Convergent finite element discretizations of the Navier-Stokes-Nernst-Planck-Poisson system

DOI:10.1051/m2an/2010013 www.esaim-m2an.org

CONVERGENT FINITE ELEMENT DISCRETIZATIONS OF THE NAVIER-STOKES-NERNST-PLANCK-POISSON SYSTEM

Andreas Prohl

and Markus Schmuck

Abstract. We propose and analyse two convergent fully discrete schemes to solve the incompressible Navier-Stokes-Nernst-Planck-Poisson system. The first scheme converges to weak solutions satisfying an energy and an entropy dissipation law. The second scheme uses Chorin’s projection method to obtain an efficient approximation that converges to strong solutions at optimal rates.

Mathematics Subject Classification. 65N30, 35L60, 35L65.

Received September 10, 2008.

Published online February 23, 2010.

1. Introduction

We consider the following electrohydrodynamic model from [11,17,22]:

Let Ω⊂R^N be a bounded Lipschitz domain forN≤3 andT >0. Find a velocity fieldu : Ω×(0, T)→R^N and a corresponding pressure function p : Ω×(0, T) → R, concentrations of positive and negative charges n^± : Ω×(0, T)→R_≥0, and a quasi-electrostatic potential ψ : Ω×(0, T)→Rsuch that

∂tu+ (u· ∇)u−Δu+∇p=f_C in Ω_T := Ω×(0, T) (1.1)

divu= 0 in Ω_T (1.2)

u= 0 on∂Ω_T :=∂Ω×(0, T) (1.3)

∂tn⁺+ div Jn⁺

= 0 in Ω_T (1.4)

J_n⁺,n= 0 on∂Ω_T (1.5)

∂tn⁻+ div Jn⁻

= 0 in Ω_T (1.6)

J_n⁻,n= 0 on∂Ω_T (1.7)

−Δψ=n⁺−n⁻ in Ω_T (1.8)

∇ψ,n= 0 on∂Ω_T (1.9)

Keywords and phrases. Electrohydrodynamics, space-time discretization, finite elements, convergence.

1 Mathematisches Institut, Universit¨at T¨ubingen, Auf der Morgenstelle 10, 72076 T¨ubingen, Germany.

prohl@na.uni-tuebingen.de

2 Department of Chemical Engineering, MIT Boston, 25 Ames Street, Cambridge, MA 02139, USA.schmuck@mit.edu

Article published by EDP Sciences c EDP Sciences, SMAI 2010

for

u(·,0) =u₀, n^±(·,0) =n^±₀, Jn^± :=∓n^±∇ψ− ∇n^±+un^±, and f_C :=−

n⁺−n⁻

∇ψ. (1.10)

Well-posedness of this model has been shown in [22]: weak solutions to the system (1.1)–(1.10) are constructed by Schauder’s fixed point theorem; the concentrations n^± are non-negative and bounded in L^∞(Ω_T) which follows from Moser’s iteration technique; in addition, weak solutions satisfy an energy and an entropy law obtained by the use of special test functions. The local existence of strong solutions is also verified in [22].

The goal of this work is to recover these characteristic properties of weak solutions in a fully discrete setting by using finite elements. A first step in this direction is [20], where boundedness, non-negativity, an energy, and an entropy law of solutions for the Nernst-Planck-Poisson sub-system (1.4)–(1.10) (foru≡0) are transferred from the continuous setting to a spatio-temporal finite element based discretization. Here, we consider the whole system (1.1)–(1.10), which requires to properly account for the interaction of the given sub-system with the fluid flow part in the discretization scheme. As will turn out, a key issue for stability and convergence of the scheme is to establish lower and upper pointwise control for involved discrete charges.

Electrokinetic flows have many applications: An important class of microfluidic and nowadays especially nanofluidic systems aims to perform basic chemical analyses and other processing steps on a fluidic chip. Fluid motion in such chemical (bio)chip systems is often achieved by using electroosmotic flow which enjoys several advantages over pressure driven flows. Briefly, the electroosmotic flow produces a nearly uniform velocity profile which results in reduced sample species dispersion as compared to the velocity gradients associated with pressure-driven flows. This characteristic property enables such applications as fluid pumping, non-mechanical valves, mixing and molecular separation. Many of these systems also employ electrophoresis; this is another electrokinetic phenomena describing the Coulomb force driven motion of suspended molecular species in the solution. Since on a chip electroosmotic and electrophoretic systems grow in complexity, the need of a detailed understanding and computational validation by experimental comparison for such flow models becomes more and more critical. Therefore related reliable numerical schemes are of great importance for design optimization.

