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Introduction
In this article, we are interested in the following boundary value problem:
{{\partial \varphi} \over {\partial t}} + \varepsilon {\Delta^2}\varphi - {1 \over \varepsilon}\Delta f(\varphi) + \alpha \varphi = 0,\quad \quad {\kern 1pt} {\rm{in}}\;{\kern 1pt} \Omega \times [0,T],{{\partial \varphi} \over {\partial \nu}} = {{\partial \Delta \varphi} \over {\partial \nu}} = 0,\quad \quad {\kern 1pt} {\rm{on}}\;{\kern 1pt} \Gamma,\varphi (x,0) = {\varphi_0}(x),\quad \quad {\kern 1pt} {\rm{in}}\;{\kern 1pt} \Omega,
in a bounded and regular domain Ω ⊂ ℝn, n ≤ 3, with boundary Γ and T > 0. The initial datum φ0(x) = φ0,1(x) + iφ0,2(x) satisfies the physical constraint |φ0| = 1, where the real part φ0,1(x) represents the initial concentration of the metallic components (the concentration of all phases is between 0 and 1), and the imaginary part
{\varphi_{0,2}}(x) = \sqrt {1 - \varphi_{0,1}^2(x)}
. Furthermore, φ = φ1 + iφ2 is the phase variable.
Equation (1) is the generalization of the original Cahn–Hilliard equation, which plays an essential role in material sciences as it describes the phase separation of binary systems in physics and chemistry. In 1958, Cahn and Hilliard [1,2,3,4] presented the equation (1) in the form of free energy, which later led to the development of the Cahn–Hilliard equation as a partial differential equation based on thermodynamic principles [5]. When a binary solution is cooled down sufficiently, the phase separation may occur in two ways: either by nucleation, in which case nuclei of the second phase appear randomly and grow, or the whole solution appears to nucleate at once, and then periodic or semi-periodic structures appear in the so-called spinodal decomposition. The pattern formation resulting from phase separation has been observed in alloys, glasses, and polymer solutions. The Cahn-Hilliard equation has many applications in material science and biology [6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23].
The function f : ℂ → ℂ belongs to 𝒞2(ℂ, ℂ) and it satisfies the following standard dissipativity assumption:
\mathop {\lim \inf}\limits_{|z| \to \infty} Re(f'(z)) > 0.
One typical choice for this function is
f(z) = |z{|^2}z - z.
In this article, we prove the existence of a unique weak solution to the steady-state problem associated with (1)–(2), using the method of fixed-point arguments. Subsequently, we consider a numerical scheme based on a finite element space discretization in space and Backward Euler discretization in time. After obtaining some error estimates for the semi-discrete solution, we demonstrate the convergence of the semi-discrete solution to the continuous one. Finally, we establish the stability of the Backward Euler scheme, which is the key to achieve the convergence of the fully discrete scheme to the continuous problem.
Our primary objective in this article is to propose a straightforward model for a grayscale multi-component phase separation that preserves the advantages of the phase separation achieved with the Cahn–Hilliard model. Specifically, it is computationally efficient and exhibits rapid convergence times. Notably, we can replicate the results of the two-phase separation by computing only two solutions (the real and imaginary parts of the order parameter), regardless of the number of phases in the initial multi-component metal.
Notations
Setting
\langle \phi \rangle = {1 \over {{\bf{Vol}}(\Omega)}}\int_\Omega \phi (x){\kern 1pt} dx,
we introduce the following spaces:
\overline {{H^{- 1}}(\Omega)} = \{\tau \in {H^{- 1}}(\Omega),{\langle \tau,1\rangle_{{H^{- 1}},{H^1}}} = 0\},\overline {L(\Omega)} = \{\tau \in {L^2}(\Omega),\langle \tau \rangle = 0\},
and
\overline {{H^1}(\Omega)} = \{\tau \in {H^1}(\Omega),\langle \tau \rangle = 0\},
which are the H−1, L2 and H1 spaces with zero spatial average, respectively.
Well-posedness of the steady state problem
In this section, we prove the existence of a weak solution for the stationary problem associated with (1)–(2):
\varepsilon {\Delta^2}\varphi - {1 \over \varepsilon}\Delta f(\varphi) + \alpha \varphi = 0\quad \quad {\rm{in}}\quad \quad \Omega,{{\partial \varphi} \over {\partial \nu}} = {{\partial \Delta \varphi} \over {\partial \nu}} = 0\quad \quad {\rm{on}}\quad \quad \Gamma.
We begin by integrating equation (5) across the domain Ω. Then, taking into account the boundary conditions, we find
\langle \alpha \varphi \rangle = 0.
We now prove the existence of a solution to the variational problem of (5)–(6) as follows.
We consider the fixed point operator
T:{L^2}(\Omega) \to {L^2}(\Omega),\quad \quad \tau \to T(\tau) = \varphi,
where τ is chosen from L2(Ω), and we consider the following equations:
{1 \over r}(\varphi - \tau) + \varepsilon {\Delta^2}\varphi - {1 \over \varepsilon}f(\varphi) + \alpha \varphi = 0\quad {\rm{in}}\quad \Omega,{{\partial \varphi} \over {\partial \nu}} = {{\partial \Delta \varphi} \over {\partial \nu}} = 0\quad {\rm{on}}\quad \Gamma,
where α is a positive constant. Integrating (8) over Ω, we find (7). Therefore, (8) can be rewritten as
({1 \over r} + \alpha)\langle \varphi \rangle = {1 \over r}\langle \tau \rangle.
The variational formulation of (10) reads as follows:
\varepsilon ((\nabla \varphi,\nabla \rho)) + {1 \over \varepsilon}((f(\varphi),\rho)) + \alpha (({(- \Delta)^{- {1 \over 2}}}(\varphi - < \varphi >),{(- \Delta)^{- {1 \over 2}}}\rho)) = 0,
for
\rho {\kern 1pt} \in {\kern 1pt} \overline {{H^1}(\Omega)}
. In addition, the functional of the variational formulation is given by
{\cal F}(\varphi,\tau) = {\cal J}(\varphi) + {1 \over {2r}}||\varphi - \tau - (\langle \varphi \rangle - \langle \tau \rangle >)||_{- 1}^2 + {\alpha \over 2}||\varphi - \langle \varphi \rangle ||_{- 1}^2,
where
{\cal J}(\varphi) = {\varepsilon \over 2}\int_\Omega |\nabla \varphi {|^2}dx + {1 \over \varepsilon}\int_\Omega F(\varphi)dx
and ||.||−1 is the norm defined in
\overline {{H^{- 1}}}
.
Lemma 1
SettingF(z) = {1 \over 4}|z{|^4} - {1 \over 2}|z{|^2}
, we have
F(z) + F(q) - 2F({{z + q} \over 2}) > - {1 \over 4}|z - q{|^2},for all z ∈ ℂ*and z is non null.
The equation (8) has a solution in H1(Ω). Furthermore, if r ≤ ɛ3 (*), then this solution is unique.
Proof
We show that there exists a unique minimizer (say φ*) of ℱ provided that (*) holds. First of all, notice that there are two positive constants c1 and c2, such that
F(\varphi) = {1 \over 4}|\varphi {|^4} - {1 \over 2}|\varphi {|^2} \ge {c_1}|\varphi {|^2} - {c_2}.
