Characteristic Finite Element Method Research Articles

An improved partitioned time stepping method based on modified characteristic FEM for the evolutionary dual-porosity-Navier–Stokes model

In this paper, we study an improved partitioned time stepping method based on the modified characteristic finite element method for the evolutionary dual-porosity-Navier–Stokes coupling model with the Beavers–Joseph interface condition. By referring to the idea of partitioned time stepping, we lead into the half-time steps to separate the original coupling problem into three subproblems, implement two-level decoupling to simplify the complexity of solving coupling systems. Before that, the matrix and microfracture equations on the half-time steps in the dual-porosity media are calculated first, with the aim of replacing the initial values of the decoupling equations. Besides, for the nonlinear convection term in Navier–Stokes equations, the modified characteristic finite element method is chosen to avoid the problem of low computational efficiency. The convergence of the decoupling algorithm is rigorously derived by using the idea of mathematical induction. Finally, the reliability of the theoretical analysis and the rationality of the scheme are verified by numerical experiments.

Unconditional stability and error estimates of the modified characteristics FEMs for the Micropolar Navier–Stokes Equations

In this paper, the unconditional stability and optimal error estimate of the velocity, pressure and angular velocity for the modified characteristics FEMs of the unsteady Micropolar Naiver–Stokes Equations (MNSE) are presented. In this method, the nonlinear equation is linearized by the characteristic finite element method for dealing with the time derivative term and the convection term. Basing on the characteristic time-discrete system, the error function is split into a temporal error and a spatial error. With a rigorous analysis to the characteristic time-discrete system, we prove that the error between the numerical solution and the solution of the time-discrete system is τ-independent, where τ denotes the time stepsize. The stability results and optimal error estimates in L2 norm and H1 norm will be given. Finally, some numerical results will be provided to confirm our theoretical analysis.

Stability analysis of a finite element approximation for the Navier-Stokes equation with free surface

In this work we study the numerical approximation of incompressible Navier-Stokes equations with free surface. The evolution of the free surface is driven by the kinematic boundary condition, and an Arbitrary Lagrangian Eulerian (ALE) approach is used to derive a (formal) weak formulation which involves three fields, namely, velocity, pressure, and the function describing the free surface. This formulation is discretised using finite elements in space and a time-advancing explicit finite difference scheme in time. In fact, the domain tracking algorithm is explicit: first, we solve the equation for the free surface, then move the mesh according to the sigma transform, and finally we compute the velocity and pressure in the updated domain. This explicit strategy is built in such a way that global conservation can be proven, which plays a pivotal role in the proof of stability of the discrete problem. The well-posedness and stability results are independent of the viscosity of the fluid, but while the proof of stability for the velocity is valid for all time steps, and all geometries, the stability for the free surface requires a CFL condition. The performance of the current approach is presented via numerical results and comparisons with the characteristics finite element method.

Modified Characteristic Finite Element Method with Second-Order Spatial Accuracy for Solving Convection-Dominated Problem on Surfaces.

We present a modified characteristic finite element method that exhibits second-order spatial accuracy for solving convection-reaction-diffusion equations on surfaces. The temporal direction adopted the backward-Euler method, while the spatial direction employed the surface finite element method. In contrast to regular domains, it is observed that the point in the characteristic direction traverses the surface only once within a brief time. Thus, good approximation of the solution in the characteristic direction holds significant importance for the numerical scheme. In this regard, Taylor expansion is employed to reconstruct the solution beyond the surface in the characteristic direction. The stability of our scheme is then proved. A comparison is carried out with an existing characteristic finite element method based on face mesh. Numerical examples are provided to validate the effectiveness of our proposed method.

A modified characteristics projection finite element method for the non‐stationary incompressible thermally coupled MHD equations

In this paper, a modified characteristics projection finite element method is proposed for the non‐stationary incompressible thermally couple MHD equations. The modified characteristics projection finite element method combines the modified characteristics finite element method with the projection method to verify the unconditional stability and optimal error estimation of velocity, pressure, magnetic field and temperature in the non‐stationary incompressible thermally couple MHD equations. In order to show the effectiveness of the proposed method, some numerical results are presented. The numerical results are consistent with our theoretical analysis, and our method is effective.

A mass conservative characteristic finite element method with unconditional optimal convergence for semiconductor device problem

We consider a mass conservative type method for semiconductor device problem by employing mixed finite element method (FEM) for electric potential equation and mass conservative characteristic FEM for both electron and hole density equations. The boundedness of numerical solution without certain time‐step restriction and optimal error estimates of full discrete scheme are proved. Numerical experiment is presented to verify the effectiveness and unconditional stability of the proposed method.

