Local projection stabilization for the Stokes system on ... - CiteSeerX

Local projection stabilization for the Stokes system on anisotropic quadrilateral meshes Malte Braack and Thomas Richter Institute of Applied Mathematics, Heidelberg University, INF 294, 69120 Heidelberg, [email protected]

Summary. The local projection stabilization for the Stokes system is formulated for anisotropic quadrilateral meshes. Stability is proven and an error analysis is given.

1 Introduction The local projection stabilization (LPS) is suitable to stabilize the saddle point structure of the Stokes system when equal-order finite elements are used, as well as convective terms for Navier-Stokes. Hence, it has already been applied with large success to different fields of computational fluid dynamics, e.g., in 3D incompressible flows [7], compressible flows [12], reactive flows [8], parameter estimation [4, 5] and optimal control problems [11]. Although locally refined meshes have been used for this stabilization technique in all of these applications, the meshes have been isotropic so far. The solution of partial differential equations on anisotropic meshes are of substantial importance for efficient solutions of problems with interior layers or boundary layers, as for instance in fluid dynamics at higher Reynolds number. It is well known that stabilized finite element schemes, e.g. streamline upwind Petrov-Galerkin (SUPG), see [10], or pressure stabilized Petrov-Galerkin (PSPG), see [9], must be modified in the case of anisotropy. Becker has shown in [2] how the PSPG stabilization should be modified on anisotropic Cartesian grids. In this work, we make the first step of formulating LPS on anisotropic quadrilateral meshes by considering the Stokes system in the domain Ω ⊂ R2 for velocity v and pressure p: −∆v + ∇p = f ,

div v = 0 ,

together with appropriate boundary conditions for v on ∂Ω. The right hand side f is supposed to be in the Hilbert space L2 (Ω). The corresponding

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M. Braack, T. Richter

Galerkin formulation is known to be unstable for equal-order interpolation due to the violation of the discrete inf-sup condition [9]. By Vh and Qh we denote the discrete test spaces for v and p, respectively, consisting of piecewise polynomials of degree r = 1 on quadrilaterals, (Q1 elements). After a short presentation of LPS on isotropic meshes, we will generalize this technique to the case of anisotropic meshes obtained by bilinear transformations from a reference quadrilateral. A stability proof an a priori estimate will be given.

2 Local projection stabilization on isotropic meshes The mesh Th is supposed to be constructed by patches of quadrilaterals. The coarser mesh T2h is obtained by one global coarsening of Th . The correspondence between these two meshes is as follows: Each quadrilateral P ∈ T2h is cut into four new quadrilaterals (dividing all lengths of edges of P by 2) in order to obtain the fine partition Th . The space Qdisc 2h consists of patch-wise polynomials of degree r − 1, but discontinuous across patches P ∈ T2h . The projection πh : L2 (Ω) → Qdisc 2h is defined as the L2 -orthogonal projection: (πh q, ξ) = (q, ξ)

∀ξ ∈ Qdisc 2h .

The idea of LPS, see [3], consists of adding the stabilization term involving the difference between the identity I and πh to the Galerkin form: sh (ph , ξ) := ((I − πh )∇ph , α∇ξ) . The stabilization parameter is chosen as α ∼ h2 on isotropic meshes. Hence, the discrete system becomes: Find {vh , ph } ∈ Vh × Qh so that a(vh , ph ; φ, ξ) + sh (ph , ξ) = (f, φ)

∀{φ, ξ} ∈ Vh × Qh ,

with the linear form: a(v, p; φ, ξ) = (∇v, ∇φ) − (p, div φ) + (div v, ξ) . In [3] the following a priori estimate was shown for piecewise bilinear elements on quasi-uniform meshes: ||∇(v − vh )|| + ||p − ph || ≤ Ch(||∇p|| + ||∇2 v||) ,

(1)

with a constant C ≥ 0 and the maximal mesh size h. The norms above denote the L2 −norms over Ω. This estimate is optimal on isotropic meshes. On anisotropic meshes, it is suboptimal: if the solution has much larger gradients in, e.g., y−direction the mesh size should be chosen as hy 0 so that for every vh ∈ Vh and ph ∈ Qh there holds sup {φ,ξ}∈Xh

a(vh , ph ; φ, ξ) + sh (ph , ξ) ≥ γ|||{vh , ph }||| . |||{φ, ξ}|||

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Proof. Taking into account that Vh − Q2h is a stable pair for the Stokes system and following the results in [3] it is sufficient that the existence of an interpolation operator ih : Qh → Q2h is ensured, so that ||ih p|| . ||p|| ||p − ih p||2 . sh (p, p)

∀p ∈ Qh , ∀p ∈ Qh .