For an overview of the applied models describing the electrokinetic flows, it is often customary to distinguish between electroosmosis (no external driving force) and electrophoresis (arising by an external force). For a complete description of these two terms we refer to [11,17]. However, we briefly introduce principle ideas in the following.

We first consider the purely electroosmotic description. When an electrolyte is brought into contact with a solid surface, a spontaneous electrochemical reaction typically occurs between the two types of media resulting in a redistribution of charges. In the cases of interest, an electric double layer (EDL) is formed that consists of a charged solid surface and a region near the surface that supports a net excess of counter-ions. By the assumption that the concentration profile in the ionic region of the EDL can be described by the Boltzmann distribution, one obtains the Poisson-Boltzmann equation for the net charge density

−Δψ=−F

L i=1

zic∞,iexp

−zieψ kT

=:−F

ρ (1.11)

which is usually considered only for one particle, i.e., L = 1. A second approximation is to consider only a symmetric electrolyte such that the right hand side in (1.11) reduces to a sinh function. In a third approxi- mation, called the Debye-H¨uckel limit, one obtains the linear form of the Poisson-Boltzmann equation in case the term ^zeψ_kT is small enough to replace the sinh by its argument. Finally, the velocity field uis described in an arbitrarily shaped micro- or nano-channel for an incompressible liquid viathe linear Stokes equation for the Coulomb driving forcef_C on the right hand side. This linear description allows now to consider separately the velocity components due to the electric field u_ψ and the pressure gradient u_p, where u=u_ψ+u_p solves

Table 1. Scheme A: (I) ={M-matrix (strongly acute mesh, h >0 small enough); existence viaBanach’s fixed point theorem

k≤Ch^N3+β,β >0

}. Scheme B bases on Chorin’s projection scheme.

Algorithm Convergence Scheme Convergence System

A₁ ^{(I), θ→0}−→ A ^{h, k→0}−→ weak solutions of (1.1)–(1.10) B ^{h, k→0}−→ strong solutions of (1.1)–(1.10)

the linear Stokes equation

−Δu=f_C− ∇p in Ω

divu= 0 in Ω (1.12)

for f_C = ρ∇ψ analogously to the electrohydrodynamic model (1.1)–(1.10) and henceρ := n⁺−n⁻. In the computational part (Sect.6) we investigate in Examples 1 and 2 the purely electroosmotic behavior especially in view of the energy and entropy property (Sect. 3) proposed in this article.

These electrophoretic phenomena are induced by applying an electric field, and result in the motion of colloidal particles or molecules suspended in ionic solutions. The application of the Stokes law f_v = 3πμdu allows to balance the electrostatic force q∇ψand the viscous drag f_v associated with its resulting motion. As a result we have

u= q∇ψ

3πμd (1.13)

whereqis the total charge on the molecule,∇Ψ is the applied field, anddis the diameter of the Stokes sphere in a continuum flow. Hence, we consider the species in the liquid to be of sphere shape as in [22]. The above considerations are made for a stationary liquid. The idea is that the electric double layer acts like a capacitor, and suggests that the dynamics can be described in terms of equivalent circuits, where the double layer remains in quasi equilibrium with the neutral bulk is discussed and validated in the thin double layer limit by asymptotic analysis of the Nernst-Planck-Poisson equations [3]. Moreover, in [12], the Nernst-Planck-Poisson equations are recently modified to account for the effect of steric constraints on the dynamics. A more detailed description of the physical derivation and motivations concerning the electrohydrodynamic model (1.1)–(1.10) is given in [21].

In this paper we investigate the incompressible Navier-Stokes-Nernst-Planck-Poisson system (1.1)–(1.10) which is a more general description of electrokinetic flows if compared to the above reduced models for elec- troosmosis and electrophoresis, see [11,17]. First, we introduce a fully implicit Scheme A which allows for non-negativity and a discrete maximum principle for the concentrations, and further validates a discrete energy and entropy law for solutions. All results for Scheme A are obtainedviaan implementable Algorithm A₁which is proven to converge to Scheme A forθ→0, whereθ >0 defines the threshold parameter of the stopping crite- rion in the fixed point iteration, see Table1. Hence, we verify existence of iterates for Scheme A by validating a contraction principle for Algorithm A₁, which requiresk≤Ch^N3+β for anyβ >0. Further, non-negativity and boundedness of iterates of the discrete Nernst-Planck equations in Algorithm A₁are obtainedviatheM-matrix property, provided a compatibility constraint (see (2.6) below) for admissible finite element spaces is met, and used meshes are strongly acute. This latter compatibility requirement accounts for the coupling of the Nernst- Planck system with the incompressible Navier-Stokes system. Then, iterates of Scheme A converge towards weak solutions of the system (1.1)–(1.10) forh, k→0. Moreover, we verify a discrete energy law for solutions of Scheme A, and in two dimensions a discrete entropy dissipation property. The latter discrete (perturbed) entropy estimate is verified in two dimensions for the couplingk≤Ch²of the mesh parameters (h, k), and initial data satisfyingn^±₀ ∈H¹(Ω). Hence, we have to require slightly more regularity on the initial concentrationsn^±₀, and a dimensional restriction compared to the continuous setting.

