Finite element methods for Maxwell's equations.

We survey finite element methods for approximating the time harmonic Maxwell equations. We concentrate on comparing error estimates for problems with spatially varying coefficients. For the conforming edge finite element methods, such estimates allow, at least, piecewise smooth coefficients. But for Discontinuous Galerkin (DG) methods, the state of the art of error analysis is less advanced (we consider three DG families of methods: Interior Penalty type, Hybridizable DG, and Trefftz type methods). Nevertheless, DG methods offer significant potential advantages compared to conforming methods.

Finite Element Methods for Maxwell's Equations

A preasymptotic error analysis of the finite element method (FEM) and some continuous interior penalty finite element method (CIP-FEM) for the Helmholtz equation in two and three dimensions is proposed. $H^1$- and $L^2$-error estimates with explicit dependence on the wave number $k$ are derived. In particular, it is shown that if $k^{2p+1}h^{2p}$ is sufficiently small, then the pollution errors of both methods in $H^1$-norm are bounded by $O(k^{2p+1}h^{2p})$, which coincides with the phase error of the FEM obtained by existent dispersion analyses on Cartesian grids, where $h$ is the mesh size, and $p$ is the order of the approximation space and is fixed. The CIP-FEM extends the classical one by adding more penalty terms on jumps of higher (up to $p$th order) normal derivatives in order to reduce efficiently the pollution errors of higher order methods. Numerical tests are provided to verify the theoretical findings and to illustrate the great capability of the CIP-FEM in reducing the pollution effect.

Preasymptotic Error Analysis of Higher Order FEM and CIP-FEM for Helmholtz Equation with High Wave Number

Solving time-harmonic wave propagation problems by iterative methods is a difficult task, and over the last two decades, an important research effort has gone into developing preconditioners for the simplest representative of such wave propagation problems, the Helmholtz equation. A specific class of these new preconditioners are considered here. They were developed by researchers with various backgrounds using formulations and notations that are very different, and all are among the most promising preconditioners for the Helmholtz equation. 
The goal of the present manuscript is to show that this class of preconditioners are based on a common mathematical principle, and they can all be formulated in the context of domain decomposition methods called optimized Schwarz methods. This common formulation allows us to explain in detail how and why all these methods work. The domain decomposition formulation also allows us to avoid technicalities in the implementation description we give of these recent methods. 
The equivalence of these methods with optimized Schwarz methods translates at the discrete level into equivalence with approximate block LU decomposition preconditioners, and we give in each case the algebraic version, including a detailed description of the approximations used. While we chose to use the Helmholtz equation for which these methods were developed, our notation is completely general and the algorithms we give are written for an arbitrary second order elliptic operator. The algebraic versions are even more general, assuming only a connectivity pattern in the discretization matrix.

A Class of Iterative Solvers for the Helmholtz Equation: Factorizations, Sweeping Preconditioners, Source Transfer, Single Layer Potentials, Polarized Traces, and Optimized Schwarz Methods

The finite element method has been widely used to discretize the Helmholtz equation with various types of boundary conditions. The strong indefiniteness of the Helmholtz equation makes it difficult to establish stability estimates for the numerical solution. In particular, discontinuous Galerkin methods for Helmholtz equation with a high wave number result in very large matrices since they typically have more degrees of freedom than conforming methods. However, hybridizable discontinuous Galerkin (HDG) methods offer an attractive alternative because they have build-in stabilization mechanisms and a reduced global linear system. In this paper, we study the HDG methods for the Helmholtz equation with first order absorbing boundary condition in two and three dimensions. We prove that the proposed HDG methods are stable (hence well-posed) without any mesh constraint. The stability constant is independent of the polynomial degree. By using a projection-based error analysis, we also derive the error estimates in L2 norm for piecewise polynomial spaces with arbitrary degree.

An analysis of HDG methods for the Helmholtz equation

This paper analyzes the error estimates of the hybridizable discontinuous Galerkin (HDG) method for the Helmholtz equation with high wave number in two and three dimensions. The approximation piecewise polynomial spaces we deal with are of order $p\geq 1$. Through choosing a specific parameter and using the duality argument, it is proved that the HDG method is stable without any mesh constraint for any wave number $\kappa$. By exploiting the stability estimates, the dependence of convergence of the HDG method on $\kappa,h$, and $p$ is obtained. Numerical experiments are given to verify the theoretical results.

A Hybridizable Discontinuous Galerkin Method for the Helmholtz Equation with High Wave Number

This paper analyzes the error estimates of the hybridizable discontinuous Galerkin (HDG) method for the Helmholtz equation with high wave number in two and three dimensions. The approximation piecewise polynomial spaces we deal with are of order $p\geq 1$. Through choosing a specific parameter and using the duality argument, it is proved that the HDG method is stable without any mesh constraint for any wave number $\kappa$. By exploiting the stability estimates, the dependence of convergence of the HDG method on $\kappa,h$ and $p$ is obtained. Numerical experiments are given to verify the theoretical results.

We present and analyze a hybridizable discontinuous Galerkin (HDG) method for the time-harmonic Maxwell equations. The divergence-free condition is enforced on the electric field, then a Lagrange multiplier is introduced, and the problem becomes the solution of a mixed curl-curl formulation of the Maxwell's problem. The method is shown to be an absolutely stable HDG method for the indefinite time-harmonic Maxwell equations with high wave number. By exploiting the duality argument, the dependence of convergence of the HDG method on the wave number k, the mesh size h and the polynomial order p is obtained. Numerical results are given to verify the theoretical analysis.

An absolutely stable $hp$-HDG method for the time-harmonic Maxwell equations with high wave number

A robust multilevel method for hybridizable discontinuous Galerkin method for the Helmholtz equation

We have explored the skyrmion phases and phase diagram of poly- and single-crystal MnSi by the measurements of the magnetoelectric coefficient αE and ac magnetic susceptibility of the MnSi/0.7Pb(Mg1/3Nb2/3)O3–0.3PbTiO3 composite. We found that the regular skyrmion lattice phase in the single-crystal sample has been averaged in the MnSi polycrystal due to random grain orientations, which results in an extended skyrmion lattice-conical mixture phase down to 25 K. The magnitude of the out-of-phase component in αE of the polycrystal, not single crystal, decreases gradually with decreasing frequency. With the changing of the driven ac field, we reveal a depinning threshold behavior in both samples. The depinning field is stronger in the polycrystal than that in the single crystal and may be responsible for the diminishing of the dissipative behavior at lower frequency due to grain boundaries and defects. The composite magnetoelectric method provides a unique approach to probe topological phase dynamics.

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Peipei Lu

Papers

A Hybridizable Discontinuous Galerkin Method for the Helmholtz Equation with High Wave Number

A Hybridizable Discontinuous Galerkin Method for the Helmholtz Equation with High Wave Number

An absolutely stable $hp$-HDG method for the time-harmonic Maxwell equations with high wave number

A robust multilevel method for hybridizable discontinuous Galerkin method for the Helmholtz equation

Comparison of skyrmion phases between poly- and single-crystal MnSi by composite magnetoelectric method