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.
In this paper, we propose and analyze a fully discrete finite element projection method for the magnetohydrodynamic (MHD) equations. A modified Crank--Nicolson method and the Galerkin finite element method are used to discretize the model in time and space, respectively, 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 second-order decoupling projection method of the V…
We present a four-field Virtual Element discretization for the time-dependent resistive Magnetohydrodynamics equations in three space dimensions, focusing on the semi-discrete formulation. The proposed method employs general polyhedral meshes and guarantees velocity and magnetic fields that are divergence free up to machine precision. We provide a full convergence analysis under suitable regularity assumptions, which is validated by some numerical tests.
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.
We present compatible finite element space discretizations for the ideal compressible magnetohydrodynamic equations. The magnetic field is considered both in div- and curl-conforming spaces, leading to a strongly or weakly preserved zero-divergence condition, respectively. The equations are discretized in space such that transfers between the kinetic, internal, and magnetic energies are consistent, leading to a preserved total energy. We also discuss further adjustments to the discretization required to additionally achieve magnetic helicity preservation. Finally, we describe new transport sta…
In this paper, we consider the numerical approximation for a diffuse interface model of the two-phase incompressible inductionless magnetohydrodynamics 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 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…
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…
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）
We present a virtual element method (VEM) for the numerical approximation of the electromagnetics subsystem of the resistive magnetohydrodynamics (MHD) model in two spatial dimensions. The major advantages of the virtual element method include great flexibility of polygonal meshes and automatic divergence-free constraint on the magnetic flux field. In this work, we rigorously prove the well-posedness of the method and the solenoidal nature of the discrete magnetic flux field. We also derive stability energy estimates. The design of the method includes three choices for the construction of the …
A priori analysis for a generalized local projection stabilized finite element approximations for the solution of an advection-reaction equation is presented in this article. The stability and a priori error estimates are established for both the conforming and the nonconforming (Crouzeix-Raviart) approximations with respect to the local projection streamline derivative norm. Finally, the validation of the proposed stabilization scheme and verification of the the derived estimates are presented with appropriate numerical experiments.
In this note, we propose a simple way of constructing HDG+ projections on polyhedral elements. The projections enable us to analyze the Lehrenfeld-Schöberl HDG (HDG+) methods in a very concise manner, and make many existing analysis techniques of standard HDG methods reusable for HDG+. The novelty here is an alternative way of constructing the projections without using $M$-decompositions as a middle step. This extends our previous results [S. Du and F.-J. Sayas, SpringerBriefs in Mathematics (2019)] (elliptic problems) and [S. Du and F.-J. Sayas, Math.~Comp.~89~(2020), 1745-1782] (elasticity) …
We construct local projections into canonical finite element spaces that appear in the finite element exterior calculus. These projections are bounded in $L^2$ and commute with the exterior derivative.
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…
In this paper we discuss a hybridised method for FEM-BEM coupling. The coupling from both sides use a Nitsche type approach to couple to the trace variable. This leads to a formulation that is robust and flexible with respect to approximation spaces and can easily be combined as a building block with other hybridised methods. Energy error norm estimates and the convergence of Jacobi iterations are proved and the performance of the method is illustrated on some computational examples.
Quasi-Optimal Error Estimates for the Incompressible Navier-Stokes Problem Discretized by Finite Element Methods and Pressure-Correction Projection with Velocity Stabilization
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 paper, the pressure correctionfinite element method is proposed for the 2D/3D time-dependent thermomicropolarfluid equations. Thefirst-order and second-order backward difference formulas (BDF) are adopted to approximate the time derivative term, stability analysis and error estimation of the first-order semi-discrete scheme are proved. Finally, some numerical examples are given to show the effectiveness and reliability of the proposed method, which can be used to simulate the problem with high Rayleigh number.
