[Paper Review] Optimal error estimates of a second-order projection finite element method for magnetohydrodynamic equations

Cheng Wang, Jilu Wang+2 more|arXiv (Cornell University)|Nov 30, 2020

Advanced Numerical Methods in Computational Mathematics39 references4 citations

TL;DR

This paper presents a second-order accurate, fully discrete finite element method for the incompressible magnetohydrodynamic (MHD) equations using a modified Crank–Nicolson scheme with semi-implicit treatments for nonlinear terms. The method employs a decoupled Van Kan-type projection Stokes solver, ensuring unconditional energy stability and proving optimal $\mathcal{O}(\tau^2 + h^{r+1})$ error estimates in the discrete $L^\infty(0,T;L^2)$ norm for velocity and magnetic fields.

ABSTRACT

In this paper, we propose and analyze a temporally second-order accurate, fully discrete finite element method for the magnetohydrodynamic (MHD) equations. A modified Crank--Nicolson method is used to discretize the model and appropriate semi-implicit treatments are applied to the fluid convection term and two coupling terms. These semi-implicit approximations result in a linear system with variable coefficients for which the unique solvability can be proved theoretically. In addition, we use a decoupling projection method of the Van Kan type \cite{vankan1986} in the Stokes solver, which computes the intermediate velocity field based on the gradient of the pressure from the previous time level, and enforces the incompressibility constraint via the Helmholtz decomposition of the intermediate velocity field. The energy stability of the scheme is theoretically proved, in which the decoupled Stokes solver needs to be analyzed in details. Optimal-order convergence of $\mathcal{O} (τ^2+h^{r+1})$ in the discrete $L^\infty(0,T;L^2)$ norm is proved for the proposed decoupled projection finite element scheme, where $τ$ and $h$ are the time stepsize and spatial mesh size, respectively, and $r$ is the degree of the finite elements. Existing error estimates of second-order projection methods of the Van Kan type \cite{vankan1986} were only established in the discrete $L^2(0,T;L^2)$ norm for the Navier--Stokes equations. Numerical examples are provided to illustrate the theoretical results.

Motivation & Objective

• To develop a temporally second-order accurate, fully discrete finite element method for the incompressible MHD equations with improved computational efficiency.
• To address the challenge of solving coupled, nonlinear MHD systems by introducing a decoupled projection method that reduces computational cost.
• To establish rigorous theoretical convergence and energy stability for the proposed scheme under realistic regularity assumptions.
• To extend existing error estimates—previously limited to $L^2(0,T;L^2)$ norms—for projection methods to the stronger $L^\infty(0,T;L^2)$ norm.
• To validate the theoretical findings with numerical experiments demonstrating optimal convergence rates and energy dissipation.

Proposed method

• A modified Crank–Nicolson scheme is applied for temporal discretization, ensuring second-order accuracy in time.
• Semi-implicit approximations are used for the fluid convection and coupling terms to linearize the system while preserving stability.
• A Van Kan-type decoupled projection method is employed in the Stokes solver, using the pressure gradient from the previous time level to compute an intermediate velocity field.
• The incompressibility constraint is enforced via Helmholtz decomposition of the intermediate velocity, enabling efficient solution of the velocity and pressure fields separately.
• The resulting linear system with variable coefficients is proven to have a unique solution through analysis of the corresponding homogeneous problem.
• Finite element spaces of degree $r$ for velocity and magnetic field, and degree $r-1$ for pressure, are used for spatial discretization.

Experimental results

Research questions

• RQ1 Can a second-order accurate, fully discrete finite element method be constructed for the incompressible MHD equations that ensures both stability and optimal convergence?
• RQ2 How can the nonlinear and coupled nature of the MHD system be handled efficiently while preserving second-order temporal accuracy?
• RQ3 Can a decoupled projection scheme be designed such that the Stokes solve remains unconditionally energy stable and computationally efficient?
• RQ4 Is it possible to derive optimal error estimates in the stronger $L^\infty(0,T;L^2)$ norm for such a decoupled scheme, rather than the weaker $L^2(0,T;L^2)$ norm?
• RQ5 What is the convergence rate of the scheme in both time and space, and does it match theoretical predictions?

Key findings

• The proposed scheme achieves optimal convergence order $\mathcal{O}(\tau^2 + h^{r+1})$ in the discrete $L^\infty(0,T;L^2)$ norm for velocity and magnetic field approximations.
• Numerical experiments confirm second-order convergence in time with $\tau = 1/10, 1/20, 1/40, 1/80$, yielding temporal convergence orders of approximately 2.01, 2.06, and 2.03.
• Spatial convergence rates of approximately 3.0 are observed for quadratic finite elements ($r=2$), matching the predicted $h^{r+1}$ rate.
• The energy of the system is observed to decay monotonically over time, confirming unconditional energy stability as proven in Theorem 2.6.
• The decoupled Stokes solver enables efficient solution by avoiding the need to solve a non-symmetric linear system at each time step.
• Theoretical analysis confirms unique solvability of the linear system with variable coefficients, relying on the fact that the homogeneous version admits only the trivial solution.