A Hybrid High-Order method for the convective Cahn-Hilliard problem in mixed form
Florent Chave Daniele Di Pietro Fabien Marche


A Hybrid High-Order method for the convective Cahn–Hilliard problem in mixed form
A HHO method for the Cahn–Hilliard problem


Florent Chave, Daniele A. Di Pietro and Fabien Marche
University of Montpellier, Institut Montpelliérain Alexander Grothendieck

34095 Montpellier, France

{florent.chave, daniele.di-pietro, fabien.marche}@umontpellier.fr

*
    Florent Chave, Daniele A. Di Pietro and Fabien Marche
    
Abstract
0.9
We discuss effects of the brane-localized mass terms on the fixed points of the toroidal orbifold T^2/Z_2 under the presence of background magnetic fluxes, where multiple lowest and higher-level Kaluza–Klein (KK) modes are realized before introducing the localized masses in general.
Through the knowledge of linear algebra, we find that, in each KK level, one of or more than one of the degenerate KK modes are almost inevitably perturbed, when single or multiple brane-localized mass terms are introduced.
When the typical scale of the compactification is far above the electroweak scale or the TeV scale, we apply this mechanism for uplifting unwanted massless or light modes which are prone to appear in models on magnetized orbifolds.






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We propose a novel Hybrid High-Order method for the Cahn–Hilliard problem with convection. The proposed method is valid in two and three space dimensions, and it supports arbitrary approximation orders on general meshes containing polyhedral elements and nonmatching interfaces. An extensive numerical validation is presented, which shows robustness with respect to the Péclet number.




§ CAHN–HILLIARD EQUATION


Let Ω⊂ℝ^d, d∈{2,3}, denote a bounded connected convex polyhedral domain with Lipschitz boundary ∂Ω and outward normal n⃗, and let t_F>0. The convective Cahn–Hilliard problem consists in finding the order-parameter c : Ω×(0,t_F] →ℝ and the chemical potential w : Ω×(0,t_F] →ℝ such that

  
    d_t c - 1/PeΔw + ∇·(u⃗ c)    = 0           in Ω×(0,t_F
    

    w    = Φ'(c) - γ^2Δc           in Ω×(0,t_F
    

    c(0)    = c_0           in Ω
    

    ∂_n⃗c=∂_n⃗w   =0           on ∂Ω× (0,t_F
  

where γ > 0 is the interface parameter (usually taking small values), Pe>0 is the Péclet number, u⃗ the velocity field such that ∇·u⃗=0 in Ω and Φ the free-energy such that Φ(c) := 1/4 (1-c^2)^2. This formulation is an extension of the Cahn–Hilliard model originally introduced in <cit.> and a first step towards coupling with the Navier–Stokes equations. 

In this work we extend the HHO method of <cit.> to incorporate the convective term in (<ref>). Therein, a full stability and convergence analysis was carried out for the non-convective case, leading to optimal estimates in (h^k+1+τ) (with h denoting the meshsize and τ the time step) for the the C^0(H^1)-error on the order-parameter and L^2(H^1)-error on the chemical potential. The convective term is treated in the spirit of <cit.>, where a HHO method fully robust with respect to the Péclet number was presented for a locally degenerate diffusion-advection-reaction problem.

The proposed method offers various assets: (i) fairly general meshes are supported including polyhedral elements and nonmatching interfaces; (ii) arbitrary polynomial orders, including the case k=0, can be considered; (iii) when using a first-order (Newton-like) algorithm to solve the resulting system of nonlinear algebraic equations, element-based unknowns can be statically condensed at each iteration. 

The rest of this paper is organized as follows: in Section <ref>, we recall discrete setting including notations and assumptions on meshes, define localy discrete operators and state the discrete formulation of (<ref>). In Section <ref>, we provide an extensive numerical validation.



§ THE HYBRID HIGH-ORDER METHOD


In this section we recall some assumptions on the mesh, introduce the notation, and state the HHO discretization.



