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(Color figure.) (a) Snapshots from a simulation of spinodal decomposition using quadratic elements, q = 2. The domain Ω is a dodecagon of diameter 3.2, and we use a uniform triangulation consisting of 12288 elements. The base level mesh consists of 12 triangles, and there are exactly 5 levels of global quadrisection refinement to obtain the final mesh. The final time is T = 0.02, and τ = 10 −6 . The interface parameter is ε = 0.0125; u 0 ≈ 3.0 × 10 −03 ; and θ = 0.0. (b) The discrete Cahn-Hilliard energy (32) corresponding to the simulation in (a), plotted as a function of time. We observe that, as predicted in Lemma 3.4, the energy is nonincreasing in time.
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In this paper we devise and analyze a mixed discontinuous Galerkin finite element method for a modified Cahn-Hilliard equation that models phase separation in diblock copolymer melts. The time discretization is based on a convex splitting of the energy of the equation. We prove that our scheme is unconditionally energy stable with respect to a spat...
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... the first frame of Fig. 4a. If θ = 0.0 and u 2 0 ≤ 1 3 -i.e, the initial mass average is in the so called chemical spinodal region -then the system modeled by (1a) -(1d) is unstable and the solutions can evolve as depicted in Fig. 4a (and also in Fig. 1, third ...
Context 2
... from a DGFE simulation of spinodal decomposition are shown in Fig. 4a, and parameters for the test are given in the caption of the same figure. Initially a very fine-scale structure, comprised of alternating layers of (nearly) pure phase regions, emerges. Afterwards, certain of these pure phase regions grow, and some shrink, a process known as coarsening. Coarsening occurs on a very slow time scale, ...
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... Initially a very fine-scale structure, comprised of alternating layers of (nearly) pure phase regions, emerges. Afterwards, certain of these pure phase regions grow, and some shrink, a process known as coarsening. Coarsening occurs on a very slow time scale, which can be seen in the plot of the fully discrete energy for the simulation given in Fig. 4b. The energy decreases rapidly initially, then decreases very slowly Table 2. Convergence tests using quadratic (q = 2) elements. The exact solution u is given in (71). The final time is T = 1.5, and the refinement paths are taken to be quadratic and linear, respectively. The parameters are ε = 0.5, θ = 0.0, Ω = (0, 1) 2 . The global ...
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Citations
... Recently, some efforts have been made to develop first-order schemes which are accurate in time and energy-stable for the diblock copolymer model (cf. [2,9,10]). These schemes are primarily based on either the nonlinear convex splitting approach (cf. ...
... These schemes are primarily based on either the nonlinear convex splitting approach (cf. [9,11]) or the linear stabilization approach (cf. [12][13][14][15][16][17][18][19][20][21][22][23][24]). ...
An efficient modified leapfrog time-marching scheme for the diblock copolymer model is investigated in this paper. The proposed scheme offers three main advantages. Firstly, it is linear in time, requiring only a linear algebra system to be solved at each time-marching step. This leads to a significant reduction in computational cost compared to other methods. Secondly, the scheme ensures unconditional energy stability, allowing for a large time step to be used without compromising solution stability. Thirdly, the existence and uniqueness of the numerical solution at each time step is rigorously proven, ensuring the reliability and accuracy of the method. A numerical example is also included to demonstrate and validate the proposed algorithm, showing its accuracy and efficiency in practical applications.
... For the initial boundary value problem of CHO equation together with regular potential [22] studied the well-posedness and existence of global and exponential attractors, and Aristotelous et. al. [1], Farrell and Pearson [10] established the numerical schemes. On the other hand, for the initial boundary value problem of CHO equation with singular potential, Giorgini et al. [11] proved some regularization properties of the (unique) solution in finite time, proved the so-called strict separation property in dimension two and established the existence of the global and exponential attractors. ...
In this paper, we are concerned with the well-posedness and large time behavior of Cauchy problem for Cahn–Hilliard equation in \(\mathbb {R}^n\) (\(n\in \mathbb {Z}^+,~n\ge 3\)). First, based on the higher-order norm estimates of solutions and the mollifier technique, we obtain the local well-posedness of strong solutions. Then, by using pure energy method, standard continuity argument together with negative Sobolev norm estimates, one proves the global well-posedness and time decay estimates provided that the \(H^{\left[ \frac{n}{2}\right] +1}\)-norm of initial data is sufficiently small.
... Choose interface parameter ε = 0.01, λ = 7.5 × 10 3 and time step Δt = 0.0005. Figures 25,26,27,28,29,30,31,32,33,34,35,and 36 show the segmentation of the initial image with different iterations and the corresponding discrete adaptive Corresponding adaptively meshes at different iterations polygonal mesh. The last two tests consist of a fluid channel that is surrounded and blocked by a porous domain. ...
