A second-order time-accurate, linear fully decoupled unconditional energy stabilization finite element method for tumor growth model

A novel fully-decoupled, second-order time-accurate, unconditionally energy stable scheme for a flow-coupled volume-conserved phase-field elastic bending energy model

Numerical approximations of the Navier–Stokes equation coupled with volume-conserved multi-phase-field vesicles system: Fully-decoupled, linear, unconditionally energy stable and second-order time-accurate numerical scheme

In this paper, we consider numerical approximations of the Cahn–Hilliard type phase-field crystal model and construct a fully discrete finite element scheme for it. The scheme is the combination of the finite element method for spatial discretization and an invariant energy quadratization method for time marching. It is not only linear and second-order time-accurate, but also unconditionally energy-stable. We prove the unconditional energy stability rigorously and further carry out various numerical examples to demonstrate the stability and the accuracy of the developed scheme numerically.

An energy-stable method for a phase-field surfactant model

Consistent energy-stable method for the hydrodynamics coupled PFC model

In this paper, we consider the Galerkin finite element method (FEM) for the Kelvin-Voigt viscoelastic fluid flow model with the lowest equal-order pairs. In order to overcome the restriction of the so-called inf-sup conditions, a pressure projection method based on the differences of two local Gauss integrations is introduced. Under some suitable assumptions on the initial data and forcing function, we firstly present some stability and convergence results of numerical solutions in spatial discrete scheme. By constructing the dual linearized Kelvin-Voigt model, stability and optimal error estimates of numerical solutions in various norms are established. Secondly, a fully discrete stabilized FEM is introduced, the backward Euler scheme is adopted to treat the time derivative terms, the implicit scheme is used to deal with the linear terms and semi-implicit scheme is applied to treat the nonlinear term, unconditional stability and convergence results are also presented. Finally, some numerical examples are presented to verify the developed theoretical analysis and show the performances of the considered numerical schemes.

This paper considers the stability and convergence of Euler implicit/explicit scheme for the incompressible magnetohydrodynamic (MHD) equations by the scalar auxiliary variable approach. The linear and nonlinear terms are treated by the implicit and explicit schemes, respectively, then the considered problem is split into two constant coefficient linear algebraic equations plus a nonlinear algebraic equation. Compared with the original discrete coefficient matrix, the computational size reduce and we can solve these subproblems conveniently and efficiently. First, an equivalent form of the MHD problem with four variables is developed, the corresponding stability and convergence results of spatial discrete scheme are presented. Second, the decoupled and linearized auxiliary variable finite element method is constructed, the discrete unconditional energy dissipation and stability of and in various norms are established. The optimal error estimates of numerical solutions in and H‐norms are also developed by the energy method and Gronwall lemma. Finally, some numerical examples are presented to verify the established theoretical findings and show the performances of the considered numerical algorithm.

Provably unconditionally stable, second-order time-accurate, mixed variational methods for phase-field models

High-order space–time finite element methods for the Poisson–Nernst–Planck equations: Positivity and unconditional energy stability

The shortcoming of permanent magnet (PM) machine is the de-excitation problem under fault condition, which is a main issue for aircraft application. Thus, modeling different kinds of faults accurately is of great importance. This paper presents the comparison of three different modeling methods dealing with stator winding open circuit (OC) fault and turn-to-turn short circuit (SC) fault. The modeling methods are field-circuit coupling model (FCCM) method, finite element analysis based electrical model (FEABEM) method and finite element analysis based mathematical model (FEABMM) method. The evaluation criteria are based on the accuracy of the methods and the simulation time. The simulation results show that the FEABMM method presents a high accuracy and a relatively shorter time.

A fully decoupled linearized finite element method with second-order temporal accuracy and unconditional energy stability for incompressible MHD equations

A reduced-dimension method for unknown Crank-Nicolson finite element solution coefficient vectors of elastic wave equation with singular source term

Decoupled, non-iterative, and unconditionally energy stable large time stepping method for the three-phase Cahn-Hilliard phase-field model

In this paper, we consider numerical approximations for solving the Cahn-Hilliard phase-field model with the Flory-Huggins-de Gennes free energy for homopolymer blends. We develop an efficient, second-order accurate, and unconditionally energy stable scheme that combines the SAV approach with the stabilization technique, in which the H1 norm is split from the total free energy and two extra linear stabilization terms are added to enhance the stability and keeping the required accuracy while using large time steps. The scheme is very easy to implement and non-iterative where 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 non-stabilized-SAV and MSAV schemes for some benchmark numerical examples in 2D and 3D, we demonstrate the stability and the accuracy of the developed scheme numerically.

A novel modeling method for in-plane eigenproblem estimation of the cable-stayed bridges