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The analysis of diffusion-reaction equations with general two-sided fractional derivative characterized by a parameter p ∈ [ 0 , 1 ] is investigated in this paper. First, we present a Petrov-Galerkin method, derive a proper weak formulation and show the well-posedness of its weak solution. Moreover, on the basis of the two-sided Jacobi polyfractonomials, a priori error analysis of Petrov-Galerkin method is derived. Further, a posteriori error analysis is established rigorously. More precisely, we develop a novel postprocessing technique to enhance the Petrov-Galerkin method by adding a small amount of computation, and analyze asymptotically exact a-posteriori error estimators. Finally, we demonstrate the theoretical results with numerical examples.

We construct a fully-discrete finite element numerical scheme for the Cahn–Hilliard phase-field model of the two-phase incompressible flow system with variable density and viscosity. The scheme is linear, decoupled, and unconditionally energy stable. Its key idea is to combine the penalty method of the Navier–Stokes equations with the Strang operator splitting method, and introduce several nonlocal variables and their ordinary differential equations to process coupled nonlinear terms. The scheme is highly efficient and it only needs to solve a series of completely independent linear elliptic equations at each time step, in which the Cahn–Hilliard equation and the pressure Poisson equation only have constant coefficients. We rigorously prove the unconditional energy stability and solvability of the scheme and carry out numerous accuracy/stability examples and various benchmark numerical simulations in 2D and 3D, including the Rayleigh–Taylor instability and rising/coalescence dynamics of bubbles to demonstrate the effectiveness of the scheme, numerically.

We consider in this paper numerical approximations for the hydro-dynamically coupled phase-field model for an immiscible binary mixture of passive/active nematic Liquid Crystals (LCs) and viscous fluids. The passive model is a highly coupled nonlinear system that consists of the incompressible Navier-Stokes equations, the Cahn-Hilliard equations, and the constitutive equation for the polarization field. The active model can be viewed as an assembly of a passive part and a phenomenological active part. We then construct an efficient and linear time marching scheme to solve the passive model and further extend it to solve the active model. Various numerical simulations are presented to demonstrate the stability and accuracy of the proposed scheme in simulating some simulations in 2D and 3D, including the drop deformation dynamics and spinodal decompositions.

We consider in this paper numerical approximations of a Cahn-Hilliard binary phase-field fluid-surfactant model. By adding a quartic form of the gradient potential, we first modify the commonly used total free energy into a form which is bounded from below and establish the energy law for the new system. Then we develop a stabilized-SAV scheme that combines the SAV approach with the stabilization technique, where a crucial linear stabilization term is added to enhance the stability thus allowing large time steps. With many desired properties such as a second-order in time, totally decoupled, linear, and non-iterative, this scheme is unconditionally energy stable and requires solving only four decoupled and linear biharmonic equations with constant coefficients at each time step. We further prove the energy stability and present numerous 2D and 3D numerical simulations to demonstrate the accuracy and stability of the developed scheme

In this paper, we consider numerical approximations to solving the Cahn-Hilliard phase field model for diblock copolymers. We combine the recently developed SAV (scalar auxiliary variable) approach with the stabilization technique to arrive at a novel stabilized-SAV method, where a crucial linear stabilization term is added to enhancing the stability and keeping the required accuracy while using the large time steps. The scheme is very easy-to-implement and fast in the sense that one only needs to solve two decoupled fourth-order biharmonic equations with constant coefficients at each time step. We further prove the unconditional energy stability of the scheme rigorously. Through the comparisons with some other prevalent schemes like the fully-implicit, convex-splitting, and non-stabilized SAV scheme for some benchmark numerical examples in 2D and 3D, we demonstrate the stability and the accuracy of the developed scheme numerically.

In this paper, we consider numerical approximations for a dendritic solidification phase field model with melt convection in the liquid phase, which is a highly nonlinear system that couples the anisotropic Allen-Cahn type equation, the heat equation, and the weighted Navier-Stokes equations together. We first reformulate the model into a form which is suitable for numerical approximations and establish the energy dissipative law. Then, we develop a linear, decoupled, and unconditionally energy stable numerical scheme by combining the modified projection scheme for the Navier-Stokes equations, the Invariant Energy Quadratization approach for the nonlinear anisotropic potential, and some subtle explicit-implicit treatments for nonlinear coupling terms. Stability analysis and various numerical simulations are presented.