Interpolated coefficient characteristic finite element method for semilinear convection–diffusion optimal control problems

In this paper, a fully discrete interpolated coefficient characteristic finite element approximation is proposed for optimal control problems governed by time-dependent semilinear convection–diffusion equations, where the hyperbolic part of the state equation is first treated by directional derivatives and then discretized by backward difference, the semilinear term is dealt with interpolation coefficient finite elements technique. A priori error estimates for the control, state and co-state variables are derived. Theoretic results are confirmed by a numerical example.

Analysis of Two-Grid Characteristic Finite Element Methods for Convection-Diffusion Equations

In this paper, two efficient two-grid algorithms for the convection-diffusion problem with a modified characteristic finite element method are studied. We present an optimal error estimate in L p -norm for the characteristic finite element method unconditionally, while all previous works require certain time-step restrictions. To linearize the characteristic method equations, two-grid algorithms based on the Newton iteration approach and the correction method are applied. The error estimate and the convergence result of the two-grid method are derived in detail. It is shown that the coarse space can be extremely coarse and achieve asymptotically optimal approximations as long as the mesh sizes H = O h 1 / 3 in the first algorithm and H = O h 1 / 4 in the second algorithm, respectively. Finally, two numerical examples are presented to demonstrate the theoretical analysis.

A multipoint flux mixed finite element method with mass-conservative characteristic finite element method for incompressible miscible displacement problem

A multipoint flux mixed finite element method with mass-conservative characteristic finite element method for incompressible miscible displacement problem

High-order unfitted characteristic finite element methods for moving interface problem of Oseen equations

Moving interface problems have broad applications in multi-phase flows and fluid–structure interactions. The design of high-order numerical methods for such problems becomes challenging, in particular, when the interface is severely deformed. We propose kth-order unfitted characteristic finite element methods (k=2,3,4) for solving two-dimensional Oseen equations with a time-varying interface. In each time step, the interface is not specified with an explicit function, but constructed numerically with a (k+1)th-order interface-tracking algorithm. We obtain optimal convergence orders of both the discrete velocity and the discrete pressure under the H1-norm. Our convergence analysis includes errors of the approximate interface and the time integration and spatial discretization of the governing equations. Numerical experiments are performed on a rotating object and a severely deformed interface and show optimal convergence orders of the method.

A fourth-order unfitted characteristic finite element method for free-boundary problems

A fourth-order unfitted characteristic finite element method (UCFEM) is proposed to solve free-boundary problems of time-dependent partial differential equations (PDEs). The boundary of the domain is implicitly driven by the solution of the PDE, while the PDE is proposed on the time-varying unknown domain. In this way, they form a coupled nonlinear system. The key ingredient of the UCFEM is to trace the domain explicitly with a fourth-order forward flow map and discretize the PDE in time with a fourth-order backward flow map. The PDE is solved with a fourth-order unfitted finite element method on a fixed mesh. Since the boundary of the domain is expressed explicitly with cubic spline functions, quadrature on cut elements can be done accurately and efficiently even the domain undergoes severe deformations. The UCFEM provides a framework of designing high-order numerical methods for free-boundary problems. We apply the UCFEM to a two-dimensional convection-diffusion equation and Navier-Stokes equations, and obtain the overall fourth order of accuracy. With extensive numerical experiments, we show that the proposed method can achieve the fourth-order convergence even on severely deformed domains.

A Fourth-Order Unfitted Characteristic Finite Element Method for Solving the Advection-Diffusion Equation on Time-Varying Domains

We propose a fourth-order unfitted characteristic finite element method to solve the advection-diffusion equation on time-varying domains. Based on a characteristic-Galerkin formulation, our method combines the cubic MARS method for interface tracking, the fourth-order backward differentiation formula for temporal integration, and an unfitted finite element method for spatial discretization. Our convergence analysis includes errors of discretely representing the moving boundary, tracing boundary markers, and the spatial discretization and the temporal integration of the governing equation. Numerical experiments are performed on a rotating domain and a severely deformed domain to verify our theoretical results and to demonstrate the optimal convergence of the proposed method.

Unconditional stability and optimal error analysis of mass conservative characteristic mixed FEM for wormhole propagation

This paper is concerned with unconditional stability and optimal convergence of mass conservative type method for simulating wormhole propagation in porous media. Specifically, mass conservative characteristic finite element method (FEM) is used for the solute transport equation, the mixed FEM is used for velocity-pressure equation and Galerkin FEM for porosity equation. By error splitting technique, we prove the error of the solution between full discrete system and time discrete system is time-independent, while the numerical solution is bounded without certain time step restriction. Moreover, the optimal L2 error estimates further hold in a general case by elliptic quasi-projection, where the unconditional r+1 order accuracy of the concentration and porosity can be obtained with no loss of accuracy for r order approximation velocity-pressure equation. Numerical experiments are presented to verify the theoretical analysis and the effectiveness of the proposed method.