Secondly,
\matrix{{{\cal F}(\varphi,\tau) \ge {\varepsilon \over 2}||\nabla \varphi |{|^2} + {{{c_1}} \over \varepsilon}||\varphi |{|^2} - {{{c_2}} \over \varepsilon}} \hfill \cr {+ {1 \over {2r}}[{1 \over 2}||\varphi - \langle \varphi \rangle ||_{- 1}^2 - ||\tau - \langle \tau \rangle ||_{- 1}^2] + {\alpha \over 2}||\varphi - \langle \varphi \rangle ||_{- 1}^2} \hfill \cr {\ge {\varepsilon \over 2}||\nabla \varphi |{|^2} + {{{c_1}} \over \varepsilon}||\varphi |{|^2} + ({1 \over {4r}} + {\alpha \over 2})||\varphi - \langle \varphi \rangle ||_{- 1}^2 + c,} \hfill \cr}
where c is a constant that depends on Ω, ɛ and c2. Consequently, we deduce from (12) that the functional ℱ (φ, τ) is coercive, and hence ℱ has a minimizing sequence φn ∈ H1(Ω). The sequence φn is now bounded in H1(Ω). Therefore, there exists a subsequence of φn that we shall not rename, such that φn converges weakly to φ* ∈ H1(Ω). Additionally, φn converges strongly to φ* in L2(Ω), due to the fact that H1(Ω) is compactly embedded in L2(Ω). We now recall that
{\cal F}({\varphi^n},\tau) = {\cal J}({\varphi^n}) + {1 \over {2r}}||{\varphi^n} - \tau - \langle {\varphi^n} - \tau \rangle ||_{- 1}^2 + {\alpha \over 2}||{\varphi^n} - \langle {\varphi^n}\rangle ||_{- 1}^2,
which implies that
\matrix{{||{\varphi^n} - \tau - \langle {\varphi^n} - \tau \rangle ||_{- 1}^2 - ||{\varphi^*} - \tau - \langle {\varphi^*} - \tau \rangle ||_{- 1}^2} \cr {= (((- {\Delta^{- 1}})({\varphi^n} - \tau - \langle {\varphi^n} - \tau \rangle),{\varphi^n} - \tau - \langle {\varphi^n} - \tau \rangle))} \cr {- (((- {\Delta^{- 1}})({\varphi^*} - \tau - \langle {\varphi^*} - \tau \rangle),{\varphi^*} - \tau - \langle {\varphi^*} - \tau \rangle))} \cr {= (((- {\Delta^{- 1}})({\varphi^n} - {\varphi^*} - \langle {\varphi^n} - {\varphi^*}\rangle),{\varphi^n} - \tau - \langle {\varphi^n} - \tau \rangle))} \cr {+ (((- {\Delta^{- 1}})({\varphi^*} - \tau - \langle {\varphi^*} - \tau \rangle,{\varphi^n} - \tau - \langle {\varphi^n} - \tau \rangle)} \cr {\le ||{\varphi^n} - {\varphi^*} - \langle {\varphi^n} - {\varphi^*}\rangle ||.||{\varphi^n} - \tau - \langle {\varphi^n} - \tau \rangle ||} \cr {+ ||{\varphi^n} - {\varphi^*} - \langle {\varphi^n} - {\varphi^*}\rangle ||.||{\varphi^n} - \tau - \langle {\varphi^n} - \tau \rangle ||,} \cr}
and
\matrix{{||{\varphi^n} - \langle {\varphi^n}\rangle ||_{- 1}^2 - ||{\varphi^*} - \langle {\varphi^*}\rangle ||_{- 1}^2} \cr {= (((- {\Delta^{- 1}})({\varphi^n} - {\varphi^*} - \langle {\varphi^n} - {\varphi^*}\rangle,{\varphi^n} - \langle {\varphi^n}\rangle))} \cr {+ (((- {\Delta^{- 1}})({\varphi^*} - \langle {\varphi^*}\rangle),{\varphi^n} - {\varphi^*} - \langle {\varphi^n} - {\varphi^*}\rangle))} \cr {\le ||{\varphi^n} - {\varphi^*} - \langle {\varphi^n} - {\varphi^*}\rangle ||.||{\varphi^n} - \langle {\varphi^n}\rangle || + ||{\varphi^n} - {\varphi^*} - \langle {\varphi^n} - {\varphi^*}\rangle ||.||{\varphi^*} - \langle {\varphi^*}\rangle ||,} \cr}
and since φn → φ* ∈ L2(Ω) strongly, then
||{\varphi^n} - \tau - \langle {\varphi^n} - \tau \rangle ||_{- 1}^2 \to ||{\varphi^*} - \tau - \langle {\varphi^*} - \tau \rangle ||_{- 1}^2
strongly, and
||{\varphi^n} - \langle {\varphi^n}\rangle || \to ||{\varphi^*} - \langle {\varphi^*}\rangle ||_{- 1}^2
strongly. Furthermore, as F is continuous, we observe that F(φn) converges to F(φ*). By applying Fatou's Lemma, we can deduce that
{\cal F}({\varphi^*},\tau) \le \lim \inf {\cal F}({\varphi^n},\tau).
Therefore, ℱ has a minimizer in H1(Ω), i.e., ∃ φ* ∈ H1(Ω) such that φ* = ar gmin ℱ (φ, τ). Furthermore, with the assistance of the trace function and Neumann boundary conditions, we can easily prove that φ* satisfies the Neumann boundary condition
{{\partial {\varphi^*}} \over {\partial \nu}} = 0
, and φ* serves as a weak solution for the variational problem. Next, we demonstrate the uniqueness of φ*; for this purpose, we prove that the functional ℱ is strictly convex. Let v and w belong to H1(Ω) such that φ = v − w. (Note: ⟨φ⟩ = ⟨v⟩ − ⟨w⟩ = 0). Now, by employing interpolation and Young's inequalities, we obtain
\matrix{{{\cal F}(v,\tau) + {\cal F}(w,\tau) - 2{\cal F}({{v + w} \over 2},\tau)} \hfill \cr {\ge {\varepsilon \over 4}||\nabla \varphi |{|^2} + ({\alpha \over 2} + {1 \over {4r}})||\varphi ||_{- 1}^2 - {1 \over {4\varepsilon}}||\varphi |{|_{- 1}}.||\nabla \varphi ||} \hfill \cr {\ge {\varepsilon \over 4}||\nabla \varphi |{|^2} + ({\alpha \over 2} + {1 \over {4r}})||\varphi ||_{- 1}^2 - {1 \over {4\varepsilon}}({1 \over {2{\varepsilon^2}}}||\varphi ||_{- 1}^2 + {{{\varepsilon^2}} \over 2}||\nabla \varphi |{|^2})} \hfill \cr {\ge ({\alpha \over 2} + {1 \over {4r}})||\varphi ||_{- 1}^2 - {1 \over {4{\varepsilon^3}}}||\varphi ||_{- 1}^2} \hfill \cr {> 0,} \hfill \cr}
under the assumption (*). As a result, ℱ is strictly convex, and the weak solution is unique.
