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On the Divergence-Free Finite Element Method for the Stokes Equations

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On the Divergence-Free Finite Element Method for the Stokes Equations On the divergence-free finite element method for the Stokes equations and the P1 Powell-Sabin divergence-free element Shangyou Zhang∗ Abstract A general framework for the conforming k- k−1 mixed element method is set up. Because such a method would always produce point wise divergence-free solutions for the velocity and the mixed element can always be reduced to a conforming k element for the vector Laplace equations, we name it the divergence-free finite element method. It is shown that the method always has a unique solution. Further, the method is guaranteed to provide optimal-order solutions if some subspace of C1- k+1 on the same triangular or hexahedral grid has the optimal-order approximation property. The framework is applied to the 1 divergence-free element on Powell-Sabin triangulations. Numerical tests are provided. Keywords. Powell Sabin triangles, mixed finite elements, Stokes, divergence-free element. AMS subject classifications (2000). 65M60, 65N30, 76D07. 1 Introduction A natural finite element method for the Stokes equations would be the k- k−1 mixed element which approximates the velocity by continuous k piecewise-polynomials and approximates the pressure by discontinuous k−1 piecewise-polynomials. One advantage of the element is to preserve the incompressibility condition of incompressible fluids ([4, 6, 27, 38, 39, 48]). Another advantage is its simplicity in computation that the mixed element can be reduced to the standard C0 finite element solving a vector Laplace equation, see Section 5 in this paper. A fundamental study on the method was done by Scott and Vogelius in 1983 ([38, 39]) that the method is stable and consequently of the optimal order on 2D triangular grids for any k ≥ 4, provided the grids have no singular vertex (i.e., edges meeting at a vertex do not form two cross lines.) For k ≤ 3, Scott and Vogelius showed the method would not be stable for general triangular grids, in [38, 39]. What is this magic number k in 3D? Scott and Vogelius posted this question explicitly after discovering that k = 4 in 2D. The problem remains open after 20 years. Even in the 2D case, there are many open problems for low order (k < 4) k- k−1 elements. It is known that for k < 4, there are spurious pressure modes in the discrete pressure spaces and the classic inf-sup condition does not hold. Nevertheless, filtering out such modes could be trivial in this case (see Section 5 in the paper), different from other mixed elements where either the pressure space is continuous, or the grid for the pressure differs from that for the velocity, or the space for the velocity is not continuous or is not k, e.g. ¡ k. By filtering out the spurious pressure modes, all types of the k- k−1 element would have a unique solution ∗Department of Mathematical Sciences, University of Delaware, DE 19716. [email protected]. 1 for the Stokes equations (see Proposition 2.1 in the paper). However, the approximation of ¢ solutions by low-order ¢ k- k−1 elements is not guaranteed, known as locking phenomenon (see the numerical tests in Section 6 of the paper.) The approximation is shown for the Hsieh- Clough-Tocher triangles with k ≥ 2 ([33]), for the quadrilateral-triangulations with k = 2 ([4]), for the uniform criss-cross grids with k = 1 ([34]) and for the 3D Hsieh-Clough-Tocher tetrahedra with k ≥ 3 ([49]). ¢ In this paper, we set up a framework for the divergence-free element, i.e., the ¢ k- k−1 ¢ element for all k and space dimensions. To be precise, the ¢ k- k−1 mixed element is replaced ¢ by the ¢ k-div k element, after filtering out the spurious pressure modes. In addition to showing the existence of the finite element solution, we show that the convergence and the optimal-order approximation of both the velocity and the pressure are guaranteed by the optimal order approximation of any one subspace of C1- ¢ k+1 scalar finite element space on the same grid. Using the frame work, one would eagerly try to answer the Scott-Vogelius question (in a little different setting) that the magic number k in 3D is 8 or less because a subspace of the C1- ¢ 9 finite element on tetrahedral grids was constructed in [47]. However, due to the complexity of 3D singular or nearly-singular edges and/or vertices, we leave the study of 3D ¢ ¢ k- k−1 elements somewhere else. Instead, in this paper, we will