Let us briefly mention why the energy based approach convinces more than an entropy based approach introduced in [20], where an (unperturbed) entropy law holds without any mesh constraint for the Nernst- Planck-Poisson sub-system (1.4)–(1.10), foru≡0. An entropy based approach does not allow a constructive existence and uniqueness proof via a fully practical fixed point algorithm, enables only quasi-non-negativity of concentrations, does not easily allow for a discrete maximum principle and requires a perturbation of the momentum equation by the entropy-providerSSSε(·) to guarantee a discrete energy law.

In the second part of this article we propose a Scheme B based on Chorin’s projection method [5], which was independently proposed by Temam [24], to construct discrete approximations, where iterates converge to the strong solution of the system (1.1)–(1.10) with optimal rates, see [18,19]. The main advantage of Scheme B is its efficiency; it is, however, that solutions of Scheme B are not known to satisfy physically relevant properties, such as a discrete maximum principle for concentrations, a discrete energy, and an entropy law.

The results are given in Section 3; Section 2 introduces notation. The proofs are given in Section 4 for Scheme A, and in Section5for Scheme B. Comparative computational studies are reported in Section6.

2. Preliminaries 2.1. Notation

We use the standard Lebesgue and Sobolev spaces [1]. To keep the notation simple, let · := · _L². The Poisson equation for homogeneous Neumann conditions, that is

−Δu=f in Ω, ∂u

∂n = 0 on∂Ω, (2.1)

is of special interest to our analysis and concerns the following regularity estimate for 1< p <∞

u_W^2,p≤CfL^p, (2.2)

which is known if we make the following assumption;cf. [8]:

(A1) Let Ω⊂R^N be an open and bounded domain with aC^1,1 boundary, or convex in the case ofN = 2.

We frequently use the following spaces [15], D˜

DD(Ω) ={u∈C₀^∞(Ω,R^N) : divu= 0 in Ω}, V^0,2(Ω) = the closure of ˜DDDinL²= ˜DDD^L

2

, V^1,2(Ω) = the closure of ˜DDDinH₀¹= ˜DDD^H

01

.

We denote the dual space of V^1,2(Ω) byV^−1,2(Ω). Subsequently, Th denotes a quasi-uniform triangulation [4]

of Ω⊂R^N forN = 2,3. LetNh= x

∈L denote the set of all nodes ofTh. We definestrongly acute meshes [6,16] as follows:

The sum of the opposite angles to the common side of any two adjacent triangles is≤π−θ, withθ >0 independent ofh.

This condition is sufficient to validatekββ :=

∇ϕβ,∇ϕβ

≤ −Cθ<0, for β=β, for the stiffness matrix in three dimensions; here,ϕβ is the nodal basis as introduced below. We make the assumption:

(A2) LetTh be a strongly acute triangulation, or forN = 2 a Delaunay triangulation.

LetPdenote the set of all polynomials in two variables of degree ≤. We introduce the following spaces Y_h =

U∈C0(Ω,R^N) : U|K∈P1(K,R^N) ∀K∈ Th

Yh =

ϕ∈C(Ω) : ϕ|K ∈P1(K) ∀K∈ Th

(2.3)

B_h =

U∈C0(Ω,R^N) : U|K ∈P(K,R^N)∩H₀¹(K,R^N) ∀K∈ Th

X_h = Y_h∪B³_h (2.4)

Mh =

Q∈L²₀(Ω)∩C(Ω) : Q|K ∈P1(K) ∀K∈ Th

, (2.5)

where

C0(Ω,R^N) :=

u∈C(Ω,R^N) : u= 0 on∂Ω L²₀(Ω) :=

u∈L²(Ω) : (u,1) = 0 . A well-known example [2] that satisfies the discrete inf-sup condition

U∈Xsuph

(divU, Q)

∇U ≥CQ ∀Q∈Mh, is the MINI-element defined byX_h in (2.4), and byMh in (2.5). Let

V_h=

V∈X_h : (divV, Q) = 0 ∀Q∈Mh

.