Proposition 3
The operator T has a unique fixed point under the two specified conditions: (*) stated above and (**) defined below,{1 \over {2\alpha - {1 \over {{\varepsilon^2}}}}} \le r \le {1 \over {\alpha - {1 \over {{\varepsilon^2}}} - {{{a^2}\varepsilon} \over 2}}}.
Proof
We show that with the help of the two specified conditions, we can restrict the operator T to a compact convex set. By applying Schauder's fixed-point theorem, we can establish the existence of at least one fixed point, denoted as φ*. Furthermore, we can conclude the uniqueness of φ* based on the property that the functional ℱ is strictly convex. To simplify matters, we denote φ = φ*. Now, let's reframe the problem as follows:
{1 \over r}((\varphi - \tau,\varphi)) + \varepsilon ||\Delta \varphi |{|^2} + {1 \over \varepsilon}((\nabla f(\varphi),\nabla \varphi)) + ((\alpha \varphi,\varphi)) = 0.
Take into account (4), we find
{1 \over r}((\varphi - \tau,\varphi)) + \varepsilon ||\Delta \varphi |{|^2} \le {1 \over \varepsilon}||\nabla \varphi |{|^2} - \alpha ||\varphi |{|^2}.
Therefore,
\matrix{{({1 \over r} + \alpha)||\varphi |{|^2} + \varepsilon ||\Delta \varphi |{|^2} \le {1 \over \varepsilon}||\nabla \varphi |{|^2} + {1 \over {2r}}\int_\Omega |\varphi {|^2}dx + {1 \over {2r}}\int_\Omega |\tau {|^2}dx} \cr {\le {1 \over \varepsilon}||\nabla \varphi |{|^2} + {1 \over {2r}}||\varphi |{|^2} + k,} \cr}
where k is a constant depending on τ, r, and Ω. It then follows that
(\alpha + {1 \over {2r}})||\varphi |{|^2} + \varepsilon ||\Delta \varphi |{|^2} \le {1 \over \varepsilon}||\nabla \varphi |{|^2} + k.
Furthermore, through the utilization of the interpolation inequality followed by Young's inequality, we obtain
\matrix{{||\nabla \varphi |{|^2} \le ||\varphi ||.||\varphi |{|_{{H^2}(\Omega)}} \le {1 \over {2{\varepsilon^2}}}||\varphi |{|^2} + {{{\varepsilon^2}} \over 2}||\Delta \varphi |{|^2} + {{{\varepsilon^2}} \over 2}{{\langle \varphi \rangle}^2}} \cr {\le {1 \over {2{\varepsilon^2}}}||\varphi |{|^2} + {{{\varepsilon^2}} \over 2}||\Delta \varphi |{|^2} + {{{a^2}{\varepsilon^2}} \over 2}{{\langle \tau \rangle}^2}.} \cr}
Thus,
(\alpha + {1 \over {2r}} - {1 \over {2{\varepsilon^3}}})||\varphi |{|^2} + {\varepsilon \over 2}||\Delta \varphi |{|^2} \le {\cal E}||\tau |{|^2} + k,
such that
{\cal E} = {{\varepsilon {a^2}} \over 2}
with
a = {{{1 \over r}} \over {{1 \over r} + \alpha}}
. Under the assumptions (*) and (**), we find
||\varphi |{|^2} = ||T(\tau)|{|^2} \le {\cal E}'||\tau |{|^2} + {\cal K}',
where ℰ′ and 𝒦′ < 1. Therefore, φ remains bounded in L2(Ω), and T now represents a mapping from the closed ball
K = B[0,M] = \varphi, \in,{L^2}(\Omega);\parallel \varphi {\parallel_{{L^2}(\Omega)}} \le M
to itself, with an appropriate constant M > 0.
Furthermore, due to the stationary problem, we obtain the following inequality:
||\Delta \varphi ||_{{L^2}(\Omega)}^2 \le c||\tau ||_{{L^2}(\Omega)}^2 + c'.
Since ⟨φ⟩ is null, we conclude that φ is uniformly bounded in H2(Ω), and it follows that B[0, M] is compact and convex in L2(Ω). It is also clear that T is continuous, which leaves us to show that T is compact. Consider the sequence
{\tau^n} \to \tau \in {L^2}(\Omega),T({\tau^n}) = {\varphi^n};φn is bounded in H1(Ω) for all n. Then, by taking a subsequence (which we do not rename), we have: φn weakly converges to φ ∈ H1(Ω), and φn strongly converges to φ in L2(Ω) using the Rellich-Kondrachov compactness theorem. In addition, since f is continuous, f (φn) converges to f (φ) almost everywhere, and f (φn) is bounded in L2(Ω); then, f (φn) weakly converges to f (φ) in L2(Ω) due to the weak dominated convergence theorem. Thus, φ = T (τ) is a weak unique solution for (8), thanks to the previous proposition, and T is a continuous operator. Finally, by applying Schauder's Theorem, the operator T has a fixed point in L2(Ω), which is the unique solution of the stationary problem.
The variational formulation of (25)–(27) is as follows:
(({\varphi_t},\phi)) = - ((\nabla w,\nabla \phi)) - \alpha ((\varphi,\phi)),((w,\psi)) = {1 \over \varepsilon}((f(\varphi),\psi)) + \varepsilon ((\nabla \varphi,\nabla \psi)),
for all ϕ, ψ ∈ H1(Ω). In our approach, we employ a quasi-uniform family of decompositions denoted as Ωh to effectively partition the domain Ω into k-simplices. Within this discretized framework, given a specific triangulation
{\Omega^h} = \bigcup\limits_{{T^h} \in {\Omega^h}} T
, we establish the conventional P1 conforming finite element space, denoted as Vh. This space, characterized by functions mh belonging to
{C^0}(\overline \Omega)
with the property that mh|T is affine for all T ∈ Ωh, plays a critical role in our numerical analysis. Notably, we observe that Vh is a subset of the more general function space H1(Ω). To facilitate our computations, we introduce the function
I_\varphi^h
, which represents a unique element within Vh and precisely replicates the values of the function φ at the nodes of the triangulation. It is important to note that our methodology aligns with the following well-established standard approximation result, affirming the reliability of our numerical approach
\parallel \varphi - I_\varphi^h{\parallel_{{L^2}(\Omega)}} + h\parallel \varphi - I_\varphi^h{\parallel_{{H^1}(\Omega)}} \le C{h^2}\parallel \varphi {\parallel_{{H^2}(\Omega)}}{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\rm{for}}\,{\rm{all}}{\kern 1pt} {\kern 1pt} \varphi \in {H^2}(\Omega).
Here, C > 0 is a constant that solely depends on Ωh. Additionally, the inverse estimate below still remains valid (refer to [24]).
||{m^h}|{|_{{C^0}(\bar \Omega)}} \le C{h^{{{- n} \over 2}}}||{m^h}|{|_{{L^2}(\Omega)}}{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \;{\rm{for}}\,{\rm{all}}{\kern 1pt} {\kern 1pt} {m^h} \in {V^h}.
The discrete version of (28)–(29) can be written as follows: Find (φh, wh) : [0, T ] → Vh ×Vh such that they satisfy the following conditions:
((\varphi_t^h,\phi)) = - ((\nabla {w^h},\nabla \phi)) - \alpha (({\varphi^h},\phi),(({w^h},\psi)) = {1 \over \varepsilon}((f({\varphi^h}),\psi)) - \varepsilon ((\nabla {\varphi^h},\nabla \psi)),
for all ϕ, ψ ∈ Vh.
