On the Convergence of Adaptive Iterative Linearized Galerkin Methods

A wide variety of different (fixed-point) iterative methods for the solution of nonlinear equations exists. In this work we will revisit a unified iteration scheme in Hilbert spaces from our previous work that covers some prominent procedures (including the Zarantonello, Ka\v{c}anov and Newton iteration methods). In combination with appropriate discretization methods so-called (adaptive) iterative linearized Galerkin (ILG) schemes are obtained. The main purpose of this paper is the derivation of an abstract convergence theory for the unified ILG approach (based on general adaptive Galerkin discretization methods) proposed in our previous work. The theoretical results will be tested and compared for the aforementioned three iterative linearization schemes in the context of adaptive finite element discretizations of strongly monotone stationary conservation laws

Spin-orbit driven Peierls transition and possible exotic superconductivity in CsW2_{2}O6_{6}

We study \textit{ab initio} a pyrochlore compound, CsW$_{2}$O$_{6}$, which exhibits a yet unexplained metal-insulator transition. We find that (1) the reported low-$T$ structure is likely inaccurate and the correct structure has a twice larger cell; (2) the insulating phase is not of a Mott or dimer-singlet nature, but a rare example of a 3D Peierls transition, with a simultaneous condensation of three density waves; (3) spin-orbit interaction plays a crucial role, forming well-nested bands. The high-$T$ (HT) phase, if stabilized, could harbor a unique $e_{g}+ie_{g}$ superconducting state that breaks the time reversal symmetry, but is not chiral. This state was predicted in 1999, but never observed. We speculate about possible ways to stabilize the HT phase while keeping the conditions for superconductivity

Lattice dynamics and electron-phonon interaction in (3,3) carbon nanotubes

We present a detailed study of the lattice dynamics and electron-phonon coupling for a (3,3) carbon nanotube which belongs to the class of small diameter based nanotubes which have recently been claimed to be superconducting. We treat the electronic and phononic degrees of freedom completely by modern ab-initio methods without involving approximations beyond the local density approximation. Using density functional perturbation theory we find a mean-field Peierls transition temperature of approx 40K which is an order of magnitude larger than the calculated superconducting transition temperature. Thus in (3,3) tubes the Peierls transition might compete with superconductivity. The Peierls instability is related to the special 2k_F nesting feature of the Fermi surface. Due to the special topology of the (n,n) tubes also a q=0 coupling between the two bands crossing the Fermi energy at k_F is possible which leads to a phonon softening at the Gamma point.Comment: 4 pages, 3 figures; to be published in Phys. Rev. Let

Adaptive local minimax Galerkin methods for variational problems

In many applications of practical interest, solutions of partial differential equation models arise as critical points of an underlying (energy) functional. If such solutions are saddle points, rather than being maxima or minima, then the theoretical framework is non-standard, and the development of suitable numerical approximation procedures turns out to be highly challenging. In this paper, our aim is to present an iterative discretization methodology for the numerical solution of nonlinear variational problems with multiple (saddle point) solutions. In contrast to traditional numerical approximation schemes, which typically fail in such situations, the key idea of the current work is to employ a simultaneous interplay of a previously developed local minimax approach and adaptive Galerkin discretizations. We thereby derive an adaptive local minimax Galerkin (LMMG) method, which combines the search for saddle point solutions and their approximation in finite-dimensional spaces in a highly effective way. Under certain assumptions, we will prove that the generated sequence of approximate solutions converges to the solution set of the variational problem. This general framework will be applied to the specific context of finite element discretizations of (singularly perturbed) semilinear elliptic boundary value problems, and a series of numerical experiments will be presented

Adaptive iterative linearization Galerkin methods for nonlinear problems

A wide variety of (fixed-point) iterative methods for the solution of nonlinear equations (in Hilbert spaces) exists. In many cases, such schemes can be interpreted as iterative local linearization methods, which, as will be shown, can be obtained by applying a suitable preconditioning operator to the original (nonlinear) equation. Based on this observation, we will derive a unified abstract framework which recovers some prominent iterative schemes. In particular, for Lipschitz continuous and strongly monotone operators, we derive a general convergence analysis. Furthermore, in the context of numerical solution schemes for nonlinear partial differential equations, we propose a combination of the iterative linearization approach and the classical Galerkin discretization method, thereby giving rise to the so-called iterative linearization Galerkin (ILG) methodology. Moreover, still on an abstract level, based on two different elliptic reconstruction techniques, we derive a posteriori error estimates which separately take into account the discretization and linearization errors. Furthermore, we propose an adaptive algorithm, which provides an efficient interplay between these two effects. In addition, the ILG approach will be applied to the specific context of finite element discretizations of quasilinear elliptic equations, and some numerical experiments will be performed

Pressure effects on crystal and electronic structure of bismuth tellurohalides

We study the possibility of pressure-induced transitions from a normal semiconductor to a topological insulator (TI) in bismuth tellurohalides using density functional theory and tight-binding method. In BiTeI this transition is realized through the formation of an intermediate phase, a Weyl semimetal, that leads to modification of surface state dispersions. In the topologically trivial phase, the surface states exhibit a Bychkov-Rashba type dispersion. The Weyl semimetal phase exists in a narrow pressure interval of 0.2 GPa. After the Weyl semimetal--TI transition occurs, the surface electronic structure is characterized by gapless states with linear dispersion. The peculiarities of the surface states modification under pressure depend on the band-bending effect. We have also calculated the frequencies of Raman active modes for BiTeI in the proposed high-pressure crystal phases in order to compare them with available experimental data. Unlike BiTeI, in BiTeBr and BiTeCl the topological phase transition does not occur. In BiTeBr, the crystal structure changes with pressure but the phase remains a trivial one. However, the transition appears to be possible if the low-pressure crystal structure is retained. In BiTeCl under pressure, the topological phase does not appear up to 18 GPa due to a relatively large band gap width in this compound

Lattice dynamics and electron-phonon coupling in transition metal diborides

The phonon density-of-states of transition metal diborides TMB2 with TM = Ti, V, Ta, Nb and Y has been measured using the technique of inelastic neutron scattering. The experimental data are compared with ab initio density functional calculations whereby an excellent agreement is registered. The calculations thus can be used to obtain electron-phonon spectral functions within the isotropic limit. A comparison to similar data for MgB2 and AlB2 which were subject of prior publications as well as parameters important for the superconducting properties are part of the discussion.Comment: 4 pages, 3 figure