The following compatibility condition of spaces

Yh∩L²₀(Ω)⊂Mh, (2.6)

accounts for coupling effects in the electrohydrodynamical system (1.1)–(1.10). We use the nodal interpolation operatorIYh : C(Ω)→Yhsuch that

IYh(ψ) :=

z∈Nh

ψ(z)ϕz

where

ϕz : z∈ Nh

⊂Yh denotes the nodal basis ofYh, andψ∈C(Ω). For functionsφ, ψ∈C(Ω), we define mass-lumping as

(φ, ψ)_h:=

ΩIYh

φψ

dx=

z∈Nh

βzφ(z)ψ(z), φ²_h:= (φ, φ)_h,

whereβz=

Ωϕzdxforz∈ Nh. For all Φ,Ψ∈Yh there holds [7]

Φ ≤ Φh≤(N+ 2)¹²Φ,

(Φ,Ψ)_h−(Φ,Ψ)≤ChΦ∇Ψ. (2.7)

Moreover, in appropriate situations we use the convention with its induced norms [·,·]_i :=

(·,·) fori= 1,

(·,·)h fori= 2, Φ²_i :=

(Φ,Φ) fori= 1, (Φ,Φ)_h fori= 2.

We define the discrete Laplace operatorsL⁽ⁱ⁾_h : H¹(Ω)→Yhfori= 1,2 by −L⁽ⁱ⁾_h φ,Φ

i= (∇φ,∇Φ) ∀Φ∈Yh. (2.8)

Note that there exists a constantC >0, such that for all Φ∈Yhandi= 1,2 there holds

L⁽ⁱ⁾_h Φ ≤Ch⁻²Φ and L⁽ⁱ⁾_h ΦL^∞ ≤Ch⁻²ΦL^∞ ∀Φ∈Yh. (2.9) The following discrete Sobolev inequalities generalize results in [10], Lemma 4.4, in the caseN = 3, fori= 1,2,

∇ΦL³≤C∇Φ^6−N6

L⁽ⁱ⁾_h Φ+ΦH¹

^N₆

∀Φ∈Y_h,

∇Φ_L⁶≤C

L⁽ⁱ⁾_h Φ+Φ_H¹

∀Φ∈Yh. (2.10)

In the sequel, we use the L²-orthogonal projections JVh : L²(Ω,R^N) → V_h, JYh : L²(Ω,R^N)→ Y_h, and JMh : L²₀(Ω)→Mh which satisfy for allu∈L²(Ω,R^N) andq∈L²₀(Ω)

(u−JVhu,V) = 0 ∀V∈V_h (2.11)

(u−JYhu,ΦΦΦ) = 0 ∀ΦΦΦ∈Y_h, (2.12)

(q−JMhq, Q) = 0 ∀Q∈Mh. (2.13)

The following estimates can be found in [10]:

u−JVhu+h∇(u−JVhu) ≤ Ch²D²u ∀u∈V^1,2(Ω)∩H²(Ω,R^N), (2.14)

u−JVhu ≤ Ch∇u ∀u∈V^1,2(Ω). (2.15)

Corresponding approximation results also hold for JYh andu∈H₀¹(Ω)∩H²(Ω).

2.2. Discrete time-derivatives and interpolations

Given a time-step size k >0, and a sequence{U^j}^J_j=1 in some Banach space X, we set dtU^j :=k⁻¹{U^j− U^j−1} for j ≥1. Note that (dtU^j, U^j) = ¹₂dtU^j2+^k₂dtU^j2, if X is a Hilbert space. Piecewise constant interpolations of{U^j}^J_j=1 are defined for t∈[tj−1, tj), and 0≤j≤J by

U(t) :=U^j−1 and U(t) :=U^j, and a piecewise affine interpolation on [tj−1, tj) is defined by

U(t) :=U+U − U k

t−tj−1 .

Further, we employ the spaces^p(0, tJ;X) for 1≤p≤ ∞. These are the spaces of functions{Φ^j}^J_j=0 with the bounded norms

Φ^jp_(0,tJ;X):=

k

J j=0

Φ^jp_X¹_p

, Φ^j∞_(0,tJ;X):= max

1≤j≤JΦ^jX.

3. Main results

We recall the notion of weak solutions for (1.1)–(1.10),cf. [22].

