate the performance of an iterative Krylov method applied to this system we will consider domain decomposition preconditioners which are naturally parallel [12, Chapter 3]. They are based on an overlapping partition of the computational domain.
Let be a partition of the triangulation . For an integer value , we define an overlapping decomposition such that is a set of all triangles from and all triangles from that have non-empty intersection with , and . With this definition the width of the overlap will be . Furthermore, if stands for the finite element space associated to , then is the local finite element space on that is a triangulation of .
Let be the set of indices of degrees of freedom of and let be the set of indices of degrees of freedom of for . Moreover, we define the restriction operator as a rectangular matrix such that if is the vector of degrees of freedom of , then is the vector of degrees of freedom of in . Abusing notation, we denote by both the operator and its associated matrix. The extension operator from to and its associated matrix are both given by . In addition, we introduce a partition of unity as a diagonal matrix such that
where is the identity matrix.
We are ready to present the first preconditioner, called Restricted Additive Schwarz (RAS) , given by
We also introduce a new preconditioner that is a modification of the above one. The modification is similar to the Optimized RAS , however we do not use Robin IC. For this, let be the matrix associated to a discretisation of (2.1) in where we impose either TVNF (2.2) or NVTF (2.3) boundary conditions in . Then, the preconditioner reads
The improvement of convergence in the case of Optimized RAS depends on the choice of the parameter. This parameter depends on the problem and discretisation. The big advantage of MRAS preconditioners is that they are parameter-free.
4.1 Partition of Unity
The above definitions of the preconditioners can be associated with any discretisation of the problem. However, each discretisation involves the construction of a relevant partition of unity , . We discuss here the construction of when the problem (2.1) is discretised by the HDG method in case , either with TVNF boundary conditions (3.6), or NVTF boundary conditions (3.31). Let us introduce the piecewise linear functions of such that
Now we define the piecewise linear functions of as follows:
Obviously . We define the partition of unity matrix as a block diagonal matrix where the first block is associated with , the second block with , and the third block with . The degrees of freedom of the BDM elements are associated with the normal components on the edges of the mesh. For these finite elements, the diagonal of is a vector obtained by interpolating at the two points of the edges. The degrees of freedom of the Lagrange multiplier finite elements are associated with the edges of the mesh. For these finite elements, the diagonal of is a vector obtained by interpolating at the midpoints of the edges. For pressure finite elements, the diagonal of is a vector obtained by interpolating at the midpoints of the elements.
5 Numerical Results
In this section, we present a series of numerical experiments aimed at confirming the theory developed in Section 3, and to give a computational comparison of the preconditioners discussed in the previous section. All experiments have been made by using FreeFem++ , which is a free software specialised in variational discretisations of partial differential equations.
5.1 Convergence Validation
The computational domain for both test cases considered here is the unit square . We present the results for , that is, the discrete space is given by for TVNF boundary conditions and for NVTF boundary conditions. We test both the symmetric method () and the non-symmetric method (). For both cases we have followed the recommendation given in [21, Section 2.5.2] and taken .
The first example aims at verifying the formulation with TVNF boundary conditions (3.6). We choose the right-hand side and the boundary datum g such that the exact solution is given by
In Figures a and b we show the results of the usual convergence order tests for the symmetric case and the non-symmetric case by plotting in log-log scale the error as a function of the size of the mesh. We notice that they validate the theory from Section 3.3. In addition, an optimal convergence rate is observed for . The proof of this last fact is lacking. This is due to the lack of regularity results concerning the solution of the continuous problem with these nonstandard boundary conditions, and hence the usual Aubin–Nitsche approach can not be advocated.Figure 1
In the second example we want to verify the formulation with NVTF boundary conditions (3.31). In this case, the right-hand side and the boundary datum g are chosen for the following exact solution:
The convergence error with respect to the size of the mesh is depicted on the log-log plots for the symmetric case and the non-symmetric case in Figures a and b, respectively. We can see that they not only validate the theory from Section 3.3, but also show an optimal convergence rate for .Figure 2
5.2 Comparison of Different Domain Decomposition Preconditioners
In this section, we compare the standard RAS preconditioner (4.1) with the newly introduced preconditioners, that is, the ones based on nonstandard IC. We call them MRAS preconditioners (4.2) and more precisely TVNF-MRAS for which is the matrix arising from the discretisation of (2.1) in with IC (2.2) on , and NVTF-MRAS for which is the matrix arising from the discretisation of (2.1) in with IC (2.3) on . As we mentioned before, our preconditioners do not depend on the used discretisation, that is why we add also similar preconditioners but based on a more standard discretisation, that is, the lowest order Taylor–Hood discretisation [18, Section II.4.2]. In all cases, they are used in conjunction with a Krylov iterative solver such as GMRES . In addition, N stands for the number of subdomains. In all tables we present the number of iterations needed to achieve an euclidean norm of the error (with respect to the one domain solution) smaller than . We have implemented the RAS preconditioner (4.1) and the MRAS (4.2), using both TVNF and NVTF interface conditions.
