Magnetohydrodynamics (MHD) is a continuum level model for conducting fluids subject to external magnetic fields, e.g. plasmas and liquid metals. The efficient and robust solution of the MHD system poses many challenges due to it's nonlinear, non self-adjoint, and highly coupled nature. In this paper, we develop a robust and accurate a posteriori error estimate for the numerical solution of the MHD equations based on the exact penalty method. The error estimate also isolates particular contributions of error in a quantity of interest (QoI) to inform discretization choices to arrive at accurate solutions. The tools required for these estimates involve duality arguments and computable residuals.
We propose and analyse a hybrid high-order (HHO) scheme for stationary incompressible magnetohydrodynamics equations. The scheme has an arbitrary order of accuracy and is applicable on generic polyhedral meshes. For sources that are small enough, we prove error estimates in energy norm for the velocity and magnetic field, and $L^2$-norm for the pressure; these estimates are fully robust with respect to small faces, and of optimal order with respect to the mesh size. Using compactness techniques, we also prove that the scheme converges to a solution of the continuous problem, irrespective of th…
This paper presents a Crank-Nicolson leap-frog (CNLF) scheme for the unsteady incompressible magnetohydrodynamics (MHD) equations. The spatial discretization adopts the Galerkin finite element method (FEM), and the temporal discretization employs the CNLF method for linear terms and the semi-implicit method for nonlinear terms. The first step uses Stokes style's scheme, the second step employs the Crank-Nicolson extrapolation scheme, and others apply the CNLF scheme. We testify that the fully discrete scheme is stable and convergent when the time step is less than or equal to a positive consta…
We revise the structure-preserving finite element method in [K. Hu, Y. MA and J. Xu. (2017) Stable finite element methods preserving $\nabla \cdot \mathbf{B}=0$ exactly for MHD models. Numer. Math.,135, 371-396]. The revised method is semi-implicit in time-discretization. We prove the linearized scheme preserves the divergence free property for the magnetic field exactly at each time step. Further, we showed the linearized scheme is unconditionally stable and we obtain optimal convergence in the energy norm of the revised method even for solutions with low regularity.
The paper focuses on a new error analysis of a class of mixed FEMs for stationary incompressible magnetohydrodynamics with the standard inf-sup stable velocity-pressure space pairs to 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…
We introduce a new hybridized discontinuous Galerkin method for the incompressible magnetohydrodynamics equations. If particular velocity, pressure, magnetic field, and magnetic pressure spaces are employed for both element and trace solution fields, we arrive at an energy stable method which returns pointwise divergence-free velocity fields and magnetic fields and properly balances linear momentum. We discretize in time using a second-order-in-time generalized-$α$ method, and we present a block iterative method for solving the resulting nonlinear system of equations at each time step. We nume…
In this work, we consider the general family of the so called ADER PNPM schemes for the numerical solution of hyperbolic partial differential equations with \textit{arbitrary} high order of accuracy in space and time. The family of one-step PNPM schemes was introduced in [Dumbser et al., JCP, 2008] and represents a unified framework for classical high order Finite Volume (FV) schemes (N=0), the usual Discontinuous Galerkin (DG) methods (N=M), as well as a new class of intermediate hybrid schemes for which a reconstruction operator of degree M is applied over piecewise polynomial data of degree…
We consider mixed finite element methods for linear elasticity where the symmetry of the stress tensor is weakly enforced. Both an a priori and a posteriori error analysis are given for several known families of methods that are uniformly valid in the incompressible limit. A posteriori estimates are derived for both the compressible and incompressible cases. The results are verified by numerical examples.
