This paper focuses on new error analysis of a class of mixed FEMs for stationary incompressible magnetohydrodynamics with the standard inf-sup stable velocity-pressure space in cooperation with Navier-Stokes equations and the Nédélec’s edge element for the magnetic field. The methods have been widely used in various numerical simulations in the last several decades, while the existing analysis is not optimal due to the strong coupling of system and the pollution of the lower-order Nédélec’s edge approximation in analysis. In terms of a newly modified Maxwell projection we establish new and optimal error estimates. In particular, we prove that the method based on the commonly-used Taylor-Hood/lowest-order Nédélec’s edge element is efficient and the method provides the second-order accuracy for numerical velocity. Two numerical examples for the problem in both convex and nonconvex polygonal domains are presented, which confirm our theoretical analysis. 

/pdf/new-analysis-of-mixed-finite-element-methods-for-2xbezrn5.pdf

New Analysis of Mixed Finite Element Methods for Incompressible Magnetohydrodynamics

/pdf/coupled-iterative-analysis-for-the-stationary-thermally-13bcv4mq.pdf

Coupled iterative analysis for the stationary thermally coupled inductionless MHD system based on charge-conservative finite element method

In this paper, we present a new error analysis of a class of charge-conservative finite element methods for stationary inductionless magnetohydrodynamics (MHD) equations. The methods use the standard inf-sup stable Mini/Taylor–Hood pairs to discretize the velocity and pressure, and the Raviart–Thomas face element for solving the current density. Due to the strong coupling of the system and the pollution of the lower-order Raviart–Thomas face approximation in analysis, the existing analysis is not optimal. In terms of a mixed Poisson projection and the corresponding estimate in negative norms, we establish new and optimal error estimates. In particular, we prove that the method with the lowest-order Raviart–Thomas face element and Mini element provides the optimal accuracy for the velocity in L2-norm, and the method with the lowest-order Raviart–Thomas face element and P2−P1 Taylor-Hood element supplies the optimal accuracy for the velocity in H1-norm and the pressure in L2-norm. Furthermore, we propose a simple recovery technique to obtain a new numerical current density of one order higher accuracy by re-solving a mixed Poisson equation. Numerical results are provided to verify the theoretical analysis. 

New error analysis of charge-conservative finite element methods for stationary inductionless MHD equations

In this paper, we consider the numerical approximation for a diffuse interface model of the two-phase incompressible inductionless magnetohydrodynamics (MHD) problem. This model consists of Cahn–Hilliard equations, Navier–Stokes equations and Poisson equation. We propose a linear and decoupled finite element method to solve this highly nonlinear and multi-physics system. For the time variable, the discretization is a combination of the first order Euler semi-implicit scheme, several first-order stabilization terms and implicit–explicit treatments for coupling terms. For the space variables, we adopt the finite element discretization. Especially, we approximate the current density and electric potential by inf–sup stable face-volume mixed finite element pairs. With these techniques, the scheme only involves a sequence of decoupled linear equations to solve at each time step. We show that the scheme is provably mass-conservative, charge-conservative and unconditionally energy stable. Numerical experiments are performed to illustrate the properties, accuracy and efficiency of the proposed scheme. 

Decoupled, linear, unconditionally energy stable and charge-conservative finite element method for an inductionless magnetohydrodynamic phase-field model

Convergence Analysis of the Fully Discrete Projection Method for Inductionless Magnetohydrodynamics System Based on Charge Conservation

In this paper, we propose a decoupled, unconditionally energy stable and charge-conservative finite element method for the inductionless magnetohydrodynamic equations. The time marching is combined with a first order semi-implicit Euler scheme, a first order perturbation term and some delicate implicit-explicit treatments for coupling terms. The finite element discretization is based on mixed conforming elements, where the hydrodynamic unknowns are approximated by stable finite element pairs, and the current density and electric potential are discretized by divergence-conforming face elements and discontinuous volume elements. This fully discrete scheme only needs to solve a linear subsystem at each time step and yields an exactly divergence-free current density directly. We show that the proposed scheme is well-posed and unconditionally energy stable. Under a weak regularity hypothesis on the exact solution, we present the error estimates for velocity, current density and electric potential. Finally, the validity, reliability, and accuracy of the proposed algorithm are supported by several numerical experiments. 

A decoupled, unconditionally energy stable and charge-conservative finite element method for inductionless magnetohydrodynamic equations

In this paper, we develop and analyze a finite element projection method for magnetohydrodynamics equations in Lipschitz domain. A fully discrete scheme based on Euler semi‐implicit method is proposed, in which continuous elements are used to approximate the Navier–Stokes equations and H(curl) conforming Nédélec edge elements are used to approximate the magnetic equation. One key point of the projection method is to be compatible with two different spaces for calculating velocity, which leads one to obtain the pressure by solving a Poisson equation. The results show that the proposed projection scheme meets a discrete energy stability. In addition, with the help of a proper regularity hypothesis for the exact solution, this paper provides a rigorous optimal error analysis of velocity, pressure and magnetic induction. Finally, several numerical examples are performed to demonstrate both accuracy and efficiency of our proposed scheme.

Error analysis of a fully discrete projection method for magnetohydrodynamic system

In this paper, we study a fully discrete finite element scheme of thermally coupled incompressible magnetohydrodynamic with temperature-dependent coefficients in Lipschitz domain. The variable coefficients in the MHD system and possible nonconvex domain may cause nonsmooth solutions. We propose a fully discrete Euler semi-implicit scheme with the magnetic equation approximated by Ne´de´lec edge elements to capture the physical solutions. The fully discrete scheme only needs to solve one linear system at each time step and is unconditionally stable. Utilizing the stability of the numerical scheme and the compactness method, the existence of weak solution to the thermally coupled MHD model in three dimensions is established. Furthermore, the uniqueness of weak solution and the convergence of the proposed numerical method are also rigorously derived. Under the hypothesis of a low regularity for the exact solution, we rigorously establish the error estimates for the velocity, temperature and magnetic induction unconditionally in the sense that the time step is independent of the spacial mesh size.

/pdf/convergence-analysis-of-a-fully-discrete-finite-element-180cle94.pdf

Convergence analysis of a fully discrete finite element method for thermally coupled incompressible MHD problems with temperature-dependent coefficients

Error analysis of a conservative finite element scheme for time-dependent inductionless MHD problem

Based on two-grid discretizations, local and parallel finite element algorithms are proposed and analyzed for the time-dependent Oseen equations. Using conforming finite element pairs for the spatial discretization and backward Euler scheme for the temporal discretization, the basic idea of the fully discrete finite element algorithms is to approximate the generalized Oseen equations using a coarse grid on the entire domain, and then correct the resulted residual using a fine grid on overlapped subdomains by some local and parallel procedures at each time step. By the theoretical tool of local a priori estimate for the fully discrete finite element solution, error bounds of the approximate solutions from the algorithms are estimated. Numerical results are also given to demonstrate the efficiency of the algorithms.

Local and parallel finite element algorithms for the time-dependent Oseen equations

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