Definition 3.1(weak solution). Assume (A1),N≤3, and 0< T <∞. We call (u, n⁺, n⁻, ψ) a weak solution of (1.1)–(1.10), if

(i) it satisfies forp= 2, ifN = 2, or forp=⁴₃, ifN = 3, that

u∈L²(0, T;V^1,2(Ω))∩L^∞(0, T;V^0,2(Ω))∩W^1,p(0, T;V^−1,2(Ω)), n^±∈L²(0, T;H¹(Ω))∩L^∞(Ω_T)∩W^1,65(0, T; (H¹(Ω))^∗),

ψ∈L²(0, T;H²(Ω))∩L^∞(0, T;H¹(Ω));

(ii) it solves the equations (1.1)–(1.8) in the weak sense for the initial data

u₀∈V^0,2(Ω), n^±₀ ∈L^∞(Ω,R≥0), (3.1) where fort→0 holds

u(·, t)u₀ inL²(Ω,R^N), n^±(·, t) n^±₀ in L²(Ω); (3.2) (iii) it satisfies the following boundary conditions in the trace sense fora.e. t∈[0, T],i.e.,

Jn^±,n

∂Ω= 0, and

∇ψ,n

∂Ω= 0, (3.3)

wheren:∂Ω→R^N is the unit normal on the boundary of Ω,

The weak solution satisfies for a.e. t∈[0, T] the energy and entropy inequalities E(t) +

t 0

e(s) +d(s)

ds≤E(0) (3.4)

W(t) +

t 0

I⁺(s) +I⁻(s)

ds≤W(0), (3.5)

whereW(t) :=WN P P(t) +WIN S(t), for WN P P :=

Ωn⁺ log

n⁺

−1 +n⁻

log n⁻

−1 +1

2|∇ψ|²+ 2 dx WIN S :=

Ω

1

2|u|²dx+

t

0 Ω|∇u|² dxds I^± :=

Ωn^±

∇ log

n^±

−ψ₂ dx E := 1

2

u²+∇ψ²

e := ∇u²+Δψ² d :=

Ω

n⁺+n⁻

|∇ψ|²dx.

The termE(t) in the above Definition3.1contains the physically motivated kinetic energyE¹(u) := ¹₂u², and the energy density of the electric fieldE²(ψ) := ¹₂∇ψ² at timet≥0; furthermore, the termd(t) denotes the total electrical energy of the system at timet≥0.

To construct discrete approximations of the weak solutions given in Definition3.1, we propose the following Scheme A. The main interest and hence the reason for the fully implicit character of the subsequently proposed Scheme A is to preserve most of the characteristic properties of weak solutions given in the above Definition3.1.

Scheme A.

(1) SetU⁰=JVhu₀, and

(N⁺)⁰,(N⁻)⁰ :=

JYhn⁺₀, JYhn⁻₀ .

(2) Forj= 1, . . . , J, letF^j_C:=−((N⁺)^j−(N⁻)^j)∇Ψ^j. Find (U^j,(N^±)^j,Ψ^j)∈V_h×[Yh]³ such that for all (V, Φ^±, Φ )∈V_h×[Yh]³,

(d_tU^j,V) + (∇U^j,∇V) + (∇dtU^j,∇V) +

(U^j−1 · ∇)U^j,V + 1

2

(divU^j−1)U^j,V

= (F^j_C,V), (3.6)

[d_t(N^±)^j,Φ^±]_i+ (∇(N^±)^j,∇Φ^±) ±

(N^±)^j∇Ψ^j,∇Φ^±

−(U^j(N^±)^j,∇Φ^±) = 0, (3.7) (∇Ψ^j,∇Φ) = [(N⁺)^j−(N⁻)^j,Φ]_i, (3.8) where:=h^α with 0< α < ^6−N₃ .

Scheme A incorporates two discretization strategies: One approach uses exact integration [·,·]₁:= (·,·), while the second strategy uses mass lumping [·,·]2:= (·,·)_h which is needed below to validate non-negativity and an entropy law of concentration iterates without any mesh-constraint involvingk, h >0 to hold. In both versions of Scheme A, the stabilization term

∇dtU^j,∇V

with=h^α is introduced to conclude anM-matrix property for the sub-system (3.7) below, and hence accounts for the problematic nature of the coupled overall system.

It is, however, that this term in turn introduces another perturbation error to the problem which dominates the consistency error related to space discretization. Since the Scheme A is fully implicit leading to a coupled nonlinear system, the use of an iterative solver is required; its implicit character allows to recover the properties of solutions from the continuous setting. See also the discussion in Section4.

Remark 3.2. Let us recall the entropy based scheme for the Nernst-Planck-Poisson equations (1.4)–(1.10) (with u ≡ 0) introduced in [20]. We use the notion of an entropy-provider: For any ε ∈ (0,1), we call SSSε:Yh→

L^∞(Ω)_d×d

an entropy-provider if for all Φ∈Yh

(i) SSSε(Φ) is symmetric and positive definite, (ii) SSSε(Φ)∇Ih

F_ε(Φ)

=∇Φ, where

F_ε(x) =

⎧⎨

⎩

xε⁻¹+ lnε−1, ifx≤ε, lnx, ifε≤x≤2,

x2+ ln 2−1, if 2≤x.