A linearized energy preserving finite element method for the dynamical incompressible magnetohydrodynamics equations
School of Mathematics and Statistics, Huazhong University of Science and Technology（华中科技大学数学与统计学院）；Department of Mathematics, City University of Hong Kong（香港城市大学数学系）
A posteriori error estimates are constructed for the three-field variational formulation of the Biot problem involving the displacements, the total pressure and the fluid pressure. The discretization under focus is the H1(Ω)-conforming Taylor-Hood finite element combination, consisting of polynomial degrees k + 1 for the displacements and the fluid pressure and k for the total pressure. An a posteriori error estimator is derived on the basis of H(div)-conforming reconstructions of the stress and flux approximations. The symmetry of the reconstructed stress is allowed to be satisfied only weakl…
Numerical Analysis of a Variable-Order Time-Fractional Incompressible Magnetohydrodynamics System
Institute of Information and Computational Technologies（信息与计算技术研究所）；Abai Kazakh National Pedagogical University（阿拜哈萨克国立师范大学）；Sarsen Amanzholov East Kazakhstan University（萨尔森·阿曼佐洛夫东哈萨克斯坦大学）
In the flow and transport numerical simulation, mesh adaptivity strategy is important in reducing the usage of CPU time and memory. The refinement based on the pressure error estimator is commonly-used approach without considering the flux error which plays important role in coupling flow and transport systems. We derive a posteriori error estimators for Enhanced Velocity Mixed Finite Element Method (EVMFEM) in the incompressible Darcy flow. We show numerically difference of the explicit residual based error estimator and implicit error estimators, where Arbogast and Chen post-processing proce…
Solving the relativistic magnetohydrodynamics equations with ADER discontinuous Galerkin methods, a posteriori subcell limiting and adaptive mesh refinement
A stress equilibration procedure for linear elasticity is proposed and analyzed in this paper with emphasis on the behavior for (nearly) incompressible materials. Based on the displacement-pressure approximation computed with a stable finite element pair, it constructs an $H (\text{div})$-conforming, weakly symmetric stress reconstruction. Our focus is on the Taylor-Hood combination of continuous finite element spaces of polynomial degrees $k+1$ and $k$ for the displacement and the pressure, respectively. Our construction leads then to reconstructed stresses by Raviart-Thomas elements of degre…
In this work, we further develop multigoal-oriented a posteriori error estimation for the nonlinear, stationary, incompressible Navier-Stokes equations. It is an extension of our previous work [B. Endtmayer, U. Langer, T. Wick: Two-side a posteriori error estimates for the DWR method, $SISC$, 2019, accepted]. We now focus on $h$ mesh refinement and $p$ enrichment for the error estimator. These advancements are demonstrated with the help of a numerical example
A fully discretization, unconditionally energy stable finite element method solving the thermodynamically consistent diffuse interface model for incompressible two-phase MHD flows with large density ratios
Structure-preserving Randomized Neural Networks for Incompressible Magnetohydrodynamics Equations
School of Mathematics and Statistics, Xi’an Jiaotong University（西安交通大学数学与统计学院）；School of Mathematics and Statistics & State Key Laboratory of Multiphase Flow in Power Engineering, Xi’an Jiaotong University（西安交通大学数学与统计学院及电力工程多相流国家重点实验室）；School of Mathematics and Statistics, Henan University（河南大学数学与统计学院）
We construct finite element methods for the incompressible magnetohydrodynamics (MHD) system that precisely preserve magnetic and cross helicity, the energy law and the magnetic Gauss law at the discrete level. The variables are discretized as discrete differential forms in a de Rham complex. We present numerical tests to show the performance of the algorithm.
Maintaining conservation laws in the fully discrete setting is critical for accurate long-time behavior of numerical simulations and requires accounting for discrete conservation properties in both space and time. This paper derives arbitrary order finite element exterior calculus spatial discretizations for the two-dimensional (2D) Navier-Stokes and drift-reduced magnetohydrodynamic equations that conserve both energy and enstrophy to machine precision when coupled with generally symplectic time-integration methods. Both continuous and discontinuous-Galerkin (DG) weak formulations can ensure …
In the seminal paper of Bank and Weiser [Math. Comp., 44 (1985), pp.283-301] a new a posteriori estimator was introduced. This estimator requires the solution of a local Neumann problem on every cell of the finite element mesh. Despite the promise of Bank-Weiser type estimators, namely locality, computational efficiency, and asymptotic sharpness, they have seen little use in practical computational problems. The focus of this contribution is to describe a novel implementation of hierarchical estimators of the Bank-Weiser type in a modern high-level finite element software with automatic code g…