 §.§ Discrete setting


We consider sequences of refined meshes that are regular in the sense of <cit.>. Each mesh 𝒯_h in the sequence is a finite collection {T} of nonempty, disjoint, polyhedral elements such that Ω = ⋃_T∈𝒯_hT and h = max_T∈𝒯_h h_T (with h_T the diameter of T). For all T∈𝒯_h, the boundary of T is decomposed into planar faces collected in the set ℱ_T. For admissible mesh sequences, card(ℱ_T) is bounded uniformly in h. Interfaces are collected in the set ℱ_h^i, boundary faces in ℱ_h^b and we define ℱ_h := ℱ_h^i∪ℱ_h^b. For all T∈𝒯_h and all F∈ℱ_T, the diameter of F is denoted by h_F and the unit normal to F pointing out of T is denoted by n⃗_TF.

To discretize in time, we consider for sake of simplicity a uniform partition (t^n)_0≤ n ≤ N of the time interval [0,t_F] with t^0=0, t^N = t_F and t^n-t^n-1 = τ for all 1≤ n≤ N. For any sufficiently regular function of time φ taking values in a vector space V, we denote by φ^n∈ V its value at discrete time t^n, and we introduce the backward differencing operator δ_t such that, for all 1≤ n≤ N,

    δ_tφ^n := φ^n - φ^n-1τ∈ V.




 §.§ Local space of degrees of freedom


For any integer l≥ 0 and X a mesh element or face, we denote by ℙ^l(X) the space spanned by the restrictions to X of d-variate polynomials of order l. Let

    U_h^k:=(×_T∈𝒯_hℙ^k+1(T))×(×_F∈ℱ_hℙ^k(F))

be the global degrees of freedoms (DOFs) space with single-valued interface unknowns. We denote by v_h = ((v_T)_T∈𝒯_h,(v_F)_F∈ℱ_h) a generic element of U_h^k and by v_h the piecewise polynomial function such that v_h|_T = v_T for all T∈𝒯_h. For any T∈𝒯_h, we denote by U_T^k and v_T = (v_T, (v_F)_F∈ℱ_T) the restrictions to T of U_h^k and v_h, respectively.
  


 §.§ Local diffusive contribution


Consider a mesh element T∈𝒯_h. We define the local potential reconstruction p_T^k+1:U_T^k →ℙ^k+1(T) such that, for all v_T := (v_T, (v_F)_F∈ℱ_T)∈U_T^k and all z∈ℙ_T^k+1,

    (∇p_T^k+1v_T, ∇ z)_T    =    -(v_T,Δ z)_T + ∑_F∈ℱ_T (v_F, ∇ z·n⃗_TF)_F,

with closure condition ∫_T (p_T^k+1v_T - v_T) = 0.
We introduce the local diffusive bilinear form a_T on U_T^k×U_T^k such that, for all (u_T,v_T)∈U_T^k×U_T^k 

    a_T(u_T,v_T) := (∇p_T^k+1u_T, ∇p_T^k+1v_T)_T + s_T(u_T,v_T),

with stabilization bilinear form s_T:U_T^k×U_T^k→ℝ such that

    s_T(u_T,v_T) := ∑_F∈ℱ_T h_F^-1(π_F^k(u_F-u_T), π_F^k(v_F-v_T))_F,

where, for all F∈ℱ_h, π_F^k:L^1(F)→ℙ^ k(F) denotes the L^2-orthogonal projector onto ℙ^ k(F).



 §.§ Local convective contribution


For any mesh element T∈𝒯_h, we define the local convective derivative reconstruction G_u⃗,T^k+1 : U_T^k →ℙ^k+1(T) such that, for all v_T := (v_T, (v_F)_F∈ℱ_T)∈U_T^k and all w∈ℙ^k+1(T),

    (G_u⃗,T^k+1v_T, w)_T    =    -(v_T,u⃗·∇ w)_T + ∑_F∈ℱ_T(v_F,(u⃗·n⃗_TF)w)_F.

The local convective contribution b_u⃗,T on U_T^k×U_T^k is such that, for all (u_T,v_T)∈U_T^k×U_T^k 

    b_u⃗,T(u_T,v_T) := (G_u⃗,T^k+1u_T, v_T)_T + s_u⃗,T(u_T,v_T).

with local upwind stabilization bilinear form s_u⃗,T:U_T^k×U_T^k→ℝ such that  

    s_u⃗,T(u_T,v_T) := ∑_F∈ℱ_T (|u⃗·n⃗_TF|-u⃗·n⃗_TF/2(u_F-u_T), v_F - v_T)_F.