Error estimates
Setting
{\varphi^h}(t) - \varphi (t) = {\theta^\varphi} + {\beta^\varphi},{\kern 1pt} {\kern 1pt} {\rm{with}}{\kern 1pt} {\kern 1pt} {\theta^\varphi} = {\varphi^h} - {\varphi_e}^h{\kern 1pt} {\kern 1pt} {\rm{and}}{\kern 1pt} {\kern 1pt} {\beta^\varphi} = {\varphi_e}^h - \varphi,{w^h}(t) - w(t) = {\theta^w} + {\beta^w},{\kern 1pt} {\kern 1pt} {\rm{with}}{\kern 1pt} {\kern 1pt} {\theta^w} = {w^h} - {w_e}^h{\kern 1pt} {\kern 1pt} {\rm{and}}{\kern 1pt} {\kern 1pt} {\beta^w} = {w_e}^h - w,
for all t ∈ [0, T ], where weh = weh(t) represents the elliptic projection of w = w(t), and φeh = φeh(t) is the elliptic projection of φ = φ(t). These projections satisfy the following conditions:
((\nabla {w_e}^h,\nabla \psi)) = ((\nabla w,\nabla \psi))\quad \quad {\rm{for}}\,{\rm{all}}{\kern 1pt} \,{\kern 1pt} \psi {\kern 1pt} \in \overline {{H^1}(\Omega)},(({w_e}^h,1)) = ((w,1)),((\nabla {\varphi_e}^h,\nabla \psi)) = ((\nabla \varphi,\nabla \psi))\quad \quad {\rm{for}}\,{\rm{all}}{\kern 1pt} \,{\kern 1pt} \psi {\kern 1pt} \in \overline {{H^1}(\Omega)},((\varphi_e^h,1)) = ((\varphi,1)).
Using the Lax-Milgram theorem and following the Poincaré inequality, it is evident that, for all
w \in \overline {{H^1}(\Omega)}
, equations (36)–(37) establish a unique solution
{w_e}^h \in \overline {{V^h}(\Omega)}
.
Likewise, for the function
\varphi \in \overline {{H^1}(\Omega)}
, equations (38)–(39) yield a unique solution
{\varphi_e}^h \in \overline {{V^h}(\Omega)}
.
Now, we proceed to define the bilinear form
s(\phi,\psi) = ((\nabla \phi,\nabla \psi)),
which is coercive on
\overline {{H^1}(\Omega)}
, i.e., there exists c0 > 0, such that
s(\phi,\phi) \ge {c_0}\parallel \phi \parallel_{\overline {{H^1}}}^2,\quad \quad {\rm{for}}\,{\rm{all}}{\kern 1pt} {\kern 1pt} \,\phi {\kern 1pt} {\kern 1pt} \in {\kern 1pt} {\kern 1pt} \overline {{H^1}(\Omega)}.
We start by estimating βφ and βw.
Lemma 4
For all φ ∈ H2(Ω), the function φeh ∈ Vh defined by(38)satisfies\parallel {\varphi_e}^h - \varphi {\parallel_{{L^2}(\Omega)}} + h\parallel {\varphi_e}^h - \varphi {\parallel_{{H^1}(\Omega)}} \le C{h^2}\parallel \varphi {\parallel_{{H^2}(\Omega)}}.
Proof
We first have the following equation:
s(\varphi_e^h,{\psi^h}) = s(\varphi,\psi),{\kern 1pt} \,\,{\kern 1pt} {\rm{for}}\,{\rm{all}}\,\,{\kern 1pt} {\kern 1pt} \psi {\kern 1pt} \in {\kern 1pt} \overline {{V^h}}.
As a direct result of (30), we deduce that
||\varphi_e^h - \varphi |{|_{\overline {{H^1}}}} \le Ch||\varphi |{|_{{H^2}(\Omega)}}.
Furthermore, for z ∈ L2(Ω), let ϕ represent the unique solution of
s(\phi,\psi) = ((z,\psi)),{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \,{\rm{for}}\,{\rm{all}}{\kern 1pt} {\kern 1pt} \psi {\kern 1pt} {\kern 1pt} \in \overline {{H^1}}.
Thus, we obtain that
\parallel \phi {\parallel_{{H^2}(\Omega)}} \le C\parallel z{\parallel_{{L^2}(\Omega)}},
where the constant C does not depend on z.
Taking now ψ = φeh − φ in (45), we infer that
\matrix{{((z,\varphi_e^h - \varphi)) = s(\phi,\varphi_e^h - \varphi) = s(\phi - I_\phi^h,\varphi_e^h - \varphi)} \hfill & {\le ||\phi - I_\phi^h|{|_{\overline {{H^1}}}}||\varphi_e^h - \varphi |{|_{\overline {{H^1}}}}.} \hfill \cr}
Moreover, by selecting z = φeh − φ and considering (30) and (44), we obtain
\matrix{{\parallel {\varphi_e}^h - \varphi \parallel_{{L^2}(\Omega)}^2} \hfill & {\le Ch\parallel \phi {\parallel_{{H^2}(\Omega)}}Ch\parallel \varphi {\parallel_{{H^2}(\Omega)}}} \hfill \cr {} \hfill & {\le C{h^2}\parallel {\varphi_e}^h - \varphi {\parallel_{{L^2}(\Omega)}}\parallel \varphi {\parallel_{{H^2}(\Omega)}}.} \hfill \cr}
This inequality, along with inequality (44), yield the result.
In a similar manner, we can establish the existence of a constant c that is solely dependent on Ωh. For all w ∈ H2(Ω), the function weh ∈ Vh, as defined in (36)–(37), complies with the following:
\parallel {w_e}^h - w{\parallel_{{L^2}(\Omega)}} + h\parallel {w_e}^h - w{\parallel_{{H^1}(\Omega)}} \le C{h^2}\parallel w{\parallel_{{H^2}(\Omega)}}.
Next, we define the discrete inverse Laplacian
D_L^{- 1,h}:\overline L \to \overline {{V^h}}
by
D_L^{- 1,h}f = {m^h}
, where
f \in \overline {L(\Omega)}
and
{m^h}{\kern 1pt} \in \overline {{V^h}}
solves
((\nabla {m^h},\nabla {\psi^h})) = ((f,{\psi^h})),\quad \quad {\rm{for}}\,{\rm{all}}{\kern 1pt} \,\,{\kern 1pt} {\psi^h}{\kern 1pt} {\kern 1pt} \in {\kern 1pt} {\kern 1pt} {V^h}.
By expressing the discrete negative semi-norm in the following manner:
\parallel m{\parallel_{- 1,h}} = {((D_L^{- 1,h}m,m))^{{1 \over 2}}} = \parallel \nabla D_L^{- 1,h}m{\parallel_{{L^2}(\Omega)}},\quad \quad {\rm{for}}\,{\rm{all}}{\kern 1pt} {\kern 1pt} \,m{\kern 1pt} {\kern 1pt} \in {\kern 1pt} {\kern 1pt} \overline {L(\Omega)},
and using an orthonormal basis of
\overline {{V^h}}
for the L2(Ω)-scalar product, it becomes evident that the subsequent interpolation inequality is satisfied
\parallel {m^h}\parallel_{{L^2}(\Omega)}^2 \le \parallel {m^h}{\parallel_{- 1,h}}\parallel {m^h}{\parallel_{{H^1}(\Omega)}},{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\rm{for}}\,{\rm{all}}{\kern 1pt} \,{\kern 1pt} {m^h}{\kern 1pt} {\kern 1pt} \in {\kern 1pt} {\kern 1pt} \overline {{V^h}}.
