On the Stationary, Potential, Subsonic Flow

I.A. Cristescu

Abstract

In this communication we are concerned with the problem of the stationary, potential, subsonic flow. Firstly, we formulate the mechanical problem and the associated minimization problem with constraints, in a functional space W endowed with a certain norm.

Section 2 contains the main original results. A density Lemma for the space W is presented, then is introduced a stable, convergent, internal approximation of W. We state the approximate minimization problem and prove a convergence theorem. A minimax problem coresponds to the approximate minimization problem with constraints (P_{h}). For this one, we deduce the existence of the saddle point, using a separation Hahn-Banach theorem. In the end is presented an iterative algorithm for determining the minimax point.

Mathematics Subject Classification: 49J40, 76M10 Key words: subsonic flow, finite element, minimax point

1 Theoretical background

Let Ω be an open, bounded set in R^{n} (n = 2, 3) with the boundary ∂Ω Lipschitz continuous. The governing equations of the potential, subsonic, stationary flow are [1]

(1.1)  div [(k − \frac{1}{2}|∇φ|^{2})^{1/γ−1} ∇φ] = 0 in Ω,

(1.2)  (k − \frac{1}{2}|∇φ|^{2})^{1/γ−1} \frac{∂φ}{∂n} = g on Γ,

(1.3)  |∇φ| < v_{cr},

where

ρ(∇φ) = ρ_{0} (1 − \frac{γ − 1}{2} \frac{|∇φ|^{2}}{c_{0}^{2}} )^{1/γ−1} = ρ_{0} (\frac{γ − 1}{c_{0}^{2}})^{1/γ−1} (k − \frac{|∇φ|^{2}}{2} )^{1/γ−1}

Balkan Journal of Geometry and Its Applications, Vol.6, No.2, 2001, pp. 27-32 \begin{matrix}c \\ ◯ \end{matrix}Balkan Society of Geometers, Geometry Balkan Press

is the density,

k = \frac{c_{0}^{2}}{γ − 1},

p = p_{0} (\frac{ρ}{ρ_{0}} )^{γ} is the pressure,

v⃗ = ∇φ is the velocity of the flow,

g ∈ H^{1/2}(Γ),

n = the external unir normal.

Remark 1.1. For |∇φ| < \sqrt{2k \frac{γ − 1}{γ + 1}} = v_{cr}, the flow is subsonic. We denote || ||_{0} =

the norm in L^{2}(Ω), || ||_{1} = the norm in H^{1}(Ω). We shall use the following result ([2]): Lemma 1.1. Let be P_{k} the set of polynomials of degree less than or equal to k in the variables x_{1}, …, x_{n}. The seminorm

W^{k+1,p}(Ω)/_{P_{k}} ∋ v̄ ↦ |v̄|_{k+1,p} = |v|_{k+1,p} = ( ∑_{|α|=k+1} ∫_{Ω} |D^{α}v|^{p} dx)^{1/p}

is a norm over the quotient space W^{k+1,p}(Ω)/_{P_{k}}.

Further, introduce the space W = {ψ ∈ H^{1}(Ω)/ ∫_{Ω} ψ(x)dx = 0}.

Lemma 1.2. The mapping W ∋ u ↦ ||u||_{W} = ||grad u||_{0} is a norm on W, equivalent to the norm || ||_{1}.

We need a result due to Pironneau ([1]), Theorem 1.1. Let be b < v_{cr}. The problem (1.1)-(1.3) is equivalent to the minimisation problem:

find φ ∈ K_{b} = {ψ ∈ W/|∇φ| ≤ b} so that J_{0}(φ) ≤ J_{0}(ψ), ∀ψ ∈ K_{b},

with

(1.4)  J_{0}(ψ) = − ∫_{Ω} (k − \frac{1}{2}|∇ψ|^{2})^{γ/γ−1} dx − \frac{γ}{γ − 1} ∫_{Γ} g γ_{0}(ψ)dσ,

where we denoted by γ_{0} the trace application. Moreover,

(1.5)  J0^{′′}(ψ)v^{2} ≥ c||v||_{W}^{2} (∀)v ∈ W.

2 Main results

Let be T_{h} a family of regular triangulations of the poligonal (polyhedral) domain Ω. This means

(2.1)  σ(h) = sup_{S∈T_{h}} \frac{ρ_{S}}{ρ′_{S}} ≤ α (∀)h,

On the Stationary, Potential, Subsonic Flow 29

where S is a simplex in R^{n}, n = 2, 3 (triangle or tethraedron), ρ_{S} = the diameter of the smallest ball containing S, ρ′_{S} = the diameter of the largest ball contained in S.