This entropy based approach, which allows for an unperturbed entropy law in [20], leads to:

Scheme A’. Fix 0< ε <1, and let

(N⁺)⁰,(N⁻)⁰

∈ Yh

₂

, such that

(N⁺)⁰−(N⁻)⁰,1

= 0. For every j≥1, find iterates

(N⁺)^j,(N⁻)^j,Ψ^j

∈ Yh

₃

, where (Ψ^j,1) = 0 such that for all

Φ₁,Φ₂,Φ₃)∈ Yh

₃ holds dt(N⁺)^j,Φ₁

h+

∇Ψ^j,SSSε

(N⁺)^j

∇Φ₁

+

∇(N⁺)^j,∇Φ₁) = 0, (3.9) dt(N⁻)^j,Φ₂

h−

∇Ψ^j,SSSε

(N⁻)^j

∇Φ₂ +

∇(N⁻)^j,∇Φ₂) = 0, (3.10) ∇Ψ^j,∇Φ3

=

(N⁺)^j−(N⁻)^j,Φ₃

h. (3.11)

In contrast, the extension of this Scheme A’ to the electro-hydrodynamic model (1.1)–(1.10) requires to apply the entropy provider in the Coulomb force termF^j_C in the way

−(SSSε

N⁺

− SSSε

(N⁻)^j)

∇Ψ^j

to verify a discrete energy law, and compensate for the lack of a discrete maximum principle in this scheme.

This, together with the weaker results mentioned in the introduction (Sect. 1) motivate to follow the energy based approach as realized in Scheme A.

With the kinetic energyE¹(U^j) := ¹₂U^j2 and the electric energy densityE²(Ψ^j) := ¹₂∇Ψ^j2 we define the energy of the electro-hydrodynamic system to be

E(U^j,Ψ^j) := 1 2

U^j2+∇Ψ^j2 .

In below, the compatibility condition (2.6) is needed to validate anL^∞(Ω_T)-bound for discrete concentrations.

We state the first main result that is verified in Section4.

Theorem 3.3 (properties of solutions for Scheme A). Assume the initial conditions of Definition 3.1(ii), and(A1). Let (A2) and(2.6)be valid, andh≤h0(Ω) be small enough, such thatk≤Ch^N/3+β for any β >0, and moreover assume h² ≤Ck in the case i = 1. Let 0 ≤ (N^±)⁰ ≤1. Then for every j ≥1, there exists a solution (U^j, Π^j,(N^±)^j,Ψ^j)∈V_h×Mh×

Yh

₃

, such that(3.6)–(3.8)hold. Furthermore, 0≤(N^±)^j≤1 (1≤j≤J),

and for i= 1,2 it holds

(i) E(U^J,Ψ^J) +

2∇U^J2+k² J j=1

E(dtU^j, dtΨ^j) +

2dt∇U^j2 +k

J j=1

∇U^j2+(N⁺)^j−(N⁻)^j2_h +k

J j=1

(N⁺)^j+ (N⁻)^j ,|∇Ψ^j|

=E(U⁰,Ψ⁰) +

2∇U⁰²,

(ii) 1

2

U^J2+∇U^J2 +k²

2 J j=1

dtU^j2+∇dtU^j2 +k

J j=1

∇U^j2

≤CE(U⁰,Ψ⁰) +

2∇U⁰², (iii) 1

2

(N⁺)^J2_i +(N⁻)^J2_i +k²

2 J j=1

dt(N⁺)^j2_i +dt(N⁻)^j2_i +k

2 J j=1

∇(N⁺)^j2+∇(N⁻)^j2

≤CE(U⁰,Ψ⁰) +1 2

(N⁺)⁰²_i +(N⁻)⁰²_i , (iv) k

J j=1

dt(N⁺)^j2_(H1)^∗ +dt(N⁻)^j2_(H1)^∗

≤C

E(U⁰,Ψ⁰) +

(N⁺)⁰²+(N⁻)⁰² ,

(v) k

J j=

U^j−U^j−2≤C(k)^1/4 ∀0≤≤J.

The estimates (i)–(iii) provide uniform bounds in time which control the long time behavior of iterates, whereas (iv) and (v) are necessary for a compactness property needed in the convergence proof.

We introduce a practical Algorithm A₁ in Section 4 that is a simple fixed-point scheme, together with a suitable stopping criterion to verify the statements of Theorem3.3.

