Pressure-improved Scott–Vogelius type elements

In this paper we consider the numerical solution of the stationary Stokes equations by conforming Galerkin finite element methods. This is a vivid research topic since many decades in numerical analysis and scientific computing. The unknowns are the vector-valued velocity field and the scalar pressure and the Galerkin discretization is based on the choice of a pair of finite element spaces: one, say \({\textbf{S}}\), for the velocity and one, say M, for the pressure approximation. It is well-known that the most intuitive choice for \({\textbf{S}}\), i.e., continuous, piecewise polynomials of degree k and for M, i.e., discontinuous, piecewise polynomials of degree \(k-1\) (which we will denote as the full pressure space) can be unstable: although the continuous problem is well-posed the Galerkin discretization may result in a singular system matrix and is not solvable (see, e.g., [24] and [7, Chap. 7] for quadrilateral meshes).

This problem motivated the development of many pairs \(\left( {\textbf{S}},M\right) \) of Stokes elements; standard strategies include enrichment of the intuitive velocity space, see, e.g., [16], reducing the intuitive pressure space, see, e.g., [5, 15, 21, 24] and combinations thereof [2, 3, 11, 7, Chap. 3, §7]. Other approaches are based on a consistent modification of the discrete variational formulation and we refer to the overviews [6, 8, 10] for detailed expositions. In any case, a “good” Stokes discretization should have the following features: (a) discrete stability in the form of a discrete inf-sup condition, (b) the divergence of the discrete velocity is zero or very small, (c) the Stokes element \(\left( {\textbf{S}},M\right) \) enjoys good approximation properties for the continuous solution depending on its regularity, (d) the element is simple and easy to be implemented.

The Scott–Vogelius element (see [21, 24]) is a very popular element which is based on an appropriate reduction of the full pressure space in the intuitive element described above. The element is inf-sup stable, the discrete velocity is divergence-free, and its implementation very simple. However, it suffers from two drawbacks; (a) the discrete inf-sup constant is not mesh-robust: if some vertex becomes nearly singular – a geometric notion which will be recalled in the paper and which is not related to shape regularity – the discrete inf-sup constant tends to zero and the discretization becomes increasingly ill-posed; (b) in the presence of super-critical vertices (defined in (3.13) below), the approximation property of the pressure space becomes sub-optimal and affects the accuracy of the discrete pressure significantly. As a remedy for drawback (a), mesh refinement strategies are proposed in the literature (see [2, Rem. 2]) or, alternatively, a very simple modification of the Scott–Vogelius element which circumvents mesh refinement is introduced in [13, Def. 3] and called the pressure-wired Stokes element. For both methods, the discrete inf-sup constant becomes mesh-robust while the divergence of the discrete velocity for the pressure-wired Stokes element is not exactly zero but small (without any regularity requirement on the solution) and controlled by a parameter \(\eta \).

Drawback (b) affects both, the Scott–Vogelius and the pressure-wired Stokes element: if the mesh contains super-critical vertices, a geometric notion which will be introduced in this paper, the discrete pressure converges only at a very sub-optimal rate. In this paper, we propose a method to modify the pressure space so that the pressure converges with optimal rate. The method is very simple: for both, the Scott–Vogelius and the pressure-wired Stokes element, the reduction of the full pressure space in the most intuitive Stokes element is formulated via a linear (local) constraint at each critical vertex written in the form \(A_{{\mathcal {T}},{\textbf{z}}}\left( q\right) =0\) for a discrete pressure function q and a critical point \({\textbf{z}}\) in the mesh \({\mathcal {T}}\). This condition is replaced at super-critical vertices by another linear side condition which we introduce in this paper. Its definition relies on the explicit knowledge of a local basis (set of critical functions) for the orthogonal complement of the reduced pressure space in the full pressure space. We define a linear injection of the reduced pressure space into the full pressure space by adding a linear combination of those critical functions related to super-critical vertices. The coefficients in this linear combination are given by local linear functionals applied to a pressure in the reduced pressure space. We emphasize that algorithmically this modification is simply realized by replacing the linear constraint \(A_{{\mathcal {T}},{\textbf{z}}}\left( q\right) =0\) at super-critical vertices with another local linear constraint and hence the computational complexity and the dimension of the pressure space stay unchanged. It turns out that the inf-sup stability and the smallness of the divergence of the discrete velocity is inherited by the resulting element with this modified pressure space. In addition, the modified pressure space satisfies optimal approximation properties.