We proceed with the concept of stable, convergent, internal approximation (W_{h}, p_{h}, r_{h}) for a normed space W ([6]). Definition 2.1. Let be (W, || ||) a normed space and (W_{h}, p_{h}, r_{h}) a family of triples so that:

W_{h} is a normed space, p_{h} : W_{h} → W is a linear, continuous application, r_{h} : W → W_{h}. The approximation (W_{h}, p_{h}, r_{h}) of W is called internal, stable and convergent if:

1. W_{h} ⊂ W, (∀)h,

2. ||p_{h}||_{L(W_{h},W)} ≤ M independently of h,

3. lim_{ρ(h)→0} ||p_{h}r_{h}u − u||_{W} = 0, (∀)u ∈ W where ρ(h) = sup_{S∈T_{h}} ρ_{s}.

We introduce the following notations: (a_{i})_{i=1,n+1} are the vertices of S;

E_{h} the set of vertices of all simplices S ∈ T_{h}; (λ_{i})_{i=1,n+1} = the baricentric coordinates;

V_{h} = {u_{h} : Ω → R | u_{h}|_{S} = ∑_{i=1}^{n+1} u_{h}(a_{i})λ_{i} } .

We denote by (u_{hM})_{M∈E_{h}} a basis in V_{h}, which verifies u_{hM}(M) = 1, (∀)M ∈ E_{h} and

u_{hM}(P) = 0, (∀)P ≠ M, P ∈ E_{h}.

Then the set of functions (w_{hM})_{M∈E_{h}}, defined by

w_{hM}(x) = u_{hM}(x) − \frac{1}{meas(Ω)} ∫_{Ω} u_{hM}(x)dx

is a basis in W_{h} = {v_{h} ∈ V_{h} | ∫_{Ω} v_{h}(x)dx = 0}.

Lemma 2.1. The set W̃ = {u ∈ C^{2}(Ω̄) | ∫_{Ω} u(x)dx = 0} is dense in (W, || ||). Lemma 2.2. Let be (T_{h}) a regular triangulation of the domain Ω,

(2.3)  \begin{matrix}p_{h} : W_{h} → W, p_{h}u_{h} = u_{h}, \\ r_{h} : W → W_{h}, r_{h}v(x) = ∑_{M∈E_{h}} w_{hM}(x)v(M). \end{matrix}

Then the approximation (W_{h}, p_{h}, r_{h}) of W is internal, stable and convergent.

Now, we are able to approximate the minimisation problem (1.4). Theorem 2.1. Let be K_{hb} = {ψ_{h} ∈ W_{h}/|∇ψ_{h}| ≤ b}. The minimisation problem (P_{h}) :

find φ_{h} ∈ K_{hb} so that J_{0}(φ_{h}) ≤ J_{0}(ψ_{h}), ∀ψ_{h} ∈ K_{hb} admits a unique solution, for any h. Moreover, lim_{ρ(h)→0} ||φ_{h} − φ||_{W} = 0 where φ denotes the unique solution of the

problem (1.4).

We denote X_{h} = {μ_{h} ∈ L^{2}(Ω) | μ_{h} = ∑_{S∈T_{h}} μ_{hS}χ_{S} }, where χ_{S} is the characteris-

tic function of the set S and ∧_{h} = {μ_{h} ∈ X_{h}/μ_{h} ≥ 0} .

We state the following two problems: Primal problem. Find φ_{h} ∈ K_{hb}, so that

(2.4)  J_{0}(φ_{h}) ≤ J_{0}(ψ_{h}), (∀)ψ_{h} ∈ K_{hb}.

The minimax problem. Find (φ_{h}, λ_{h}) ∈ W_{h} × Λ_{h}, such that

(2.5)  L(φ_{h}, μ_{h}) ≤ L(φ_{h}, λ_{h}) ≤ L(ψ_{h}, λ_{h}), (∀)(ψ_{h}, μ_{h}) ∈ W_{h} × Λ_{h},

where

(2.6)  L(ψ_{h}, μ_{h}) = J_{0}(ψ_{h}) + ∫_{Ω} μ_{h} (|∇ψ_{h}|^{2} − b^{2}) dx

is the Lagrangean associated to the primal problem. Remark 2.1. According to [4], if (φ_{h}, λ_{h}) is a saddle point for the Lagrangean L, then φ_{h} is solution for the primal problem. Theorem 2.2. The minimax problem (2.5) has a solution.