A reduced-order characteristic finite element method based on POD for optimal control problem governed by convection–diffusion equation

This paper proposes a reduced-order characteristic finite element (ROCFE) approximation with several unknowns for optimal control problem governed by convection–diffusion equation. It inherits the advantages of the characteristic methods in solving convection-dominated diffusion equations. For this problem, the usual snapshots extraction way cannot be used, so we propose a new snapshots extraction way. Then the proper orthogonal decomposition (POD) is used on unsteady state and co-state systems, which efficiently reduces the number of unknowns and computational costs. Optimal a priori error estimates are derived for the state, co-state and control. Some numerical experiments are carried out to show the accuracy and efficiency of the ROCFE method.

Decoupled modified characteristic finite element method for the time‐dependent Navier–Stokes/Biot problem

In this article, a decoupled modified characteristic finite element method is proposed for the time-dependent Navier–Stokes/Biot problem. In the numerical scheme, the implicit backward Euler scheme is used for the time discretization, whereas the coupling terms are treated explicitly. At each time step, we only need to solve two decoupled problems, one is the Navier–Stokes equations solved by the modified characteristic finite element method, and the other is Biot equations. The stability and the error estimates are established for the proposed fully discrete scheme. Numerical experiments are provided to illustrate the theory.

A characteristic finite element method for the time-fractional mobile/immobile advection diffusion model

In this paper, we study the solute transport in heterogeneous media described by the time-fractional mobile/immobile advection diffusion model, where the integer and the fractional time derivatives are employed to characterize the movement of the particles in the mobile and immobile zone, respectively. We propose a fully discrete characteristic finite element scheme for the model, in which the modified method of characteristics is applied to handle the domination of advection. The optimal L2 error estimate is derived with first-order accuracy in time and second-order accuracy in space. Several practical numerical experiments are presented to validate the effectiveness and accuracy of the proposed method.

Numerical analysis of incompressible wormhole propagation with mass-preserving characteristic mixed finite element procedure

In this article, we construct a new combined characteristic mixed finite element procedure to simulate the incompressible wormhole propagation. In this procedure, we use the classical mixed finite element method to solve the pressure equation and a modified mass-preserving characteristic finite element method for the solute transport equation, and solve the porosity function straightly by the given concentration. This combined method not only keeps mass balance globally but also preserves maximum principle for the porosity. We considered the corresponding convergence and derive the optimal L2-norm error estimate. Finally, we present some numerical examples to confirm theoretical analysis.

Decoupled modified characteristic finite element method with different subdomain time steps for nonstationary dual–porosity–Navier–Stokes model

In this paper, we propose and analyze a decoupled modified characteristic finite element method with different subdomain time steps for the mixed stabilized formulation of nonstationary dual-porosity-Navier-Stokes model. Based on partitioned time-stepping methods, the mixed system with a stabilization term is decoupled, which means that the Navier-Stokes equations and two different Darcy equations are solved independently at each time step of subdomains. In particular, we solve the Navier-Stokes equations by the modified characteristic finite element method, which overcomes the computational inefficiency caused by the nonlinear term. In order to increase the efficiency, different time steps are used to different subdomains. We prove the error convergence of solutions by mathematical induction, whose proof implies the uniform L∞-boundedness of the fully discrete velocity solution in conduit flow. Finally, some numerical tests are presented to show efficiency of the proposed method.

Numerical simulation for clustering and pattern formation in active colloids with mass-preserving characteristic finite element method

We consider the numerical approximations to simulate the active colloids dynamic process. One linear, decoupled, mass-preserving scheme is proposed, in which mass-preserving characteristic finite element method is utilized for the density equation of active colloids, and the traditional Galerkin finite element methods is used for the polarization dynamics equation and self-secreted chemical density equation. This method not only improves the computational efficiency but also keeps the mass conservation. We analyze the convergence of this method and give the corresponding error estimate under some regularity assumptions of the solution. Numerical experiments are carried out to support the theoretical analysis. The proposed method is also applied to simulate the active colloids dynamic process for the clustering and pattern formation in chemoattraction and chemorepulsion, respectively.

A finite point algorithm for soil water-salt movement equation

In this paper, we propose the meshless finite point method for solving a type of fluid flow problem. The moving least square function is combined with the collocation method to treat nonlinear one- and two-dimensional soil water-salt movement equations. An adaptive windward scheme is used to stabilize the numerical solution in regions with a large gradient change. Numerical examples with the comparison among the proposed method, finite element method and characteristic finite element method show that the meshless finite point method is more accurate and is used to eliminate the numerical oscillation phenomenon.