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Register a new userAnalysis of mixed interior penalty discontinuous Galerkin methods for the Cahn-Hilliard equation and the Hele-Shaw flow
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This paper proposes and analyzes two fully discrete mixed interior penalty discontinuous Galerkin (DG) methods for the fourth order nonlinear Cahn-Hilliard equation. Both methods use the backward Euler method for time discretization and interior penalty discontinuous Galerkin methods for spatial discretization. They differ from each other on how the nonlinear term is treated, one of them is based on fully implicit time-stepping and the other uses the energy-splitting time-stepping. The primary goal of the paper is to prove the convergence of the numerical interfaces of the DG methods to the interface of the Hele-Shaw flow. This is achieved by establishing error estimates that depend on $epsilon^{-1}$ only in some low polynomial orders, instead of exponential orders. Similar to [14], the crux is to prove a discrete spectrum estimate in the discontinuous Galerkin finite element space. However, the validity of such a result is not obvious because the DG space is not a subspace of the (energy) space $H^1$ and it is larger than the finite element space. This difficult is overcome by a delicate perturbation argument which relies on the discrete spectrum estimate in the finite element space proved in cite{Feng_Prohl04}. Numerical experiment results are also presented to gauge the theoretical results and the performance of the proposed fully discrete mixed DG methods.
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The phase separation of an isothermal incompressible binary fluid in a porous medium can be described by the so-called Brinkman equation coupled with a convective Cahn-Hilliard (CH) equation. The former governs the average fluid velocity $mathbf{u}$, while the latter rules evolution of $varphi$, the difference of the (relative) concentrations of the two phases. The two equations are known as the Cahn-Hilliard-Brinkman (CHB) system. In particular, the Brinkman equation is a Stokes-like equation with a forcing term (Korteweg force) which is proportional to $mu ablavarphi$, where $mu$ is the chemical potential. When the viscosity vanishes, then the system becomes the Cahn-Hilliard-Hele-Shaw (CHHS) system. Both systems have been studied from the theoretical and the numerical viewpoints. However, theoretical results on the CHHS system are still rather incomplete. For instance, uniqueness of weak solutions is unknown even in 2D. Here we replace the usual CH equation with its physically more relevant nonlocal version. This choice allows us to prove more about the corresponding nonlocal CHHS system. More precisely, we first study well-posedness for the CHB system, endowed with no-slip and no-flux boundary conditions. Then, existence of a weak solution to the CHHS system is obtained as a limit of solutions to the CHB system. Stronger assumptions on the initial datum allow us to prove uniqueness for the CHHS system. Further regularity properties are obtained by assuming additional, though reasonable, assumptions on the interaction kernel. By exploiting these properties, we provide an estimate for the difference between the solution to the CHB system and the one to the CHHS system with respect to viscosity.
We present a second-order-in-time finite difference scheme for the Cahn-Hilliard-Hele-Shaw equations. This numerical method is uniquely solvable and unconditionally energy stable. At each time step, this scheme leads to a system of nonlinear equations that can be efficiently solved by a nonlinear multigrid solver. Owing to the energy stability, we derive an $ell^2 (0,T; H_h^3)$ stability of the numerical scheme. To overcome the difficulty associated with the convection term $ abla cdot (phi boldsymbol{u})$, we perform an $ell^infty (0,T; H_h^1)$ error estimate instead of the classical $ell^infty (0,T; ell^2)$ one to obtain the optimal rate convergence analysis. In addition, various numerical simulations are carried out, which demonstrate the accuracy and efficiency of the proposed numerical scheme.
In this paper we analyze the convergence properties of two-level and W-cycle multigrid solvers for the numerical solution of the linear system of equations arising from hp-version symmetric interior penalty discontinuous Galerkin discretizations of second-order elliptic partial differential equations on polygonal/polyhedral meshes. We prove that the two-level method converges uniformly with respect to the granularity of the grid and the polynomial approximation degree p, provided that the number of smoothing steps, which depends on p, is chosen sufficiently large. An analogous result is obtained for the W-cycle multigrid algorithm, which is proved to be uniformly convergent with respect to the mesh size, the polynomial approximation degree, and the number of levels, provided the number of smoothing steps is chosen sufficiently large. Numerical experiments are presented which underpin the theoretical predictions; moreover, the proposed theoretical assumptions are not fully satisfied.
In this paper, the optimal choice of the interior penalty parameter of the discontinuous Galerkin finite element methods for both the elliptic problems and the Biots systems are studied by utilizing the neural network and machine learning. It is crucial to choose the optimal interior penalty parameter, which is not too small or not too large for the stability, robustness, and efficiency of the numerical discretized solutions. Both linear regression and nonlinear artificial neural network methods are employed and compared using several numerical experiments to illustrate the capability of our proposed computational framework. This framework is an integral part of a developing automated numerical simulation platform because it can automatically identify the optimal interior penalty parameter. Real-time feedback could also be implemented to update and improve model accuracy on the fly.
We study a Cahn-Hilliard-Hele-Shaw (or Cahn-Hilliard-Darcy) system for an incompressible mixture of two fluids. The relative concentration difference $varphi$ is governed by a convective nonlocal Cahn-Hilliard equation with degenerate mobility and logarithmic potential. The volume averaged fluid velocity $mathbf{u}$ obeys a Darcys law depending on the so-called Korteweg force $mu abla varphi$, where $mu$ is the nonlocal chemical potential. In addition, the kinematic viscosity $eta$ may depend on $varphi$. We establish first the existence of a global weak solution which satisfies the energy identity. Then we prove the existence of a strong solution. Further regularity results on the pressure and on $mathbf{u}$ are also obtained. Weak-strong uniqueness is demonstrated in the two dimensional case. In the three-dimensional case, uniqueness of weak solutions holds if $eta$ is constant. Otherwise, weak-strong uniqueness is shown by assuming that the pressure of the strong solution is $alpha$-H{o}lder continuous in space for $alphain (1/5,1)$.
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