For the proof, is used the following separation theorem of Hahn-Banach type [5]: Let be V a topological vector space. Suppose T and S are two convex sets in V such

that T has at least one interior point and S does not contain any interior point of T . Then there exists a functional F ∈ V^{∗}, F ≠ 0 and α ∈ R such that

F (x) ≤ α ≤ F (y), (∀)x ∈ T, (∀)y ∈ S.

Remark 2.2. The relation L(φ_{h}, μ_{h}) ≤ L(φ_{h}, λ_{h}), (∀)μ_{h} ∈ ∧_{h} can be rewritten as:

∫_{Ω} (μ_{h} − λ_{h})J_{1}(φ_{h})dx ≤ 0, (∀)μ_{h} ∈ ∧_{h},

where J_{1}(φ_{n}) = |∇φ_{n}|^{2} − b^{2}.

Taking in consideration the variational characterization of the projection on a convex set in a Hilbert space, we infer

λ_{h} = P_{∧_{h}}(λ_{h} + ρJ_{1}(φ_{h}), (∀)ρ > 0,

where P_{∧_{h}} projection operator on ∧_{h}. We proceed with an iterative algorithm for determining the saddle point (φ_{h}, λ_{h}) ∈ W_{h} × ∧_{h}. Theorem 2.3. Let be (φ_{hn})_{n} ⊂ W_{h}, (λ_{hn}) ⊂ ∧_{h} the sequences computed by the following steps: λ_{h0} ∈ ∧_{h} is arbitrary,

(2.7)  L(φ_{hn}, λ_{hn}) ≤ L(ψ_{h}, λ_{hn}), (∀)ψ_{h} ∈ W_{h},

(2.8)  λ_{h,n+1} = P_{∧_{h}}(λ_{hn} + ρ_{n}J_{1}(φ_{hn})).

Then, for ρ_{h} > 0 suitable chosen, φ_{hn} ⟶^{n→∞} φ_{h} in W_{h}. Remark 2.3. The inequality (2.7) is equivalent to the variational equation

\frac{γ}{γ − 1} ∫_{Ω} (k − \frac{1}{2}|∇φ_{hn}|^{2})^{\frac{1}{γ−1}} ∇φ_{hn}∇ψ_{h}dx + 2 ∫_{Ω} λ_{hn}∇φ_{hn}∇ψ_{h}dx =

= \frac{γ}{γ − 1} ∫_{Γ} gψ_{h}dσ, ∀ψ_{h} ∈ W_{h}

Remark 2.4. By the variational characterization of the projection, from eq. (2.8) infer

< λ_{h,n+1}, μ_{h} − λ_{h,n+1} >_{L^{2}(Ω)}≥< λ_{hn} + ρ_{n}J_{1}(φ_{hn}), μ_{h} − λ_{h,n+1} >_{L^{2}(Ω)}, (∀)μ_{h} ∈ ∧_{h},

which is a variational inequality of the form

a (u, v − u) ≤< f, v − u >, (∀) v ∈ K,

where a : X_{h} × X_{h} → R is a bilinear, simetric, coercive form and K = ∧_{h} is a convex set. The variational inequation has an unique solution and is equivalent to the minimization problem:

find u ∈ K, so that J (u) ≤ J (v) , (∀) v ∈ K,

where J (v) = \frac{1}{2}a (v, v) − < f, v >.

Acknowledgements. A version of this paper was presented at the Third Conference of Balkan Society of Geometers, Workshop on Electromagnetic Flows and Dynamics, July 31 - August 3, 2000, University POLITEHNICA of Bucharest, Romania.

References

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[2] P.G. Ciarlet, The finite element method for elliptic problems, North Holland, 1978.

[3] P.G.Ciarlet and P.A.Raviart, General Lagrange and Hermite interpolation in ℜ^{n} with application to finite elements, Arch. Rational Mech. Anal., 46(1972), 3, 177- [4] J.Cea, Lectures on Optimization. Theory and algorithms, Springer Verlag, 1978.

[5] R.B. Holmes, Geometric Functional Analysis and Its Applications, Springer Verlag, 1975.

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[7] I.A.Cristescu, On the stationary, potential, subsonic flow, Results in Mathematics, Birkhauser, 37(2000).

University Politehnica of Bucharest Department of Mathematics I Splaiul Independenţei 313 RO-77206 Bucharest, Romania