For conforming finite element approximations of the Laplacian eigenfunctions, a fully computable guaranteed error bound in the $L^2$ norm sense is proposed. The bound is based on the a priori error estimate for the Galerkin projection of the conforming finite element method, and has an optimal speed of convergence for the eigenfunctions with the worst regularity. The resulting error estimate bounds the distance of spaces of exact and approximate eigenfunctions and, hence, is robust even in the case of multiple and tightly clustered eigenvalues. The accuracy of the proposed bound is illustrated…
The Ericksen model for nematic liquid crystals couples a director field with a scalar degree of orientation variable, and allows the formation of various defects with finite energy. We propose a simple but novel finite element approximation of the problem that can be implemented easily within standard finite element packages. Our scheme is projection-free and thus circumvents the use of weakly acute meshes, which are quite restrictive in 3D but are required by recent algorithms for convergence. We prove stability and $Γ$-convergence properties of the new method in the presence of defects. We a…
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
The $L^2$-orthogonal projection $Π_h:L^2(Ω)\rightarrow\mathbb{V}_h$ onto a finite element (FE) space $\mathbb{V}_h$ is called $H^1$-stable iff $\|\nablaΠ_h u\|_{L^2(Ω)}\leq C\|u\|_{H^1(Ω)}$, for any $u\in H^1(Ω)$ with a positive constant $C\neq C(h)$ independent of the mesh size $h>0$. In this work, we discuss local criteria for the $H^1$-stability of adaptively refined meshes. We show that adaptive refinement strategies for quadrilateral meshes in 2D (Q-RG and Q-RB), introduced originally in Bank et al. 1982 and Kobbelt 1996, are $H^1$-stable for FE spaces of polynomial degree $p=2,\ldots,9$.
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 this paper the projection hybrid FV/FE method presented in Busto et al. 2014 is extended to account for species transport equations. Furthermore, turbulent regimes are also considered thanks to the $k-\varepsilon$ model. Regarding the transport diffusion stage new schemes of high order of accuracy are developed. The CVC Kolgan-type scheme and ADER methodology are extended to 3D. The latter is modified in order to profit from the dual mesh employed by the projection algorithm and the derivatives involved in the diffusion term are discretized using a Galerkin approach. The accuracy and stabil…
Identities that relate projections of Raviart-Thomas finite element vector fields to discrete gradients of Crouzeix-Raviart finite element functions are derived under general conditions. Various implications such as discrete convex duality results and a characterization of the image of the projection of the Crouzeix-Ravaiart space onto elementwise constant functions are deduced.
The multimesh finite element method is a technique for solving partial differential equations on multiple non-matching meshes by enforcing interface conditions using Nitsche's method. Since the non-matching meshes can result in arbitrarily cut cells, additional stabilization terms are needed to obtain a stable variational formulation. In this contribution we extend the multimesh finite element method to the Navier-Stokes equations based on the incremental pressure correction scheme. For each step in the pressure correction scheme, we derive a multimesh finite element formulation with suitable …
Error analysis of projection methods for non inf-sup stable mixed finite elements. The transient Stokes problem
Universidad de Valladolid（瓦拉多利德大学）；Universidad de Sevilla（塞维利亚大学）；Universidad Autónoma de Madrid（马德里自治大学）；Instituto de Ciencias Matemáticas CSIC-UAM-UC3M-UCM（CSIC-UAM-UC3M-UCM数学科学研究所）
Micromechanical homogenization is often carried out with Fourier-accelerated methods that are prone to ringing artifacts. We here generalize the compatibility projection introduced by Vondřejc, Zeman & Marek [Comput. Math. Appl. 68, 156 (2014)] beyond the Fourier basis. In particular, we formulate the compatibility projection for linear finite elements while maintaining Fourier-acceleration and the fast convergence properties of the original method. We demonstrate that this eliminates ringing artifacts and yields an efficient computational homogenization scheme that is equivalent to canonical …
The local bounded commuting projection operators of nonstandard finite element de Rham complexes in two and three dimensions are constructed systematically. The assumptions of the main result are mild and can be verified. For three dimensions, the result can be applied to the standard finite element de Rham complex, Hermite complex, Argyris complex and Neilan's Stokes complex. For two dimensions, the result can be applied to the Hermite--Stenberg complex and the Falk--Neilan Stokes complex.