In this article, we first use the nonlocal conserved Allen-Cahn dynamics to reformulate the phase-field model with variable density and viscosity for the two-phase incompressible flow system, and then construct a novel fully-decoupled, linear, and unconditionally energy stable scheme to solve the model. The development of the scheme is based on a combination of a novel decoupling technique, the penalty method of the Navier–Stokes equation, the quadratization method of linearizing the double-well potential, and the operator Strang-splitting method. The scheme is highly efficient, and it only needs to solve a series of fully-decoupled linear elliptic equations at each time step, in which the Allen-Cahn equation and the pressure Poisson equation only have constant coefficients. We rigorously prove the unconditional energy stability of the scheme and conduct numerical simulations in 2D and 3D to demonstrate the accuracy, stability, and effectiveness of the scheme numerically.

In this article, residual-type a posteriori error estimates are studied for finite volume element (FVE) method of parabolic equations. Residual-type a posteriori error estimator is constructed and the reliable and efficient bounds for the error estimator are established. Residual-type a posteriori error estimator can be used to assess the accuracy of the FVE solutions in practical applications. Some numerical examples are provided to confirm the theoretical results. © 2016 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 33: 259–275, 2017

In this work, we first propose a new hydrodynamically-coupled phase-field model for the diblock copolymer melt that is derived based on the Allen-Cahn dynamics where the volume fraction of two composing monomers is conserved by using a nonlocal Lagrange multiplier. Formally, the system is a highly nonlinear system that consists of the incompressible Navier–Stokes equations and a nonlocal-type Allen-Cahn type equation. Then, to solve the model, we develop a linear and second-order time-marching scheme via the Invariant Energy Quadratization approach with the stabilization technique for the nonlinear potentials, as well as the projection method for the Navier–Stokes equations. The unconditional energy stability of the numerical method is proved, and several experiments of 2D and 3D are performed to validate the accuracy and energy stability of the developed numerical scheme.

A two-grid finite volume element method, combined with the modified method of characteristics, is presented and analyzed for semilinear time-dependent advection-dominated diffusion equations in two space dimensions. The solution of a nonlinear system on the fine-grid space (with grid size h) is reduced to the solution of two small (one linear and one nonlinear) systems on the coarse-grid space (with grid size H) and a linear system on the fine-grid space. An optimal error estimate in H1 -norm is obtained for the two-grid method. It shows that the two-grid method achieves asymptotically optimal approximation, as long as the mesh sizes satisfy h = O(H2). Numerical example is presented to validate the usefulness and efficiency of the method. © 2013 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2013

Two-grid methods are studied for solving a two dimensional nonlinear parabolic equation using finite volume element method. The methods are based on one coarse-grid space and one fine-grid space. The nonsymmetric and nonlinear iterations are only executed on the coarse grid and the fine-grid solution can be obtained in a single symmetric and linear step. It is proved that the coarse grid can be much coarser than the fine grid. The two-grid methods achieve asymptotically optimal approximation as long as the mesh sizes satisfy h=O(H3|lnH|). As a result, solving such a large class of nonlinear parabolic equations will not be much more difficult than solving one single linearized equation.

In this paper, we develop a finite volume element method with affine quadratic bases on right quadrangular prism meshes for three-dimensional elliptic boundary value problems. The optimal H1-norm error estimate of second order accuracy is proved under certain assumptions about the meshes. Numerical results are presented to illustrate the theoretical analysis.

The two-grid method is studied for solving a two-dimensional second-order nonlinear hyperbolic equation using finite volume element method. The method is based on two different finite element spaces defined on one coarse grid with grid size H and one fine grid with grid size h, respectively. The nonsymmetric and nonlinear iterations are only executed on the coarse grid and the fine grid solution can be obtained in a single symmetric and linear step. It is proved that the coarse grid can be much coarser than the fine grid. A prior error estimate in the H1-norm is proved to be O(h+H3|lnH|) for the two-grid semidiscrete finite volume element method. With these proposed techniques, solving such a large class of second-order nonlinear hyperbolic equations will not be much more difficult than solving one single linearized equation. Finally, a numerical example is presented to validate the usefulness and efficiency of the method.