It is also observed that
\parallel f{\parallel_{- 1,h}} \le {c_p}\parallel f{\parallel_{{L^2}(\Omega)}},\quad \quad {\rm{for}}\,{\rm{all}}{\kern 1pt} {\kern 1pt} f{\kern 1pt} \in {\kern 1pt} \overline {L(\Omega)},
where cp is the Poincaré constant. Moreover, we define
\delta (t) = {1 \over {{\kern 1pt} Vol{\kern 1pt} (\Omega)}}(({\theta^\varphi}(t),1)),\quad \quad {\rm{for}}\,{\rm{all}}{\kern 1pt} \,{\kern 1pt} {\kern 1pt} t \ge 0,
so that ((θφ − δ, 1)) = 0.
In the remaining part of this section, the final time T ∈ (0, ∞) is defined, and we express
{\cal Z}(t) = \parallel {\theta^\varphi}\parallel_{{H^1}(\Omega)}^2 + \parallel \theta_t^\varphi - {\delta_t}\parallel_{- 1,h}^2.
We now prove the following lemma.
Lemma 5
Let (φ, w) be a solution of(28)–(29)with sufficient regularity, and let (φh, wh) be a solution of(32)–(33). If R < ∞,
\matrix{{\mathop {\sup}\limits_{t{\kern 1pt} \in {\kern 1pt} [0,T]} \parallel \varphi (t){\parallel_{{C^0}(\overline \Omega)}} < R,} \hfill \cr {\mathop {\sup}\limits_{t{\kern 1pt} \in {\kern 1pt} [0,T]} \parallel {\varphi_t}(t){\parallel_{{C^0}(\overline \Omega)}} \le R,} \hfill \cr {\mathop {\sup}\limits_{t{\kern 1pt} \in {\kern 1pt} [0,T]} \parallel {\varphi^h}(0){\parallel_{{C^0}(\overline \Omega)}} < R,} \hfill \cr}
and
\parallel {\varphi^h}(t){\parallel_{{L^\infty}(\Omega)}} \le R,{\rm{for every}}{\kern 1pt} {\kern 1pt} t \in [0,{T^h}],where Th ∈ (0, T ] is the maximal time, then\matrix{{{\cal Z}(t) + \int_0^t [||{\theta^w}||_{{H^1}}^2 + ({1 \over 2} - {\alpha^2})||\theta_t^\varphi ||_{{H^1}}^2]ds} \hfill \cr {\le C{\cal Z}(0) + C'\int_0^t [||{\beta^\varphi}||_{{L^2}}^2 + ||\beta_{tt}^\varphi ||_{{L^2}}^2]ds} \hfill \cr {+ C'\int_0^t [||{\beta^w}||_{{L^2}}^2 + ||\beta_t^w||_{{L^2}}^2]ds\quad \quad for{\rm{}}all{\kern 1pt} {\kern 1pt} \,t{\kern 1pt} \in {\kern 1pt} [0,{T^h}].} \hfill \cr}
In particular, if ϕ ≡ 1, we obtain
{\delta_t}(t) = {1 \over {{\bf{Vol}}(\Omega)}}((\theta_t^\varphi,1)) = - {1 \over {{\bf{Vol}}(\Omega)}}[((\beta_t^\varphi,1)) + \alpha (({\theta^\varphi} + {\beta^\varphi},1))].
Due to equation (39), we can derive the following:
((\beta_t^\varphi,1)) = 0.
Differentiating (55) with respect to time, we get
((\theta_{tt}^\varphi,1)) = {1 \over {{\bf{Vol}}(\Omega)}}{\delta_{tt}}(t) = - {1 \over {{\bf{Vol}}(\Omega)}}[((\beta_{tt}^\varphi,1)) + \alpha ((\theta_t^\varphi,1)) + ((\beta_t^\varphi,1))],
which yields,
{\delta_{tt}}(t) = {\alpha \over {{\bf{Vol}}(\Omega)}}((\theta_t^\varphi,1)).
Similarly, by subtracting (29) from (33) now, we get:
(({w^h},\psi)) - ((w,\psi)) = {1 \over \varepsilon}((f({\varphi^h}),\psi)) - {1 \over \varepsilon}((f(\varphi),\psi)) - \varepsilon ((\nabla {\varphi^h},\psi)) + \varepsilon ((\nabla \varphi,\nabla \psi)).
Hence,
- (({\theta^w},\psi)) + \varepsilon ((\nabla {\theta^\varphi},\nabla \psi)) = (({\beta^w},\psi)) - {1 \over \varepsilon}((f({\varphi^h}) - f(\varphi),\psi)),
on [0,T], for all ψ ∈ Mh|.
Furthermore, using φ = θw in (54) and
\psi = \theta _t^\varphi
in (58), and then summing the results, we obtain
\matrix{{||\nabla {\theta^w}||_{{L^2}(\Omega)}^2 + {1 \over 2}{d \over {dt}}||\nabla {\theta^\varphi}||_{{L^2}(\Omega)}^2} \hfill & {= - 2((\theta_t^\varphi,{\theta^w})) - ((\beta_t^\varphi,{\theta^w})) + \alpha (({\theta^\varphi},{\theta^w})) + \alpha (({\beta^\varphi},{\theta^w}))} \hfill \cr {} \hfill & {+ (({\beta^w},\theta_t^\varphi)) - ((f({\varphi^h}) - f(\varphi),\theta_t^\varphi)).} \hfill \cr}
In addition, the function f is Lipschitz with constant Lf, therefore
\parallel f({\varphi^h}) - f(\varphi){\parallel_{{L^2}(\Omega)}} \le {L_f}\parallel {\varphi^h} - \varphi {\parallel_{{L^2}(\Omega)}}.
We now estimate ((θw, 1)). We choose ψ ≡ 1 in (58) and use ((βw, 1)) = 0, so the estimate (59) yields
|(({\theta^w},1))| \le {L_f}[||{\theta^\varphi}|{|_{{L^2}(\Omega)}} + ||{\beta^\varphi}|{|_{{L^2}(\Omega)}}]{\bf{Vo}}{{\bf{l}}^{{{- 1} \over 2}}}(\Omega){\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} on{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} [0,{T^h}].
Thanks to inequalities (60) and (58), the triangle inequality, and the generalized Poincaré inequality, we find
\parallel v\parallel_{{L^2}(\Omega)}^2 \le {c'_p}\parallel v\parallel_{{H^1}(\Omega)}^2,{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\rm{for}}\,{\rm{all}}\,{\kern 1pt} {\kern 1pt} {\kern 1pt} v \in {H^1}(\Omega),
and we deduce (53).
Besides, we have that
ab \le \varepsilon {a^2} + {(4\varepsilon)^{- 1}}{b^2},\quad \quad {\rm{for}}\,{\rm{all}}{\kern 1pt} \,{\kern 1pt} a,b \ge 0,{\kern 1pt} {\kern 1pt} \forall {\kern 1pt} \varepsilon > 0.
