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In this work, we propose and analyze two two-level hybrid Schwarz preconditioners for solving the Helmholtz equation with high wave number in two and three dimensions. Both preconditioners are defined over a set of overlapping subdomains, with each preconditioner formed by a global coarse solver and one local solver on each subdomain. The global coarse solver is based on the localized orthogonal decomposition (LOD) technique, which was proposed in [27,28] originally for the discretization schemes for elliptic multiscale problems with heterogeneous and highly oscillating coefficients and Helmholtz problems with high wave number to eliminate the pollution effect. The local subproblems are Helmholtz problems in subdomains with homogeneous boundary conditions (the first preconditioner) or impedance boundary conditions (the second preconditioner). Both preconditioners are shown to be optimal under some reasonable conditions, that is, a uniform upper bound of the preconditioned operator norm and a uniform lower bound of the field of values are established in terms of all the key parameters, such as the fine mesh size, the coarse mesh size, the subdomain size and the wave numbers. It is the first time to show that the LOD solver can be a very effective coarse solver when it is used appropriately in the Schwarz method with multiple overlapping subdomains. Numerical experiments are presented to confirm the optimality and efficiency of the two proposed preconditioners.

We formulate and analyze a multiscale method for an elliptic problem with an oscillatory coefficient based on a skeletal (hybrid) formulation. More precisely, we employ hybrid discontinuous Galerkin approaches and combine them with the localized orthogonal decomposition methodology to obtain a coarse-scale skeletal method that effectively includes fine-scale information. This work is the first step in reliably merging hybrid skeletal formulations and localized orthogonal decomposition to unite the advantages of both strategies. Numerical experiments are presented to illustrate the theoretical findings.

We propose a multigrid method to solve the linear system of equations arising from a hybrid discontinuous Galerkin (in particular, a single face hybridizable, a hybrid Raviart--Thomas, or a hybrid Brezzi--Douglas--Marini) discretization of a Stokes problem. Our analysis is centered around the augmented Lagrangian approach and we prove uniform convergence in this setting. Numerical experiments underline our analytical findings.

We have explored the skyrmion phases and phase diagram of poly and single-crystal MnSi by the measurements of the magnetoelectric coefficient alfaE and ac magnetic susceptibility of the MnSi/PMN-PT composite. We found that the regular skyrmion lattice phase in 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 alfaE of the polycrystal, not single crystal, decreases gradually with decreasing frequency. With the changing of the driven ac field, it reveals a depinning threshold behavior in both samples. The depinning field is stronger in the polycrystal than that in single crystal and maybe responsible for the diminishing of dissipative behavior at lower frequency due to grain boundaries and defects. The composite magnetoelectric method provides a unique approach to probe topological phase dynamics., Comment: 14 pages, 5 figures,submitted

The vortex phases in type-II superconductors are very important since they determine many magnetic and electric properties of the parent compound. However, a universal tool to characterize the vortex phases is still lacking. We demonstrate in a type-II superconductors YBa2Cu3O7-x polycrystal sample that its vortex phases and phase boundaries can be comprehensively studied by a magnetoelectric technique. In this method, a thin piezoelectric material 0.7Pb(Mg1/3Nb2/3)O3-0.3PbTiO3(PMN-PT) is mechanically bonded with YBa2Cu3O7-x to form a laminate structure and act as a strain gauge. The phase diagram of the YBa2Cu3O7-x polycrystalline was explored by this method. Surprisingly, it can accurately estimate the Hc1, irreversible line, Hc2 and distinguish among vortex glass, vortex liquid, non-vortex states. Moreover, it can probe the dynamic response under different frequencies and observe the threshold phenomena of vortex liquid phase. It can even account for the density of vortices in the vortex solid phase. Our technique is readily extended to investigate the vortex phases in other type-II superconductors., Comment: 23 pages, 6 figures