In this paper, we develop a polygonal mesh adaptation algorithm for a fully implicit scheme based on discontinuous Galerkin (DG) finite element methods in space and backward Euler method in time to solve the Allen-Cahn equation. The mathematical framework and a procedure for solving this nonlinear equation are given. We extend the DG discretization with a polygonal mesh adaptation method to save computation time and capture the thin interfaces more accurately. A criterion based on the local value of the phase field function gradient is used to select the target element for refinement and coarsening, and then a 4-node polygonal mesh refinement strategy is adopted by connecting the midpoint of each edge to the barycenter of the target element. Using numerical tests, including motion by the mean curvature, curvature-driven flow, the Allen-Cahn equation with a logarithmic free energy, the Allen-Cahn equation with advection, and the application for image segmentation, we verify the accuracy, efficiency, and capabilities of the adaptive DG on polygonal meshes and confirm the decreasing property of the discrete energy.
... To this end, fully non-conforming discontinuous Galerkin (DG) methods are a suitable choice. For the use of DG methods in the numerical solution of the mixed Cahn-Hilliard system, we refer to [12,13,14] for the LDG method and [15,16,17] for the IPDG method. ...
... However, to the best of our knowledge, the theoretical analysis of the hybridized IPDG methods in [2,4] is missing from the literature. We remark that the additional facet unknowns in our HDG scheme precludes the use of discrete functional analysis tools used previously in [15,16] to analyze the IPDG method. Fortunately, appropriate analogues of these tools have been extended to the HHO setting in [21], and using similar techniques we will show that they hold also in the HDG setting. ...
... Returning to (52), using (53), the continuity of the bilinear form a D (30), (16), (54) and (49), we obtain ...
The mixed form of the Cahn-Hilliard equations is discretized by the hybridizable discontinuous Galerkin method. For any chemical energy density, existence and uniqueness of the numerical solution is obtained. The scheme is proved to be unconditionally stable. Convergence of the method is obtained by deriving a priori error estimates that are valid for the Ginzburg-Lindau chemical energy density and for convex domains. The paper also contains discrete functional tools, namely discrete Agmon and Gagliardo-Nirenberg inequalities, which are proved to be valid in the hybridizable discontinuous Galerkin spaces.
... e modified Cahn-Hilliard equation was introduced by Shahriari [18] to suppress the phase coarsening. A discontinuous Galerkin method coupled with convex splitting for the modified Cahn-Hilliard equation was proposed by Aristotelous et al. [19]. Other related researches for the modified Cahn-Hilliard equation can be seen in Refs. ...
A decoupled unconditionally stable numerical scheme for the modified Cahn–Hilliard–Hele–Shaw system with logarithmic potential is proposed in this paper. Based on the convex-splitting of the associated energy functional, the temporal discretization of the scheme is given. The fractional step method is used to decouple the nonlinear modified Cahn–Hilliard equation from the pressure gradient. Then, at each time step, one only needs to solve Poisson’s equation which is obtained by using an incremental pressure-stabilization technique. In terms of logarithmic potential, using the regularization procedure can make the domain extended from − 1,1 to − ∞ , ∞ for F ϕ . Finally, the proof of the energy stability and the calculation of the error estimate are presented. Numerical examples are recorded to illustrate the accuracy and performance for the proposed scheme.
... Meanwhile, DG might have high performance in simulating the tumor boundaries with highly irregular properties [44]. It can be seen that there exists an increasing interest in applying the DG methods for solving the phase-field related models, see [2,18,29,57]. However, as far as the author knows, if the logarithmic Flory-Huggins potential is used, then for either the Cahn-Hilliard-Brinkman-Ohta-Kawaski model studied in this article or those widely concerned models like the Cahn-Hilliard-Brinkman or Cahn-Hilliard-Navier-Stokes system, how to use the DG method for the spatial discretization to obtain the decoupled and energy stable fully discrete scheme has not been resolved successfully. ...
In this article, we consider the Cahn-Hilliard-Brinkman-Ohta-Kawasaki tumor growth system, which couples the Brinkman flow equations in the porous medium and the Cahn-Hilliard type equation with the nonlocal Ohta-Kawasaki term. We first construct a fully-decoupled discontinuous Galerkin method based on a decoupled, stabilized energy factorization approach and implicit-explicit Euler method in the time discretization, and strictly prove its unconditional energy stability. The optimal error estimate for the tumor interstitial fluid pressure is further obtained. Numerical results are also carried out to demonstrate the effectiveness of the proposed numerical scheme and verify the theoretical results. Finally, we apply the scheme to simulate the evolution of brain tumors based on patient-specific magnetic resonance imaging, and the obtained computational results show that the proposed numerical model and scheme can provide realistic calculations and predictions, thus providing an in-depth understanding of the mechanism of brain tumor growth.