It then follows from (60)–(62) that
\matrix{{||{\theta^w}|{|_{{H^1}(\Omega)}} + {d \over {dt}}||{\theta^\varphi}|{|_{{H^1}(\Omega)}}} \hfill & {\le {C_1}(||{\beta^\varphi}||_{{L^2}(\Omega)}^2 + ||{\beta^w}||_{{L^2}(\Omega)}^2 + ||\beta_t^\varphi ||_{{L^2}(\Omega)}^2)} \hfill \cr {} \hfill & {\le {C_2}(||{\theta^\varphi}||_{{L^2}(\Omega)}^2 + ||\theta_t^\varphi ||_{{L^2}(\Omega)}^2),} \hfill \cr}
where the constants C1 and C2 depend on Vol(Ω), cp, Lf, and α.
We now must calculate the value of
\theta_t^\varphi
, so if we differentiate equation (59) with respect to time, we get
((\theta_{tt}^\varphi,\phi)) + ((\nabla \theta_t^w,\nabla \phi)) = - ((\beta_{tt}^\varphi,\phi)) + \alpha ((\theta_t^\varphi + \beta_t^\varphi,\phi)).
In addition, if we differentiate (58) with respect to time, we get
- ((\theta_t^w,\psi)) + \varepsilon ((\nabla \theta_t^\varphi,\nabla \psi)) = ((\beta_t^\varphi,\psi)) - {1 \over \varepsilon}(({[f({\varphi^h}) - f(\varphi)]_t},\psi)).
Next, we select
\phi = D_L^{- 1,h}(\theta_t^\varphi - {\delta_t})
in equation (64) and
\psi = \theta_t^\varphi - {\delta_t}
in equation (65). When we combine these equations, we obtain
\matrix{{((\theta_{tt}^\varphi,D_L^{- 1,h}(\theta_t^\varphi - \delta t)) + \varepsilon ||\theta_t^\varphi ||_{{H^1}(\Omega)}^2 = - (((\beta_{tt}^\varphi,D_L^{- 1,h}(\theta_t^\varphi - {\delta_t}))) + ((\beta_t^w,\theta_t^\varphi - {\delta_t}))} \cr {+ \alpha ((\theta_t^\varphi + \beta_t^\varphi,D_L^{- 1,h}(\theta_t^\varphi - {\delta_t}))) - (({{[f({\varphi^h}) - f(\varphi)]}_t},\theta_t^\varphi - {\delta_t})).} \cr}
In the first term on the left-hand side, we can express
\theta_{tt}^\varphi = (\theta_{tt}^\varphi - {\delta_{tt}}) + {\delta_{tt}}.
We should note that
(({\delta_{tt}} + \beta_{tt}^\varphi,1)) = \alpha ((\theta_t^\varphi,1))
according to (57). As for the nonlinear terms, we have
{[f({\varphi^h}) - f(\varphi)]_t} = f'({\varphi^h})[\varphi_t^h - {\varphi_t}] + {\varphi_t}[f'({\varphi^h}) - f'(\varphi)]
and
\alpha {[{\varphi^h} - \varphi]_t} = \alpha (\theta_t^\varphi + \beta_t^\varphi).
With the help of the interpolation inequality (49) applied to
{v^h} = \theta_t^\varphi - {\delta_t}
i.e.
\parallel \theta_t^\varphi - {\delta_t}{\parallel_{{L^2}(\Omega)}} \le \parallel \theta_t^\varphi - {\delta_t}{\parallel_{- 1,h}}\parallel \theta_t^\varphi {\parallel_{{H^1}(\Omega)}},
and the inequalities (50) and (63), along with the Poincaré Inequality, we obtain
\matrix{{{d \over {dt}}||\theta_t^\varphi - {\delta_t}||_{- 1,h}^2 + ||\theta_t^\varphi |{|_{{H^1}(\Omega)}}} \cr {\le {C_3}[||{\delta_{tt}} + {\beta^\varphi}tt||_{{L^2}(\Omega)}^2 + ||\beta_t^w||_{{L^2}(\Omega)}^2 + ||\beta_t^\varphi ||_{{L^2}(\Omega)}^2 + 2||{\beta^\varphi}|| + 2||{\delta_t}||_{{L^2}(\Omega)}^2]} \cr {+ \;{C_4}(||\theta_t^\varphi - {\delta_t}||_{- 1,h}^2 + ||{\theta^\varphi}||_{{L^2}(\Omega)}^2),{\kern 1pt} {\kern 1pt} {\kern 1pt} {\rm{on}}\,{\kern 1pt} {\kern 1pt} [0,{T^h}],} \cr}
for some constants C3 and C4 which depend on R, cp, Lf′,
\mathop {\sup |L'|}\limits_{[- R,R]}
and
\mathop {\sup |f'|}\limits_{[- R,R]}
.
Finally, we add (63) and (69), using the modified Poincaré inequality (61) and the triangular inequality, we get
\parallel \theta_t^\varphi {\parallel_{{L^2}{{(\Omega)}^2}}} \le \parallel \theta_t^\varphi - {\delta_t}\parallel_{{L^2}(\Omega)}^2 + |{\delta_t}{|^2},
the interpolation inequality (68), and inequality (62), we obtain that
\matrix{{{1 \over 2}||\theta_t^\varphi ||_{{H^1}(\Omega)}^2 + ||{\theta^w}|{|_{{H^1}(\Omega)}} + {d \over {dt}}{\cal Z}(t) \le {C_5}(||{\beta^w}||_{{L^2}(\Omega)}^2 + ||{\beta^\varphi}||_{{L^2}(\Omega)}^2 + |{\delta_t}{|^2} + ||\beta_t^w||_{{L^2}(\Omega)}^2 + ||\beta_t^\varphi ||_{{L^2}(\Omega)}^2} \cr {+ ||\beta_{tt}^\varphi ||_{{L^2}(\Omega)}^2 + |{\delta_{tt}}{|^2}) + {C_6}(||\theta_t^\varphi - {\delta_t}|{|_{- 1,h}} + ||{\theta^\varphi}||_{{L^2}(\Omega)}^2 + ||\theta_t^\varphi ||_{{L^2}(\Omega)}^2} \cr {\le {C_5}(||{\beta^w}||_{{L^2}(\Omega)}^2 + ||{\beta^\varphi}||_{{L^2}(\Omega)}^2 + |{\delta_t}{|^2} + ||\beta_t^w||_{{L^2}(\Omega)}^2 + ||\beta_t^\varphi ||_{{L^2}(\Omega)}^2 + ||\beta_{tt}^\varphi ||_{{L^2}(\Omega)}^2 + |{\delta_{tt}}{|^2})} \cr {+ {C_6}(||\theta_t^\varphi - {\delta_t}||_{- 1,h}^2 + ||{\theta^\varphi}||_{{H^1}(\Omega)}^2).} \cr}
Moreover, due to equation (33), we have
|{\delta_t}{|^2} \le {\alpha^2}||{\theta^\varphi}||_{{L^2}(\Omega)}^2
and
|{\delta_{tt}}{|^2} \le {\alpha^2}||\theta_t^\varphi ||_{{L^2}(\Omega)}^2,
which lead to the the following inequality:
\matrix{{({1 \over 2} - {\alpha^2})||\theta_t^\varphi ||_{{H^1}(\Omega)}^2 + ||{\theta^w}||_{{H^1}(\Omega)}^2 + {d \over {dt}}||{\cal Z}(t)||} \cr {\le C(||{\beta^w}||_{{L^2}(\Omega)}^2 + ||{\beta^\varphi}||_{{L^2}(\Omega)}^2 + ||\beta_t^w||_{{L^2}(\Omega)}^2 + ||\beta_{tt}^\varphi ||_{{L^2}(\Omega)}^2) + C'{\cal Z}(t).} \cr}
Therefore, we conclude (52) by applying Gronwall's lemma.