The paper is organized as follows. We introduce the Stokes problem, its variational form, and the Galerkin discretization in Sect. 2. Section 3 is devoted to the Scott–Vogelius element with the modified pressure space. We introduce the linear functional which serves as the side constraint for the reduction of the intuitive pressure space, prove that the inf-sup stability of the Scott–Vogelius element is inherited, and the divergence of the discrete velocity remains zero. This modification is particularly simple for the Scott–Vogelius element since it can be realized as a postprocessing step applied to the original Scott–Vogelius solution. In Sects. 3.4 and 3.5 we prove that this postprocessed pressure converges at optimal rate. In Sect. 4 we introduce the pressure-improvement strategy for the pressure-wired Stokes element and define the modified pressure space. In contrast to the Scott–Vogelius element, the discretization of the Stokes equation employs the modified pressure space and directly yields the final discrete solution. We prove in Sect. 4.1 that the inf-sup stability of the original pressure-wired Stokes element is inherited to its modified version. In Sect. 4.2 we will show that our pressure-improvement strategy applied to the pressure-wired Stokes element leads to a pressure space with optimal approximation properties. This result is applied to the modified element in Sect. 3.5 and optimal convergence rates for the discrete solution are shown. It remains to investigate the divergence of the corresponding discrete velocity which is considered in Sect. 4.3. The key role is played by the derivation of an explicit basis representation for the orthogonal complement of the modified pressure space (Lemma 4.11). This is used in Theorem 4.10 to prove that the smallness of the velocity divergence is controlled by the parameter \(\eta \). For this result, no regularity assumption on the exact solution is imposed. However, if the continuous solution has some regularity, we derive additional convergence rates for the smallness of the divergence.

Let \(\Omega \subset {\mathbb {R}}^{2}\) be a bounded Lipschitz domain with polygonal boundary \(\partial \Omega \). We consider the numerical solution of the Stokes equationwith homogeneous Dirichlet boundary conditions for the velocity and the usual normalization condition for the pressure, namelyThroughout this paper, standard notation applies for real-valued Lebesgue and Sobolev spaces. Let \(H_{0}^{1}\left( \Omega \right) \) be the closure of the space of infinitely smooth, compactly supported functions with respect to the \(H^{1}\left( \Omega \right) \) norm. Its dual space is given by \(H^{-1}\left( \Omega \right) :=H_{0}^{1}\left( \Omega \right) ^{\prime }\). The scalar product and norm in \(L^{2}\left( \Omega \right) \) are written asVector-valued and \(2\times 2\) tensor-valued analogues of these function spaces are denoted by bold and blackboard bold letters, e.g., \({\textbf{H}}^{s}\left( \Omega \right) =\left( H^{s}\left( \Omega \right) \right) ^{2}\) and \({\mathbb {H}}^{s}\left( \Omega \right) =\left( H^{s}\left( \Omega \right) \right) ^{2\times 2}\) and analogously for other quantities.

The \({\textbf{L}}^{2}\left( \Omega \right) \) scalar product and norm for vector-valued functions are given bywith the standard Euclidean scalar product \(\left\langle \cdot ,\cdot \right\rangle \). In a similar fashion, we define the scalar product and norm in \({\mathbb {L}}^{2}\left( \Omega \right) \) bywhere \(\left\langle {\textbf{G}},{\textbf{H}}\right\rangle =\sum _{i,j=1} ^{2}G_{i,j}H_{i,j}\). Finally, let \(L_{0}^{2}\left( \Omega \right) :=\left\{ u\in L^{2}\left( \Omega \right) :\int _{\Omega }u=0\right\} \). We introduce the bilinear forms \(a:{\textbf{H}}^{1}\left( \Omega \right) \times {\textbf{H}}^{1}\left( \Omega \right) \rightarrow {\mathbb {R}}\) and \(b:{\textbf{H}} ^{1}\left( \Omega \right) \times L^{2}\left( \Omega \right) \rightarrow {\mathbb {R}}\) bywith the derivative \(\nabla {\textbf{v}}\) and the divergence \({\text {div}} {\textbf{v}}\) of any \({\textbf{v}} \in {\textbf{H}}^{1} \left( \Omega \right) \). Given a source \({\textbf{F}}\in {\textbf{H}}^{-1}\left( \Omega \right) \), the variational form of the stationary Stokes problem seeks \(\left( {\textbf{u}},p\right) \in {\textbf{H}}_{0}^{1}\left( \Omega \right) \times L_{0}^{2}\left( \Omega \right) \) such thatConcerning the well-posedenss of (2.2), we refer, e.g., to [12] for details. In this paper, we consider a conforming Galerkin discretization of (2.2) by a pair \(\left( {\textbf{S}},M\right) \) of finite dimensional subspaces of the continuous solution spaces \(\left( {\textbf{H}}_{0}^{1}\left( \Omega \right) ,L_{0}^{2}\left( \Omega \right) \right) \). For any given \({\textbf{F}}\in {\textbf{H}}^{-1}\left( \Omega \right) \), the weak formulation seeks \(\left( {\textbf{u}}_{{\textbf{S}} },p_{M}\right) \in {\textbf{S}}\times M\;\)such thatIt is well known that the bilinear form \(a\left( \cdot ,\cdot \right) \) is symmetric, continuous, and coercive so that problem (2.3) is well-posed if the bilinear form \(b\left( \cdot ,\cdot \right) \) satisfies the inf-sup condition for \(\left( {\textbf{S}}, M \right) \).

