Finite Element Least-squares Methods for a Compressible Stokes System

The least-squares functional related to a vorticity variable or a velocity flux variable is considered for two-dimensional compressible Stokes equations. We show ellipticity and continuity in an appropriate product norm for each functional.


Introduction.
Let Ω be a convex polygonal domain in R 2 .Consider the stationary compressible Stokes equations with zero boundary conditions for the velocity u = (u 1 ,u 2 ) t and pressure p as follows: where the symbols ∆, ∇, and ∇• stand for the Laplacian, gradient, and divergence operators, respectively (∆u is the vector of components ∆u i ); the number µ is a viscous constant; f is a given vector function; β = (U , V ) t is a given C 1 function.The system (1.1) may be obtained by linearizing the steady-state barotropic compressible viscous Navier-Stokes equations without an ambient flow (see [8,9] for more detail).Since the continuity equation is of hyperbolic type containing a convective derivative of p, we further assume that the boundary condition for the pressure is given on the inlet of the boundary where the characteristic function β points into Ω, that is, where Γ in = {(x, y) ∈ ∂Ω | β • n < 0} with the outward unit normal n to ∂Ω.Hence the boundary ∂Ω consists of Γ in and Γ out where Γ out = {(x, y)|β • n ≥ 0}.There was a study on a mixed finite element theory for a compressible Stokes system (see, e.g., [8]), but there are a few trials dealing with a compressible Stokes system like (1.1) using leastsquares method.Some papers focused on a H −1 least-squares method (see, e.g., [6,9]).Least-squares approach was developed for the incompressible Stokes and Navier-Stokes equations in [1,2,7].The purpose of this paper is to apply the philosophy of firstorder system least-squares (FOSLS) methodology developed in [5] to a compressible stationary Stokes system.We consider two basic first-order systems.The first one is induced by a vorticity variable, and the second one is induced by a velocity flux variable which is further extended to the system's associated curl and trace equations.This extended system is not a system of first order but a mixture system of first-and secondorder equations due to the continuity equation ∇ • u + β • ∇p = g.In order to provide ellipticity for each functional, we assume the H 1 and H 2 regularity assumptions for the compressible Stokes equations.As usual in FOSLS approach, we first show that the H −1 and L 2 FOSLS functional is elliptic in the product norm w + u 1 + q + p 0,β for the functional involving vorticity variable and U + u 1 + p for the functional involving flux variable.We also show that the extended functional related to velocity flux variable is elliptic in the product norm U 1 + u 1 + p 1,β .Then we provide the error estimates for using finite element methods.The outline of the paper is as follows.In Section 2, we discuss least-squares system and other preliminaries.The continuity and ellipticity of least-squares functionals are discussed in Section 3.These can be done by employing regularity estimates for (1.1).The finite element approximations are briefly discussed in Section 4.
2. Least-squares system for compressible Stokes equations, and other preliminaries.For the development of least-squares theory, we will adopt the notation introduced in [5] and introduce the necessary definitions in this section.A new independent variable related to the 4-vector function of gradients of the displacement vectors, u i , i = 1, 2 will be given.It will be convenient to view the original n-vector functions as column vectors and the new 4-vector functions as either block column vectors or matrices.The velocity variable u = (u 1 ,u 2 ) t is a column vector with scalar components u i , so that the gradient ∇u t is a matrix with columns ∇u i .For a function U with 2-vector components which is a matrix with entries U ij = ∂u j /∂x i , 1 ≤ i, j ≤ 2. Then we can define the trace operator tr as (2.3) Define the curl as and the divergence as We also define the tangential operator n × componentwise The inner products and norms on the block column vector functions are defined in the natural componentwise way; for example, (2.7) We use standard notations and definitions for the Sobolev spaces H s (Ω) n , associated inner products (•, •) s , and respective norms • s , s ≥ 0. When s = 0, H 0 (Ω) n is the usual L 2 (Ω) n , in which case the norm and inner product will be denoted by • 0 = • and (•, •), respectively.The space H s 0 (Ω) is the set of functions in H s (Ω) vanishing on the boundaries.From now on, we will omit the superscript n and Ω if the dependence of vector norms on dimension is clear by context.We use H −1 0 (Ω) to denote the dual spaces of H 1 0 (Ω) with norm defined by Define the product spaces H s 0 (Ω) 2 and L 2 (Ω) 2 in usual way with standard product norms.Let (2.9) Define a space where k is either 1 or 0, which is a Hilbert space with norm We frequently use the notation constant C Ω to denote that it depends on Ω only, but it may be a different constant.If a constant depends on another variable, we specify it in each place.Throughout this paper, we assume the following regularity.
Assumption 1. Assume that µ and β are such that (1.1) has a unique solution which satisfies the following a priori estimate: where k is either 0 or 1; C 0 := C 0 (µ, Ω) is a constant depending on µ, β, and Ω.Note that one may find (2.12) for k = 1 in [10, Theorem 1.3] for β = (1, 0) t and one may get (2.12) for k = 0 by following the arguments in [10, Section 3].In fact, using triangle inequality and the assumption (2.12), one may get the improved a priori estimates: where k is 1 or 0 and C 0 := C 0 (µ, Ω) is a constant depending on µ, β, and Ω.
2.1.Velocity-vorticity-pressure formulation.Note that As in [4] for Stokes equations, introducing the vorticity variable w = ∇×u, the first equation of the compressible Stokes equations (1.1) using the second equation of (1.1) is By setting q = ∇•u, the equivalent first-order system is now (2.16)