Motivated by the entropy estimate for (1.1)–(1.10), we recover the proof from there in a fully discrete setting.

Therefore, we introduce the entropy functional J →W^J :=E

U^J,Ψ^J +

2∇U^J2+

Ω

IYh

F

(N⁺)^J

+F

(N⁻)^J

+ 2

dx, (3.12) whereF(x) :=x(lnx−1), and herewith we extend the version in [20].

Theorem 3.4 (entropy law for Scheme A). Let n^±₀ ∈H¹(Ω),(A2),(2.6),N = 2,i= 2, andk≤Ch² be valid for some T :=tJ>0. Suppose that δ≤(N^±)⁰≤1 for some0< δ < ¹₂, and let

U^j,(N^±)^j,Ψ^j_J

j=1 solve the SchemeA. Then, for all 0≤j < j≤J,

W^j +k² 2

j

l=j+1

∇dtΨ^l2+dt∇U^l2 +k

j

l=j+1

(N⁺)^l,∇ Ψ^l+Ih

F((N⁺)^l)² +∇U^l2+

(N⁻)^l,∇ Ψ^l− Ih

F((N⁻)^l)²

≤W^j+Chδ⁻⁴

E(U⁰,Ψ⁰) +∇(N⁺)⁰²_h+∇(N⁻)⁰²_h2

. (3.13)

Asymptotically, the dissipation ofW^j in (3.13) is then guaranteed for δ:=h¹⁴⁻ and ¹₄ > >0 arbitrarily small.

The main convergence result concerning Scheme A is:

Theorem 3.5 (convergence of Scheme A). Assume the initial conditions of Definition 3.1(ii). Suppose (A1), (A2), (2.6), and0< tJ <∞. Let0≤(N⁺)⁰,(N⁻)⁰≤1,(n⁺₀, n⁻₀)∈

L^∞(Ω)₂

, as well as

U⁰u₀ in L²(Ω,R^N), (N⁺)⁰ n⁺₀,(N⁻)⁰ n⁻₀ inL²(Ω),

h→0limh^α/2∇JVhu₀= 0.

Let(UUU,Π,N^±,ΨΨΨ )be constructed from the solution

U^j,Π^j,(N^±)^j,Ψ^j_J

j=1⊂V_h×Mh× Yh

₃

of SchemeA by piecewise affine interpolation as outlined in Section 2.2. Then, for h, k→0 such thath²≤Ck in the case i= 1, ork≤Ch^N3+βfor someβ >0in the casei= 2, there exists a convergent subsequence

UUU,Π,N^±,ΨΨΨ

k,h

whose limit is a weak solution of(1.1)–(1.10).

In Section5, we analyze a time-splitting scheme based on Chorin’s projection method [5,25]; we refer to [18], and [9] for a recent review of this and related methods to solve the incompressible Navier-Stokes equations. In this scheme, the computation of iterates is fully decoupled in every time-step, which leads to significantly reduced computational effort. However, this strategy sacrifices the discrete energy and entropy inequalities, which are relevant tools to characterize long-time asymptotics and convergence towards weak solutions. Therefore, instead, the related numerical analysis requires the existence of (local) strong solutions which is verified for the system (1.1)–(1.10) in [22].

Definition 3.6(strong solution). Let 0< T≤ ∞. The weak solutions (u, n⁺, n⁻, ψ) are called strong solutions of (1.1)–(1.10), if

(i)

u ∈ L^∞(0, T;V^1,2(Ω)∩H^2,2(Ω))∩W^1,2(0, T;V^1,2(Ω)) n^± ∈ L^∞(0, T;H²(Ω))∩W^1,∞(0, T;L²(Ω))∩W^1,2(0, T;H¹(Ω))

ψ ∈ C([0, T];H²(Ω))∩W^1,∞(0, T;W^1,2(Ω)) p ∈ L^∞(0, T; (H¹∩L²₀)(Ω)),

(ii) the initial conditions

u₀∈V^1,2(Ω)∩H²(Ω), n^±₀ ∈H²(Ω), (3.14) are attained fort→0,

u(·, t)→u₀ in H¹(Ω,R^N), n^±(·, t)→n^±₀ in H¹(Ω). (3.15) A slightly weakened notion of strong solution is studied in [22], and (local) existence of strong solutions as defined here then follows by a simple bootstrapping argument. Recall that in dimension N = 3 the time T =T(u₀)>0 is in general finite, see [15,22,25] for example. Moreover, forN ≤3 andt∈[0, T], the following energy and entropy identities hold for strong solutions,