Let \({\mathcal {T}}\) be a conforming, shape-regular triangulation of the domain \(\Omega \) into closed triangles \(K\in {\mathcal {T}}\) with diameter \(h_{K}\). The set of vertices is given by \({\mathcal {V}}\left( {\mathcal {T}}\right) \) and the additional subscripts \({\mathcal {V}}_{\Omega }\left( {\mathcal {T}}\right) \), \({\mathcal {V}}_{\partial \Omega }\left( {\mathcal {T}}\right) \) specify whether a vertex is located in the domain or on its boundary. The set of edges is denoted by \({\mathcal {E}}\left( {\mathcal {T}}\right) \) and the same subscript convention as for the vertices applies. For a vertex \({\textbf{z}}\in {\mathcal {V}}\left( {\mathcal {T}}\right) \), the local vertex patch is given bywith the local mesh width \(h_{{\textbf{z}}}:=\max \left\{ h_{K}:K\in {\mathcal {T}}_{{\textbf{z}}}\right\} \). For any vertex \({\textbf{z}}\in {\mathcal {V}}\left( {\mathcal {T}}\right) \), we fix a local counterclockwise numbering of the \(N_{{\textbf{z}}}:={\text {card}}{\mathcal {T}}_{{\textbf{z}}}\) triangles inA triangle neighborhood of some triangle \(K\in {\mathcal {T}}\) is given byThe shape-regularity constantrelates the local mesh width \(h_{K}\) with the diameter \(\rho _{K}\) of the largest inscribed ball in an element \(K\in {\mathcal {T}}\). The global mesh width is given by \(h_{{\mathcal {T}}}:=\max \left\{ h_{K}:K\in {\mathcal {T}}\right\} \). For a subset \(M\subset {\mathbb {R}}^{2}\), we denote the area of M by \(\left| M\right| \). Let \({\mathbb {P}}_{k}(K)\) denote the space of polynomials on \(K\in {\mathcal {T}}\) with total degree smaller than or equal to \(k\in {\mathbb {N}}_{0}\) and defineAs usual \(\overset{\circ }{K}\) denotes the open interior of K. For any \(q\in {\mathbb {P}}_{k}\left( {\mathcal {T}}\right) \), we write

The pressure-wired Stokes element introduced in [13] generalises the standard Scott–Vogelius element by restricting the discrete pressure space additionally at nearly singular vertices \({\textbf{z}}\in {\mathcal {V}}\left( {\mathcal {T}}\right) \) (called \(\eta \)-critical vertices) to guarantee a mesh-robust inf-sup stability. In general, the divergence of the discrete velocity field no longer vanishes pointwise while its marginality has been analyzed in detail in [13, Sec. 5]. This section discusses a modified pressure space with a parameter \(\eta \ge 0\) introduced below in full analogy to Sect. 3 for optimal approximation properties.

Definition 3.11 generalises to \(\eta \)-super critical vertices: a vertex \(\mathbf {z\in }{{\mathcal {S}}}{{\mathcal {C}}}_{{\mathcal {T}}}\left( \eta \right) \) is called a Robinson vertex if (i): \({\mathcal {T}} _{{\textbf{z}}}^{*}\cap {\mathcal {T}}_{{\textbf{y}}}^{*}=\emptyset \) for all \({\textbf{y}}\in {{\mathcal {S}}}{{\mathcal {C}}}_{{\mathcal {T}}}\left( \eta \right) {\setminus }\left\{ {\textbf{z}}\right\} \) and (ii): \(\omega _{{\textbf{z}}}^{*}\cap {\mathcal {C}} _{{\mathcal {T}}}\left( \eta \right) =\left\{ {\textbf{z}}\right\} \) hold.

As already announced in the introduction, there is a significant loss in accuracy of the pressure approximation for the pressure-wired Stokes element if the mesh contains \(\eta \)-super critical vertices. As a remedy, we introduce a modification in complete analogy to Sect. 3 for the Scott–Vogelius element.