This conditions are fulfilled for the nodal interpolation onto Q2h , due to the scaling argument on a patch P ∈ T2h . This can be verified easily for the case θ = 0: 2 p − ic ||p − ih p||2P = hx hy ||b h p||K b

b p||2 ≤ hx hy ||(I − πh )∇b b K

= hx hy (||(I − πh )∂xbpb||2Kb + ||(I − πh )∂ybpb||2Kb )

. (hx + σhy )2 ||(I − πh )∂x p||2K + h2y ||(I − πh )∂y p||2K . Due to Assumption (A1), hx + σhy ≤ (1 + σ0 )hx , it follows ||p − ih p||2 ≤ (1 + σ0 )2 sh (p, p) . For θ 6= 0, the arguments are the same. ⊓ ⊔ 4.3 A priori estimate In the remainder Pof this work, expressions as for instance hx ||∂x u||, can always be replaced by K∈T2h hK,x ||∂x u||K . At first we need to bound the stabilization term applied to the anisotropic H 1 -stable projection Bh : Lemma 1. The stabilization term has the following interpolation property: sh (Bh p, Bh p)1/2 . (1 + σ0 )2 hx ||∂x p|| + (1 + σ0 )hy ||∂y p|| . Proof. sh (Bh p, Bh p) = (M (I − πh )∇Bh p, M ∇Bh p) = (M (I − πh )∇Bh p, M (I − πh )∇Bh p) = ||M (I − πh )∇Bh p||2 ≤ h2x (||∂x Bh p||2 + ||πh ∂x Bh p||2 ) + h2y (||∂y Bh p||2 + ||πh ∂y Bh p||2 ) . Due to the L2 −stability of πh and Proposition 1: sh (Bh p, Bh p)1/2 . hx ||∂x Bh p|| + hy ||∂y Bh p|| . hx (1 + σ0 )||∂x p|| + h2y h−1 x ||∂y p|| + (1 + σ0 )hx σ0 ||∂x p|| +hy (1 + σ0 )||∂y p||) ≤ hx (1 + σ0 )2 ||∂x p|| + hy (hy hx−1 + 1 + σ0 )||∂y p|| . The assertion follows due to hy h−1 ⊔ x ≤ 1. ⊓

Local projection stabilization on anisotropic meshes

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Proposition 3. Under the conditions (A1), (A2) and transformations of type (7) the following estimate holds for v ∈ H 2 (Ω) and p ∈ H 1 (Ω): ||∇(v − vh )|| + ||p − ph || . σ12 hx ||∂x p|| + ||∂x2 v|| + ||∂xy v|| + σ1 hy ||∂y p|| + ||∂y2 v|| ,

with σ1 := 1 + σ0 .

Proof. As usual we split the error ||∇(v − vh )|| + ||p − ph || in the interpolation part ||∇(v −Bh v)||+||p−Bh p|| and projection part ||∇(vh −Bh v)||+||ph −Bh p||. Due to Proposition 1 the interpolation part can be bounded by the right hand side of the estimate in Proposition 3. What remains is to bound the projection error. Due to the stability of the bilinear form it holds: ||∇(vh − Bh v)|| + ||ph − Bh p|| |a(vh − Bh v, ph − Bh ph ; φ, ξ) + sh (ph − Bh p, ξ)| . . sup |||{φ, ξ}||| {φ,ξ}∈Vh ×Qh Using the perturbed Galerkin orthogonality for discrete φ, ξ, a(vh − v, ph − p; φ, ξ) = −sh (ph , ξ) , we obtain a(vh − Bh v, ph − Bh p; φ, ξ) + sh (ph − Bh p, ξ) = a(v − Bh v, p − Bh p; φ, ξ) − sh (Bh p, ξ) . The last term is bounded by |sh (Bh p, ξ)| ≤ sh (Bh p, Bh p)1/2 |||{0, ξ}||| ≤ sh (Bh p, Bh p)1/2 . Hence, Lemma 1 gives the desired bound for the stabilization part. Finally, the Galerkin part can be bounded as: |a(v − Bh v, p − Bh p; φ, ξ)| . (||∇(v − Bh v)|| + ||p − Bh p||) |||{φ, ξ}||| , consisting once more of the previously addressed interpolation part. ⊓ ⊔ This result separates the partial derivatives and partial mesh sizes much more properly than the isotropic version in the estimate (1). Let us shortly discuss this result in the situation of a boundary layer with the usual local property |∂y2 v| >> |∂x2 v| + |∂xy v| and hx >> hy . Due to the multiplication of |∂y2 v| with the smaller mesh size hy the estimate in Proposition 3 is properly tuned. The shearing parameter σ1 enters moderately.

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4.4 Summary and outlook We extended the local projection stabilization (LPS) of the Stokes system to anisotropic quadrilateral meshes. In particular, we allow for high aspect ratios, shearing and bilinear effects. Stability and an a priori estimate is given. The result of this paper is still limitated to the Stokes system. For the application to Navier-Stokes, also the convective term has to be stabilized. Although LPS is designed for doing so, the additional terms have to be estimated for the anisotropic version. This will be subject of forthcoming work.

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