Theorem 6
Let (φ, w) represent a solution to(28)–(29)such that φ, φt, φtt, w, wt ∈ L2(0, T, H2(Ω)), and let (φh, wh) denote the solution to(32)–(33).
We start by differentiating equations (36)–(38) with respect to time, we find that the elliptic projections of φt and wt are respectively (φe)t and (we)t. The same applies to φtt and wtt. Given that φ ∈ C1([0, T ], H2(Ω)) and due to the Sobolev continuous injection property,
{H^2}(\Omega ) \subset {C^0}(\overline \Omega)
, we can conclude that
\varphi,{\varphi_t}{\kern 1pt} {\kern 1pt} \in {\kern 1pt} {\kern 1pt} {C^0}([0,T],{C^0}(\overline \Omega)).
Using the inverse estimate (31), we have
\matrix{{||{\varphi^h}(0) - \varphi (0)|{|_{{C^0}(\overline \Omega)}} \le {C_0}{h^{- {n \over 2}}}(||{\varphi^h}(0) - \varphi (0)|{|_{{L^2}(\Omega)}} + ||\varphi (0) - I_\varphi^h(0)|{|_{{L^2}(\Omega)}})} \cr {+ {{C'}_0}{h^l}||\varphi (0)|{|_{{H^2}(\Omega)}},} \cr}
where l is a real number in (0, 1) ensuring that H2(Ω) is a subset of C0,l(Ω). Thanks to Lemma 4, as well as equations (30) and (70), we obtain
||{\varphi^h}(0) - \varphi (0)|{|_{{C^0}(\overline \Omega)}} < (C{C_0}{h^{2 - {n \over 2}}} + {C'_0}{h^l})||\varphi (0)|{|_{{H^2}(\Omega)}}.
Taking h small enough, we get
||{\varphi^h}(0)|{|_{{C^0}(\overline \Omega)}} < R.
We now assert that 𝒵 (0) ≤ Ch4, where 𝒵 is defined in Lemma 5. Hence
{\cal Z}(0) = \parallel \theta_t^\varphi (0) - {\delta_t}(0)\parallel_{- 1,h}^2.
We follow a similar approach to the proof in Lemma 5. Starting with equation (55) being valid at t = 0, we then substitute
\phi = D_L^{- 1,h}({\theta^\varphi}t(0) - \delta t(0))
into (54), yielding
\matrix{{((\theta_t^\varphi (0),D_L^{- 1,h}(\theta_t^\varphi (0) - {\delta_t}(0)))) + ((\nabla {\theta^w}(0),\nabla D_L^{- 1,h}(\theta_t^\varphi (0) - {\delta_t}(0)))) =} \cr {- ((\beta_t^\varphi (0),D_L^{- 1,h}(\theta_t^\varphi (0) - {\delta_t}(0)))) + \alpha (({\theta^\varphi}(0) + {\beta^\varphi}(0),D_L^{- 1,h}(\theta_t^\varphi (0) - {\delta_t}(0)))).} \cr}
We subsequently use (70) to obtain
((\theta_t^\varphi (0),D_L^{- 1,h}(\theta_t^\varphi (0) - {\delta_t}(0)))) = - ((\beta_t^\varphi (0),D_L^{- 1,h}(\theta_t^\varphi (0) - {\delta_t}(0)))).
Consequently,
||\theta_t^\varphi (0) - {\delta_t}(0)|{|_{- 1,h}} \le ||\beta_t^\varphi (0) + {\delta_t}(0)|{|_{- 1,h}} \le C||\beta_t^\varphi (0) + {\delta_t}(0)|{|_{{L^2}(\Omega)}} \le C{h^2}||{\varphi_t}(0)|{|_{{H^2}(\Omega)}},
where we used Lemma 4, (50) and (53). Therefore, 𝒵 (0) ≤ Ch4, which validates our assertion.
In addition, Lemmas 4 and 5, combined with the estimation in (47) and the regular assumption regarding φ and w, yield the following inequality:
{\cal Z}(t) \le C{h^4},\quad \quad {\rm{for}}\,{\rm{all}}{\kern 1pt} {\kern 1pt} t{\kern 1pt} {\kern 1pt} \in {\kern 1pt} [0,{T^h}].
This inequality, in particular, implies the subsequent result
\parallel {\theta^\varphi}(t){\parallel_{{L^2}(\Omega)}} \le C{h^2},\quad \quad {\rm{for}}\,{\rm{all}}{\kern 1pt} \,{\kern 1pt} t{\kern 1pt} \in {\kern 1pt} [0,{T^h}].
In addition, arguing as in (71), we then deduce that
\sup ||{\varphi^h}(t) - \varphi (t)|{|_{{C^0}(\bar \Omega)}} \to 0,{\kern 1pt} {\kern 1pt} {\kern 1pt} {\rm{as}}{\kern 1pt} {\kern 1pt} {\kern 1pt} h \to 0.
Consequently, by choosing a sufficiently small value of h, we have Th = T . Also, Lemma 4, Lemma 5, and (47) collectively establish the results presented in our theorem.
Stability of the Backward Euler scheme
In this section, we examine the backward Euler scheme with respect to time. After showing that the functional energy decreases during time discretization, we can conclude that our scheme maintains stability. Our initial assumption is that the time step ηt > 0 remains constant.
The numerical scheme is as follows:
(({{\varphi_h^n - \varphi_h^{n - 1}} \over {\eta t}},\phi)) = - ((\nabla v,\nabla \phi)) - \alpha ((\varphi_h^n,\phi)),((v_h^n,\psi)) = {1 \over \varepsilon}((f(\varphi_h^n),\psi)) + \varepsilon ((\nabla \varphi_h^n,\nabla \psi)),
for all ϕ, ψ ∈ Vh.
In what follows, we show the existence, uniqueness, and stability of sequences
((\varphi_h^n),(w_h^n))
.
Theorem 7
For every\varphi_h^0 \in {V^h}
, there exist two sequences,
(\varphi_h^n)and(v_h^n)
, generated by equations(73)–(74), which satisfy the following:{\cal J}(\varphi_h^n) + {\alpha \over 2}||\varphi_h^n|{|^2} + {1 \over {2\eta t}}||\varphi_h^n - \varphi_h^{n - 1}||_{- 1}^2 \le {\cal J}(\varphi_h^{n - 1}) + {\alpha \over 2}||\varphi_h^{n - 1}|{|^2},{\kern 1pt} {\kern 1pt} {\rm{for}}\,{\rm{all}}{\kern 1pt} {\kern 1pt} {\kern 1pt} n \ge 1.