Velocity-flux-pressure formulation.
As in [5] for Stokes equations, introducing the velocity flux variable U = ∇u t , the compressible Stokes equations (1.1) may be written as the following equivalent first-order system: (2.17) We consider the following extended equivalent system for (2.17): (2.18) 3. Least-squares functionals.The main objective in this section is to establish ellipticity and continuity of least-squares functionals based on (2.16), (2.17), and (2.18) in appropriate Sobolev spaces.

Velocity, vorticity, and pressure.
The first-order least-squares functional corresponding to (2.16) is and let The FOSLS variational problem for the compressible Stokes equations corresponding to (2.16) is to minimize the quadratic functional G 0 over ᐂ 0 : find (w, u,q,p) ∈ ᐂ 0 such that Theorem 3.1.Under the assumption (2.12), there are two positive constants c and C, dependent on δ and Ω, such that for all (w, u,q,p) ∈ ᐂ 0 , cM 0 (w, u,q,p) ≤ G 0 (w,u,q,p; 0, 0) ≤ CM 0 (w, u,q,p). (3.5) Proof.Upper bound in (3.5) is a simple consequence of the triangle inequality and Cauchy-Schwarz inequality.For any (w, u,q,p) ∈ ᐂ 0 , using (2.13), triangle inequality, and (•), we have where Ĉ0 is a constant that depends on µ, β, and Ω.Using (3.6), we have where C is a constant depending on Ω and the Poincare constant.Now, cancelling w on both sides and squaring the remainder, we have w 2 ≤ CG 0 (w, u,q,p; 0, 0), (3.8)where C is a constant depending on Ω and the Poincare constant.Now, using (3.6), we have where C is a constant depending on Ω. Cancelling q on both sides and squaring the remainder, we have q ≤ CG 0 (w, u,q,p). (3.10) Finally, combining (3.6), (3.8), and (3.10) yields the lower bound.This completes the proof.

Velocity, flux, and pressure.
The first-order least-squares functional corresponding to (2.17) is The extended least-squares functional corresponding to (2.18) is (3.15) The least-squares variational problem for the compressible Stokes equations corresponding to (2.17) or (2.18) is to minimize the quadratic functional G i over ᐂ i : find Theorem 3.2.Under the assumption (2.12), there are two positive constants c and C, dependent on µ, β, and Ω, such that for all (U, u,p) ∈ ᐂ 1 , cM 1 (U, u,p) ≤ G 1 (U, u,p; 0, 0) ≤ CM 1 (U, u,p). (3.17) Proof.Upper bound in (3.17) is a simple consequence of the triangle inequality and Cauchy-Schwarz inequality.To limit arguments, it is enough to show that lower bound in (3.17) holds for ᐂ = H(div; Ω) 2 ×H 1 0 (Ω) 2 ×Q(Ω).Using (2.12) and triangle inequality, we have where Ĉ0 is a constant that depends on µ and Ω.Note that where C is a constant depending on Ω.Now cancelling U on both sides, squaring the remainder, and using (3.19), we have where C is a constant depending on µ, β, and Ω.Finally, combining (3.19) and (3.20) yields the lower bound.This completes the proof.
The following lemma is basically proved in [5,Lemma 3.2].
Due to the above lemma, one may get the following theorem.

Finite element approximations.
In this section, we provide the finite element approximation of the minimization of the least-squares functionals G 0 only.Note that an obvious modification in this section also provides the finite element error analysis for the least-squares functionals G 1 and G 2 .Let T : H −1 0 (Ω) 2 → H 1 0 (Ω) 2 be the solution operator (u = T f) for the following elliptic boundary value problem with zero boundary condition

.1)
Leth be a family of triangulations of Ω by standard finite element subdivisions of Ω into quasi-uniform triangles with h = max{diam(K) : K ∈h }.
(4.4) Theorem 4.1.Suppose that the assumption in Theorem 3.1 holds.Assume that (w, u, q, p) ∈ ᐂ 0 is the solution of the minimization problem for G 1 in (3.4) and (w h , u h ,q h ,p h ) is the unique minimizer of G 0 over ᐂ 0,h .Then  Then, using (4.1),Theorem 3.1, the orthogonality of the error (w −w h , u−u h ,q−q h ,p − p h ) to ᐂ 0,h , with respect to the above inner product, and the Schwarz inequality, we have the conclusion.