... Alternatively, the Cahh-Hilliard problem can be split into a coupled of lower-order differential equations and mixed formulation can be employed for discretization at the expense of introducing additional unknowns, see, e.g., [46,58,10,62,32,64]. Recently, in [52] isogeometric analysis has been employed to discretize the Cahn-Hilliard problem, whereas in [63] the same approach has been used to discretize the advective Cahn-Hilliard equation. ...
We consider the $C^1$-Virtual Element Method (VEM) for the conforming numerical approximation of some variants of the Cahn-Hilliard equation on polygonal meshes. In particular, we focus on the discretization of the advective Cahn-Hilliard problem and the Cahn-Hilliard inpainting problem. We present the numerical approximation and several numerical results to assess the efficacy of the proposed methodology.
... where φ 0 := 1 |Ω| Ω φ 0 (x)dx. More works on the modified Cahn-Hilliard equation can be found in [18][19][20][21]. For the modified Cahn-Hilliard equation, when θ = 0, the equation becomes the classical Cahn-Hilliard equation [22][23][24]. ...
In this paper, a decoupled finite element method for a modified Cahn-Hilliard-Hele-Shaw system with double well potential is constructed. The proposed scheme is based on the convex splitting of the energy functional in time and uses the mixed finite element discretzation in space. The computation of the velocity u is separated from the computation of the pressure p by using an operator-splitting strategy. A Possion equation is solved to update the pressure at each time step. Unconditional stability and error estimates are analyzed in detail theoretically. Furthermore, the theoretical part is verified by several numerical examples, whose results show that the numerical examples are consistent with the results of the theoretical part.
... This counter-intuitive fact has been recently analyzed in [61], where the authors prove that the convex-splitting scheme for the CH model is exactly the same as the fullyimplicit for a different model that is a nontrivial perturbation of the original CH, and the gain of stability is at the expense of a possible loss of accuracy. In any case, the convex-splitting technique is a mainstream ingredient in the construction of energy-stable schemes for the CH equation, and has been successfully applied in various discretization strategies such as finite differences [33,36], finite volumes [27], finite elements [7,9,28,58], spectral methods [38] and discontinuous Galerkin schemes [2] (see [57] for an extensive review of energy-stable schemes for the CH equation). A recent and promising strategy to design energy-stable schemes is the so-called scalar auxiliary variable [50,51]. ...
We propose finite-volume schemes for the Cahn-Hilliard equation that unconditionally and discretely satisfy the boundedness of the phase field and the free-energy dissipation. Our numerical framework is applicable to a variety of free-energy potentials including the Ginzburg-Landau and Flory-Huggins, general wetting boundary conditions and degenerate mobilities. Its central thrust is the finite-volume upwind methodology, which we combine with a semi-implicit formulation based on the classical convex-splitting approach for the free-energy terms. Extension to an arbitrary number of dimensions is straightforward thanks to their cost-saving dimensional-splitting nature, which allows to efficiently solve higher-dimensional simulations with a simple parallelization. The numerical schemes are validated and tested in a variety of prototypical configurations with different numbers of dimensions and a rich variety of contact angles between droplets and substrates.
... The total free energy is the standard Cahn-Hilliard free energy supplemented with a nonlocal term, which indicates the long-range interaction between the chain molecules. There has been a lot of works on numerical analysis of OK models, such as [8][9][10][11][12][13][14]. In this paper, we consider the MPFC model with long-range interaction, which is a pseudo-gradient flow (see Section 2 for details) of a new free energy functional, in which a nonlocal potential term for long-range interaction is added to the general Swift-Hohenberg type functional. ...
In this paper, we consider numerical approximations for the modified phase field crystal model with long-range interaction, which describes the micro-phase separation in diblock copolymers. The model is a nonlinear damped wave equation with a nonlocal term that includes both diffusive dynamics and elastic interaction. To develop easy-toimplement time-stepping schemes with unconditional energy stabilities, we employ the scalar auxiliary variable (SAV) approach to achieve two highly efficient and linear numerical
schemes based on the second-order Crank–Nicolson and backward differentiation
formula. These schemes lead to decoupled linear equations with constant coefficients
at each time step and their unconditional energy stabilities are proved rigorously. The
stabilization technique is adopted to further improve the stability of the numerical schemes. Various 2D and 3D numerical experiments are performed to demonstrate the
accuracy, stability, and efficiency.