Uniform convergence of the geometric multigrid V-cycle is proven for HDG methods with a new set of assumptions on the injection operators from coarser to finer meshes. The scheme involves standard smoothers and local solvers which are bounded, convergent, and consistent. Elliptic regularity is used in the proofs. The new assumptions admit injection operators local to a single coarse grid cell. Examples for admissible injection operators are given. The analysis applies to the hybridized local discontinuous Galerkin method, hybridized Raviart-Thomas, and hybridized Brezzi-Douglas-Marini mixed element methods. Numerical experiments are provided to confirm the theoretical results., Comment: arXiv admin note: text overlap with arXiv:2011.14018

We introduce a homogeneous multigrid method in the sense that it uses the same embedded discontinuous Galerkin (EDG) discretization scheme for Poisson's equation on all levels. In particular, we use the injection operator developed in [LuRK2020] for HDG and prove optimal convergence of the method under the assumption of elliptic regularity. Numerical experiments underline our analytical findings., Comment: arXiv admin note: substantial text overlap with arXiv:2011.14018

We introduce a homogeneous multigrid method in the sense that it uses the same HDG discretization scheme for Poisson's equation on all levels. In particular, we construct a stable injection operator and prove optimal convergence of the method under the assumption of elliptic regularity. Numerical experiments underline our analytical findings.

Large magnetoelectric response in thin films is highly desired for high-throughput and high-density microelectronic applications. However, the d0 rule in single-phase compounds usually results in a weak interaction between ferroelectric and magnetic orders; the magnetoelectric coupling via elastic resonance in composites restricts their thin-film integration in broadband. Here, we effectuate a concurrence of ferroelectric-like and antiferromagnetic phase transitions in Sr2IrO4/SrTiO3 superlattices by artificial design periodically non-equivalent interfaces, where a maximum magnetoelectric coefficient of ~980 mV cm-1 Oe-1 can be measured. Evidenced by synchrotron X-ray absorption and electron energy loss spectroscopies, a lopsided electron occupation occurs at the interfacial Ti ions. From perturbative calculations and numerical results, a strong coupling of antiferromagnetism and asymmetric electron occupation mediated by spin-orbit interaction leads to a large bulk magnetoelectric response. This atomic tailoring of the quantum order parameters in 3d and 5d oxides provides an alternative pathway towards strong magnetoelectric effects with thin-film integrations.

We have developed a sensitive technique to probe the magnetic skyrmion phases and dynamics by employing the interfacial coupling effect in a magnetoelectric composite configuration. The study on a MnSi single crystal sample using this technique provides clear evidences for the skyrmion lattice phase and coexistence of skyrmion and conical phase. Above the Curie temperature TC, a region with strong spin fluctuation is revealed as well. By tuning the density of Skyrmion or disorder, a transition from the skyrmion lattice to skyrmion-conical coexisting phase is observed. The observation is in good agreement with a theoretical model which predicts the dissipation behavior in the coexistence phase., Comment: 18 pages, 5 figures

This paper considers the time-harmonic Maxwell equations with impedance boundary condition.We present $H^2$-norm bound and other high-order norm bounds for strong solutions. The $H^2$-estimate have been derived in [M. Dauge, M. Costabel and S. Nicaise, Tech. Rep. 10-09, IRMAR (2010)] for the case with homogeneous boundary condition. Unfortunately, their method can not be applied to the inhomogeneous case. The main novelty of this paper is that we follow the spirit of the $H^1$-estimate in [R. Hiptmair, A. Moiola and I. Perugia, Math. Models Methods Appl. Sci., 21(2011), pp. 2263-2287] and modify the proof by applying two inequalities of Friedrichs' type to make the $H^1$-estimate move into $H^2$-estimate and $W^{m, p}$-estimate.Finally, the dependence of the regularity estimates on the wave number is obtained, which will play an important role in the convergence analysis of the numerical solutions for the time-harmonic Maxwell equations.

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.