A Convergent Linearized Lagrange Finite Element Method for the Magneto-hydrodynamic Equations in Two-Dimensional Nonsmooth and Nonconvex Domains

Li, Buyang; Wang, Jilu; Xu, Li

doi:10.1137/18m1205649

Cited by 27 publications

(6 citation statements)

References 44 publications

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“…Our future work includes the study of fast solvers for the proposed HDG method for the Maxwell operator. We will also consider the application of analysis procedure presented in this work to the numerical solution of incompressible magnetohydrodynamics [17].…”

Section: Resultsmentioning

confidence: 99%

Analysis of a hybridizable discontinuous Galerkin method for the Maxwell operator

Chen

Cui

2019

ESAIM: M2AN

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In this paper, we study a hybridizable discontinuous Galerkin (HDG) method for the Maxwell operator. The only global unknowns are defined on the inter-element boundaries, and the numerical solutions are obtained by using discontinuous polynomial approximations. The error analysis is based on a mixed curl-curl formulation for the Maxwell equations. Theoretical results are obtained under a more general regularity requirement. In particular for the low regularity case, special treatment is applied to approximate data on the boundary. The HDG method is shown to be stable and convergence in an optimal order for both high and low regularity cases. Numerical experiments with both smooth and singular analytical solutions are performed to verify the theoretical results.Here, n is the outward normal unit vector to its boundary Γ, f ∈ [L 2 (Ω)] 3 is an external source filed, g T ∈ H − 1 2 (div τ ; Γ) ∩ [H δ (Γ)] 3 is a given function with δ ∈ (0, 1/2], and H − 1 2 (div τ ; Γ) is the range space of "tangential trace" of space H(curl; Ω), see [4] for the detailed description of space H − 1 2 (div τ ; Γ). The computational electromagnetics is of great importance in many areas of engineering and science, such as aerospace industry, telecommunication, medicine, biology, etc. Many computational techniques have been developed for solving Maxwell equations in both frequency and time domains. The main difficulties and challenges in designing accurate, robust and efficient numerical methods for Maxwell equations of the formulation (1) are as follows.• The irregular geometries in the real-word electromagnetic problems may lead to low regularity of true solutions, which makes the designing of stable and accurate numerical scheme and its error analysis more complicated. quantities, and so on. The first work on HDG method for time-harmonic Maxwell equations was founded in [22]. Two types of HDG schemes were introduced therein and numerical experiments were presented to show the performance of proposed schemes. At a later time, HDG methods for the time-harmonic Maxwell equations with zero frequency were introduced and analyzed in [6]. The regularity assumptions for the convergence analysis are much more restrictive than (2), i.e.,with t, s, r ≥ 1. This assumption is too strong to be satisfied (hence not realistic in practice). Actually, it can not even be met for problems defined on bounded simply-connected Lipschitz polyhedral domains. Later in [5], an HDG method was proved to converge optimally in both energy norm and L 2 -norm for u under the regularity (2), and a posteriori error estimates were derived.Recently in [8,18], the authors studied the HDG methods for time-harmonic Maxwell equations with impedance boundary conditions and large wavenumber. The error analysis in both work was derived by using the regularity results given in [13,9], where the constants in error estimates were shown to be explicitly dependent on the wave number. We also refer to [3] for a nonconforming finite element method solving the two-dimensional curl-curl ...

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Section: Resultsmentioning

confidence: 99%

Analysis of a hybridizable discontinuous Galerkin method for the Maxwell operator

Chen

Cui

2019

ESAIM: M2AN

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“…For examples, Gunzburger et al [12] showed the existence, uniqueness and finite element approximations for MHD equations in three‐dimension‐domain; Johanna [22] investigated a finite difference scheme for incompressible magnetohydrodynamics equations in two‐dimensional case; Wang et al [30] developed a new two‐level finite element algorithm based on the Newton iterative method for the two‐ and three‐dimensional cases; Gao and Qiu [8] presented a semi‐implicit finite element method and established the optimal error estimates under low assumption on the exact solutions and domain geometries; Adler et al [1] investigated multigrid preconditioners based on specialized relaxation schemes for MHD system; Zhang et al [33, 34] developed fractional‐step algorithms to construct fully decoupled, linear and unconditionally energy stable time discretization scheme; Wang et al [28, 29] proposed unconditionally energy stable schemes by using a modified Crank–Nicolson method and established rigorous error estimates for MHD system and Cahn–Hilliard‐MHD system. Other numerical schemes and analysis for the MHD system with constant density could be founded in [15, 18] and references cited therein. However, in addition to the above mentioned difficulties, there are very few works to construct efficient numerical algorithm for MHD equations with variable density due to lack of the maximum principle for

\rho

in hyperbolic equation.…”

Section: Introductionmentioning

confidence: 99%

Stability and temporal error estimate of scalar auxiliary variable schemes for the magnetohydrodynamics equations with variable density

Chen,

He,

Chen

2023

Numerical Methods Partial

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In this article, we construct first‐ and second‐order semidiscrete schemes for the magnetohydrodynamics (MHD) equations with variable density based on scalar auxiliary variable (SAV) approach. These schemes are decoupled, unconditionally energy stable and only solve a sequence of linear differential equations at each time step. We carry out a rigorous error analysis for the first‐order SAV scheme in two‐dimensional case. Some numerical experiments are presented to verify the accuracy and stability.

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“…Later, He developed 𝐻 1 -conforming FEMs in [18] for solving the time-dependent MHD equations and proved error estimates of the numerical scheme. More works on 𝐻 1 -conforming FEMs for the MHD equations can be found in [1,10,13,19,24].…”

Section: Introductionmentioning

confidence: 99%

Optimal error estimates of a Crank–Nicolson finite element projection method for magnetohydrodynamic equations

Wang

Xia

et al. 2022

ESAIM: M2AN

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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 Van Kan type [Van Kan, SIAM J. Sci. Statist. Comput. 7 (1986) 870–891] 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. Error estimates are proved in the discrete L∞(0, T; L2) norm for the proposed decoupled finite element projection scheme. Numerical examples are provided to illustrate the theoretical results.

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A Convergent Linearized Lagrange Finite Element Method for the Magneto-hydrodynamic Equations in Two-Dimensional Nonsmooth and Nonconvex Domains

Cited by 27 publications

References 44 publications

Analysis of a hybridizable discontinuous Galerkin method for the Maxwell operator

Analysis of a hybridizable discontinuous Galerkin method for the Maxwell operator

Stability and temporal error estimate of scalar auxiliary variable schemes for the magnetohydrodynamics equations with variable density

Optimal error estimates of a Crank–Nicolson finite element projection method for magnetohydrodynamic equations

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