Notice that the actual computation of G_u⃗,T^k+1 is not required, as one can simply use its definition to expand the cell-based term in the bilinear form b_u⃗,T.



 §.§ Discrete problem


Denote by U_h,0^k := {v_h = ((v_T)_T∈𝒯_h,(v_F)_F∈ℱ_h)∈U_h^k | ∫_Ω v_h = 0} the zero-average DOFs subspace of U_h^k. We define the global bilinear forms a_h and b_u⃗,h on U_h^k×U_h^k such that, for all (u_h,v_h)∈U_h^k×U_h^k

    a_h(u_h,v_h) := ∑_T∈𝒯_h a_T(u_T,v_T),     b_u⃗,h(u_h,v_h) := ∑_T∈𝒯_h b_u⃗,T(u_T,v_T).

The discrete problem reads: For all 1≤ n ≤ N, find (c_h^n,w_h^n)∈U_h,0^k×U_h^k such that

  
       (δ_t c_h^n,φ_h) + 1/Pea_h(w^n_h,φ_h) + b_u⃗,h(c^n_h,φ_h) = 0           ∀φ_h∈U_h^k
    

       (w_h^n,ψ_h) = (Φ'(c_h^n),ψ_h) + γ^2 a_h(c^n_h,ψ_h)           ∀ψ_h∈U_h^k
  

where c_h^0∈U_h,0^k solves a_h(c_h^0,φ_h) = -(Δ c_0,φ_h) for all φ_h∈U_h^k.



§ NUMERICAL TEST CASES


In this section, we numerically validate the HHO method.



 §.§ Disturbance of the steady solution


For the first test case, we use a piecewise constant approximation (k=0), discretize the domain Ω = (0,1)^2 by a triangular mesh (h=1.92· 10^-3) with γ = 5· 10^-2, τ = γ^2 and Pe = 1. The initial condition for the order-parameter and the velocity field are given by

    c_0(x⃗) := tanh (2x_1 - 12√(2)γ^2),   u⃗(x⃗) := 20·[  x_1(x_1-1)(2x_2-1); -x_2(x_2-1)(2x_1-1) ],  ∀x⃗∈Ω.

The result is depicted in Figure <ref> and shows that the method is well-suited to capture the interface dynamics subject to a strong velocity fields. 






 §.§ Thin interface between phases


For the second example, we also use a piecewise constant approximation (k=0) with a Cartesian discretization of the domain Ω = (0,1)^2, where h=1.95· 10^-3. The interface parameter is taken to be very small γ = 5· 10^-3, the time step is τ = 1· 10^-5 and Pe=50. The initial condition for the order-parameter is taken to be a random value between -1 and 1 inside a circular partition of the Cartesian mesh and -1 outside. The velocity field is given by

    u⃗(x⃗) := 1/2(1+tanh (80-200(x_1-0.5,x_2-0.5) _2)) ·[ 2x_2-1; 1-2x_1 ],   ∀x⃗∈Ω.

See Figure <ref> for the numerical result. The method is robust with respect to γ and is also well-suited to approach the thin high-gradient area of the order-parameter.






 §.§ Effect of the Péclet number


The Péclet number is the ratio of the contributions to mass transport by convection to those by diffusion: when Pe is greater than one, the effects of convection exceed those of diffusion in determining the overall mass flux. In the last test case, we compare several time evolutions obtained with different values of the Péclet number (Pe∈{1,50,200}), starting from the same initial condition. We use a Voronoi discretization of the domain Ω = (0,1)^2, where h=9.09· 10^-3, and use piecewise linear approximation (k=1). We choose γ = 1· 10^-2, τ = 1· 10^-4 and t_F = 1. The initial condition is given by a random value between -1 and 1 inside a circular domain of the Voronoi mesh and -1 outside. The convective term is given by

    u⃗(x⃗) := [ sin(π x_1) cos(π x_2); -cos(π x_1)sin(π x_2) ],   ∀x⃗∈Ω.

Snapshots of the order parameter at several times are shown on Figure <ref> for each value of the Péclet number. For each case, the method takes into account the value of Pe and appropriately models the evolution of the order parameter by prevailing advection to diffusion when Pe≫ 1.



spmpsci



The work of D. A. Di Pietro and F. Marche was partially supported by Agence Nationale de la Recherche grant HHOMM (ref. ANR-15-CE40-0005).