In addition, if ηt < ηt*, where
\eta {t^*} = {{4\varepsilon} \over m}
and
m = {1 \over \varepsilon} + {{\eta t{\alpha^2}} \over 2} + \varepsilon {\eta^2}t{\alpha^2}{\bf{Vo}}{{\bf{l}}^2}(\Omega)
, then these sequences are uniquely defined.
We can see that
{\Pi^h}(w) \ge {\varepsilon \over 2}||\nabla \varphi |{|^2} + ({{{c_1}} \over \varepsilon} + {\alpha \over 2})||w|{|^2} + C.
Since Πh(.) is continuous, it follows that that there exists a solution to the variational problem (76). This solution satisfies Euler-Lagrange's equation
\varepsilon ((\nabla \varphi,\nabla \phi)) + {1 \over \varepsilon}((f(\varphi),\phi)) + \alpha ((D_L^{- 1,h}\varphi,\phi)) + {1 \over {2\eta t}}((\varphi - \varphi_h^{n - 1},\phi)) - ((\phi,1)) = 0,
for all ϕ ∈ Vh.
We set
\varphi_h^n = \varphi
and
v_h^n = \phi - D_L^{- 1,h}({1 \over {\eta t}}(\varphi - \varphi_h^{n - 1}) - \alpha \varphi)
, and we see that
((\varphi_h^n),(w_h^n))
satisfies (73)–(74). By construction, we have
{{\cal J}^h}(\varphi_h^n) \le {{\cal J}^h}(\varphi_h^{n - 1}),
and as a result, we can derive (75).
To establish uniqueness, we consider
{\kappa^\varphi} = {(\varphi_h^n)^1} - {(\varphi_h^n)^2}
and
{\kappa^v} = {(v_h^n)^1} - {(v_h^n)^2}
as the discrepancies between two solutions
({(\varphi_h^n)^i},{(w_h^n)^i})
(where i = 1, 2) of (73)–(74) with respect to a given
\varphi_h^{n - 1}
. Then, (κφ, κv) satisfies
(({\kappa^\varphi},\phi)) = - \eta t((\nabla {\kappa^v},\nabla \phi)) - \eta t.\alpha (({\kappa^\varphi},\phi)),(({\kappa^v},\psi)) = {1 \over \varepsilon}((f{((\varphi_h^n))^1}) - f({(\varphi_h^n)^2}),\psi)) + \varepsilon ((\nabla {\kappa^\varphi},\nabla \psi)),
for all ϕ, ψ ∈ Vh.
By choosing φ = κv and ψ = κφ and subtracting the resulting equations, we obtain
\eta t||\nabla {\kappa^v}|{|^2} + \smallint ||\nabla {\kappa^\varphi}|{|^2} + {1 \over \varepsilon}Re((f(\varphi_h^{n,1}) - f(\varphi_h^{n,2}),{\kappa^\varphi})) - \eta t.\alpha (({\kappa^\varphi},{\kappa^v})) = 0.
Set
\varphi_h^{n,1} = z
,
\varphi_h^{n,2} = z'
in Proposition 3.1 of the reference [25], and observe that
\matrix{{|Re((f(z) - f(z'),{\kappa^\varphi}))| \ge {c_0}\int_\Omega [|{\kappa^\varphi}{|^4} + |z{|^2}.|{\kappa^\varphi}{|^2} + 2Re(\bar z{\kappa^\varphi})]dx - ||{\kappa^\varphi}|{|^2}} \cr {\ge - ||{\kappa^\varphi}|{|^2},} \cr}
hence
\eta t||\nabla {\kappa^v}|{|^2} + \varepsilon ||\nabla {\kappa^\varphi}|{|^2} \le {1 \over \varepsilon}||{\kappa^\varphi}|{|^2} + \eta t\alpha ||{\kappa^\varphi}||.||{\kappa^v}||,
which yields
\eta t||\nabla {\kappa^v}|{|^2} + \varepsilon ||\nabla {\kappa^\varphi}|{|^2} \le ({1 \over \varepsilon} + {{{\alpha^2}.\eta t} \over 2})||{\kappa^\varphi}|{|^2} + {{\eta t} \over 2}||{\kappa^v}|{|^2}.
Let now ϕ = ψ = Vol(Ω) in (79) and (80) and proceeding as above, we have
\langle {\kappa^\varphi}\rangle \le \eta t.\alpha.{\bf{Vol}}(\Omega)||{\kappa^\varphi}||
and
\langle {\kappa^v}\rangle \le {{{k_f}} \over {\varepsilon.{\rm{Vol}}(\Omega)}}||{\kappa^\varphi}||.
Next, by choosing ϕ = m.κφ in (79), we infer that
m{||{\kappa^\varphi}||}^{2} + m.\alpha.\eta t{||{\kappa^\varphi}||}{^2} = - m.\eta t((\nabla {\kappa^\varphi},\nabla {\kappa^\varphi})) \le {{\eta t} \over 2}||\nabla {\kappa^v}|{|^2} + {{{m^2}\eta t} \over 2}||\nabla {\kappa^v}||.
We then deduce the following inequality,
(\varepsilon - {{{m^2}\eta t} \over 4})||{\kappa^\varphi}||_{{H^1}(\Omega)}^2 \le 0.
At the end, since ((θφ, 1)) = 0, the smallness assumption on ηt implies that κφ = 0, and using (3.61) we can see that κw = 0.
Conclusions
In this article, we proposed a complex version of the Cahn-Hilliard-Oono type equation, with applications in grayscale phase separation. Instead of considering the Cahn-Hilliard-Oono system for long interaction phase separation as proposed in [26], we suggested examining a multi-phase metal treated as grayscale, where the concentration of each phase ranges between 0 and 1. We utilized the complex version of the Cahn-Hilliard equation, revealing that the real part of the solution represents the resulting separation.
We established the existence of a unique solution for the stationary problem using Schauder's fixed point theorem. Furthermore, we considered a numerical scheme based on finite element space discretization in space and Backward Euler discretization in time. After deriving error estimates for the semi-discrete solution, we demonstrated the convergence of the semi-discrete solution to the continuous one. Additionally, we proved the stability of the backward Euler scheme, enabling convergence of the fully discrete scheme to the continuous problem.
It is worth noting that numerical simulations are crucial for showcasing the efficiency of the model under investigation in future works. Furthermore, exploring the mathematical aspects of the evolution problem (well-posedness, attractors, convergence) will be a significant focus in future studies. Moreover, for long interaction phase separation and to streamline numerical simulations, we can use the complex version of the Cahn-Hilliard-Oono equation.
Declarations
Conflict of interest
Not applicable.
Funding
Not applicable.
Author's contribution
H.F.-Conception and Design, Methodology, Investigation, Drafting and Editing. M.F.-Conception and Design, Investigation, Drafting and Editing. W.S.-Investigation, Drafting and Editing. Y.A.-Investigation, Drafting and Editing. All authors reviewed the results and approved the final version of the manuscript.
Acknowledgement
The authors wish to thank the referees for their careful reading of the article and useful comments.
Data availability statement
All data that support the findings of this study are included within the article.
Using of AI tools
The authors declare that they have not used Artificial Intelligence (AI) tools in the creation of this article.