We start with the second example from the previous section. However, now we consider the symmetric () formulation with TVNF boundary conditions (3.6). The mesh is uniform and contains 125 000 triangles for a total of 565 003 degrees of freedom for the Taylor–Hood discretisation and 689 000 degrees of freedom for the HDG discretisation. We use a random initial guess for the GMRES iterative solver. The overlapping decomposition into subdomains can be uniform (Unif) or generated by METIS (MTS) and it has two layers of size h in the overlap.
The first thing that we can notice from Table 1 is the important convergence improvement in case of RAS applied to a system resulting from a HDG discretisation in comparison to the RAS applied to the system resulting from the Taylor–Hood discretisation despite the fact that the number of degrees of freedom is slightly bigger in the first case. The change in discretisation presumably leads to better conditioned systems to solve. Also the MRAS preconditioner with both discretisations performs better than the standard RAS method which fully justifies the use of the new IC no matter the discretisation method. Moreover, as expected, the number of iterations increases with respect to the number of the subdomains and this behaviour is common to the three preconditioners. It is worth noticing that this increase is slower than the expected linear one.
We also plot the convergence of the error for the different discretisations in Figure 3. We observe that in all cases the MRAS preconditioner (4.2) shortens the plateau region in the convergence curves significantly which leads, automatically, to an important reduction in the number of iterations.Figure 3
Now we consider the Poiseuille problem and we choose the right-hand side and the TVNF boundary condition such that the exact solution is given by
We use the same mesh as in the previous case and a random initial guess for the GMRES iterative solver, but this time the overlapping decomposition has three layers of size h in the overlap.
The conclusions stay the same as in previous example since the results from Table 2 are similar to the previous ones. We consider a different problem, however on the same mesh. Hence the global matrix is the same in both cases. Thus, we can notice a reduction in the number of iterations caused by the increase of the width of the overlap.Figure 4
The last example is on a T-shaped domain , and we impose mixed boundary conditions given by
In Figure 5 we plot the numerical solution obtained with the HDG discretisation using on a coarse mesh. In this case, we used a mesh containing 379 402 triangles, which gives linear systems of a size 1 712 352 for the Taylor–Hood discretisation and 2 089 735 for the HDG discretisation. The initial guess in the GMRES iterative solver is zero. The overlapping decomposition into subdomains is generated by METIS and it has two layers of size h in the overlap.
According to Table 3 the conclusions remain the same, that is, the standard RAS method performs far better when applied to a HDG discretisation with respect to a Taylor–Hood one and the MRAS preconditioners are better than the standard RAS preconditioner for both discretisations. Figure 6 again confirms the superiority of the MRAS preconditioner over the RAS preconditioner.Figure 6
In this paper, we introduced a HDG method for the Stokes equations that naturally discretises nonstandard boundary value problems such as those with TVNF and NVTF boundary conditions. This approach can be extended naturally to the case of incompressible, or nearly incompressible, elasticity. We proved the well-posedness and convergence with respect to the norm (3.7) of this method and in the numerical experiments from Section 5.1 we validated the theory and observed the optimal convergence.
To solve the discretised problem we introduced two different kinds of preconditioners with nonstandard boundary conditions whose optimality has been proved by algebraic techniques. We compared the newly introduced preconditioners to the more standard RAS preconditioner and numerical tests from Section 5.2 clearly show their superiority for different test cases in two space dimensions. Moreover, our numerical experience seems to hint that the linear systems arising from the HDG discretisation are better conditioned than those obtained using the Taylor–Hood element. This can be seen from the fact that the RAS preconditioner already performs far better when applied to the HDG method than when using Taylor–Hood.
We observed, as expected, that the Schwarz preconditioners are not scalable with respect to the number of subdomains. However, this can be fixed by using an appropriate coarse spaces [12, Chapter 4]. A suitable choice of a coarse space will be a subject of future research.
Funding source: Centre for Numerical Analysis and Intelligent Software
Award Identifier / Grant number: EP/G036136/1
Funding statement: This research has been partially funded by the Engineering and Physical Sciences Research Council of Great Britain under the Centre for Numerical Algorithms and Intelligent Software (NAIS) for the evolving HPC platform grant EP/G036136/1.
We thank Frédéric Hecht from Laboratory J. L. Lions for comments that greatly improved the FreeFem++ codes.
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