apply the framework only ¢ to the 2D ¢ 1 element on Powell-Sabin triangulations (cf. [31, 32]), i.e., a 1 divergence-free element. The stability of this ¢ 1 element has not been fully studied previously, but the element was used in computation, cf. [13]. We remark that the inf-sup condition is no longer used in the analysis of this framework for the divergence-free element. Since the discrete pressure space is div ¢ k, the inf-sup condition always holds, except the inf-sup constant might be arbitrarily close to 0. For example, when a nearly-singular vertex approaches to singular (cf. [38, 39]), the inf-sup constant approaches to zero. Nevertheless, when the vertex become a singular one, the inf-sup constant would jump back from zero to a regular one, because the extra spurious pressure mode is filtered out automatically in the divergence-free element method. On the other side, even if the inf- sup condition holds uniformly on some quasiuniform grids (independent of the grid size), the convergence of the divergence-free element method may still fail. This is because the traditional discrete pressure space of ¢ k−1 piecewise polynomials (modulo the global constant) is replaced by a subspace, div ¢ k, while the latter may lack local support and may not have enough, or the optimal order, approximation property. In this paper, we will also show how to find the divergence-free element solutions by the ¢ classic iterative penalty method. We will show how the mixed ¢ k- k−1 element can always be reduced to a conforming ¢ k element for the vector Laplace equations. Therefore, in this case, we do not really have a mixed element. We name the method as the divergence-free finite element method. We note that the name of divergence-free mixed element is used often for non-conforming ([9, 30, 45]) or discontinuous Galerkin methods ([12, 15, 21]), or even for weakly divergence-free methods ([46]). But here, the discrete solutions for the velocity is truly divergence-free, including the points on the inter-element boundary. The rest of the paper is divided into 5 sections. In Section 2, we define the divergence-free finite element method and show that the method always has a unique solution. In Section 3, we set a general framework for the convergence of the divergence-free finite element method. In Section 4, we apply the general theory to the ¢ 1 Powell-Sabin element. In Section 5, we will ¢ ¢ show how to reduce the ¢ k- k−1 mixed element to the C0- k element, and how to apply the classic iterative penalty method to solve the discrete, but positive definite, linear systems of equations. In Section 6, we provide a numerical test using the Powell-Sabin ¢ 1 divergence-free element. 2 2 The divergence-free finite element In this section, we shall define a class of finite elements for the stationary Stokes equations, where the finite element solution for the velocity is divergence-free point wise. The resulting linear systems of equations, by such elements, are shown to have a unique solution. We consider the stationary Stokes problem: Find functions u (the fluid velocity) and p (the pressure) on a 2D or 3D polygonal domain Ω ⊂ Rd such that −∆u + rp = f in Ω; div u = 0 in Ω; (2.1) u = 0 on @Ω; where f is the body force. Via the integration by parts, we get a variational problem for the d Stokes equations: Find u 2 H1;0(Ω) and p 2 L2;0(Ω) := L2(Ω)=C = fp 2 L2 j Ω p = 0g such that R d a(u; v) + b(v; p) = (f; v); 8v 2 H1;0(Ω) ; (2.2) b(u; q) = 0; 8q 2 L2;0(Ω): d d Here H1;0(Ω) is the subspace of the Sobolev space H1(Ω) (cf. [14]) with zero boundary trace, and a(u; v) = ru · rv dx; ZΩ b(v; p) = − div u p dx; ZΩ (f; v) = f v dx: ZΩ Let Ω be quasi-uniformly triangulated ([14]): Ωh = fT j T is a triangle or a tetrahedron; jT j ≤ hg : Let Vh be the space of of continuous piecewise polynomial of degree k on Ω with the homoge- neous boundary condition: d Vh = uh 2 C(Ω) j uhjT 2 Pk(T ) 8T 2 Ωh; and uhj@Ω = 0 : (2.3) n o Then we define Ph = fdiv uh j uh 2 Vhg : (2.4) Since Ω ph = Ω div uh = @Ω uh · n = 0 for any ph 2 Ph, we conclude that R R R d Vh ⊂ H1;0(Ω) ; Ph ⊂ L2;0(Ω); i.e., the mixed-finite element pair (Vh; Ph) is conforming. Then the finite element discretization of problem (2.2) reads: Find uh 2 Vh and ph 2 Ph such that a(uh; v) + b(v; ph) = (f; v); 8v 2 Vh; (2.5) b(uh; q) = 0; 8q 2 Ph: We note that by (2.4), Ph is a subspace of discontinuous, piecewise polynomials of degree (k − 1) or less. Except a few cases such as the high-order tetrahedral elements on Hsieh- Clough-Tocher grids (cf.
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