E(t) +

t

0 e(s) +d(s) ds=E(0), (3.16) W(t) +

t

0 I⁺(s) +I⁻(s) ds=W(0), (3.17) see [22].

For convenience, we say that a quadruple (u, p, n^±, ψ) is in S, if it satisfies the regularity properties i) of Definition 3.6. To approximate the strong solutions of Definition3.6, we propose the following time-splitting:

Scheme B.Letj≥1 and

u^j−1,(n^±)^j−1

, determine

u^j,(n^±)^j, ψ^j

as follows:

1. Start withu⁰=u₀,and (n^±)⁰=n^±₀. 2. Letj≥1. Computeψ^j−1∈H¹(Ω) from

−Δψ^j−1 = (n⁺)^j−1−(n⁻)^j−1 in Ω

∇ψ^j−1,n = 0 on∂Ω. 3. Compute (n^±)^j∈H¹(Ω) via

1 k

(n^±)^j−(n^±)^j−1

−Δ(n^±)^j±div((n^±)^j∇ψ^j−1) + (u^j−1· ∇)(n^±)^j = 0 in Ω

∇(n^±)^j±(n^±)^j∇ψ^j−1,n = 0 on∂Ω. 4. Find ˜u^j ∈H₀¹(Ω,R^N) by solving

1 k

u˜^j−u^j−1

−Δ˜u^j+ (u^j−1· ∇)˜u^j = −

(n⁺)^j−(n⁻)^j

∇ψ^j−1

˜

u^j = 0 on∂Ω.

5. Determine the tuple u^j, p^j

∈V^0,2×H¹∩L²₀that solves the system 1

k

u^j−u˜^j

+∇p^j = 0, divu^j= 0 on Ω, (3.18)

u^j,n = 0 on∂Ω. (3.19)

Step 5 is known as Chorin’s projection step. Using the div-operator in (3.18) amounts to solving a Laplace- Neumann problem for the pressure iterate,

−Δp^j=−1

kdiv ˜u^j in Ω, ∂np^j|= 0 on∂Ω, (3.20) followed by an algebraic update for the present solenoidal velocity field,

u^j = ˜u^j−k∇p^j in Ω. (3.21)

The goal of the second part (Sect. 5) of this paper is to analyze Scheme B by investigating its stability and approximation properties. Therefore we propose a series of auxiliary problems to separately account for inherent time-discretization, decoupling effects, and those attributed to the quasi-compressibility constraint (3.20). For this purpose, the following notation is useful.

We say that 1. the quadruple

u, p, n^±, ψ

:=

ξi

₄

i=1 ∈ S satisfies property (P1), if the following is satisfied for i∈ {1,3},

k J j=1

dtξ^j_i²_H1+dtξ₄^j2_H2

+ max

1≤j≤J

dtξ_i^j2+ξ_i^j2_H2+ξ₂^j2_H1+ξ₄^j2_H2

≤C;

2. the quadruple ξ_i^j₄

i=1 ∈ S satisfies property (P2)_l, for l ∈ {0,1}, if the following approximation properties are satisfied:

0≤j≤Jmax √

τ^ju(tj)−ξ₁^j+τ_l^jp(tj)−ξ^j_2H⁻¹+ψ(tj)−ξ₄^j+ψ(tj)−ξ₄^j_H¹+n^±(tj)−ξ^j₃ +√

k

u(tj)−ξ₁^jH¹+

τ_l^jp(tj)−ξ^j₂+n^±(tj)−ξ₃^jH¹

≤Ck,

where

τ_l^j :=

1, ifl= 0, min

1, tj

, ifl= 1.

With a slight abuse of notation, iterates{ξi}⁴_i=1 are here considered as continuous piecewise affine, continuous interpolants in time of corresponding time iterates. The property (P2)₀ is used in the analysis of Scheme B.

The generic constantCis independent ofk, and depends only on the given data. In the following theorem, we state the main result concerning optimal convergence behaviour of the solution obtainedviaScheme B.

Theorem 3.7 (convergence of Scheme B). Suppose (A1), the initial and boundary conditions from Defini- tion 3.6, let u₀∈V^1,2∩H²,n^±₀ ∈H²(Ω), and0≤tJ ≤T. Then the solution

u^j,(n⁺)^j,(n⁻)^j, ψ^j_J

j=1⊂Sof SchemeBsatisfies the properties (P1) and(P2)₁ for sufficiently small time-stepsk≤k0(tJ).

The proof of this result is based on optimal estimates for Chorin’s scheme in the context of incompressible Navier-Stokes equations in [18], and a combining analysis of further splitting, regularization, and perturbation effects in Scheme B. If we additionally include the error effects of a corresponding space discretization which uses the setup of (2.3)–(2.5), we arrive at the following: