We present a local Fourier analysis of multigrid methods for the two-dimensional curl-curl formulation of Maxwell's equations. Both the hybrid smoother proposed by Hiptmair and the overlapping block smoother proposed by Arnold, Falk and Winther are considered. The key to our approach is the identification of two-dimensional eigenspaces of the discrete curl-curl problem by decou-pling the Fourie...

We use the deformation method for moving grids to construct invertible transformation. We review the existing method by harmonic maps in 2D and discuss the difficulty it encounters in 3d domains. The proposed method is based on a div-curl-ode system, which can be used as constraints for a class of optimization problems. Preliminary numerical results demonstrated the effectiveness of the method....

Weak-gravitational-lensing distortions to the intensity pattern of 21-cm radiation from the dark ages can be decomposed geometrically into curl and curl-free components. Lensing by primordial gravitational waves induces a curl component, while the contribution from lensing by density fluctuations is strongly suppressed. Angular fluctuations in the 21-cm background extend to very small angular s...

Abstract In different branches of physics, we frequently deal with vector del operator (~∇). This del operator is generally used to find curl or divergence of a vector function or gradient of a scalar function. In many important cases, we need to know the parent vector whose curl or divergence is known or require to find the parent scalar function whose gradient is known. But the task is not ve...

We utilize group-theoretical methods to develop a matrix representation of differential operators that act on tensors of any rank. In particular, we concentrate on the matrix formulation of the curl operator. A self-adjoint matrix of the curl operator is constructed and its action is extended to a complex plane. This scheme allows us to obtain properties, similar to those of the traditional cur...

This paper deals with the isotropic realizability of a given regular divergence free field j in R3 as a current field, namely to know when j can be written as σ∇u for some isotropic conductivity σ > 0, and some gradient field ∇u. The local isotropic realizability in R3 is obtained by Frobenius’ theorem provided that j and curl j are orthogonal in R3. A counter-example shows that Frobenius’ cond...

These lectures introduce some of the basic theory of cosmic microwave background (CMB) polarization with the primary aim of developing the theory of CMB polarization from inflationary gravitational waves, as well as some of the related theory of weak gravitational lensing (cosmic shear) of CMB polarization. We begin with production of polarization by Thomson scattering. We then discuss tensor-h...

The topographical control of western boundary currents within a basin and zonal jets in a channel is investigated in terms of the potential vorticity (PV) and barotropic vorticity (BV: the curl of the depthintegrated velocity) budgets using isopycnic, adiabatic wind–driven experiments. Along the western boundary, the wind-driven transport is returned across latitude lines by the bottom pressure...

Triticum aestivum PBW-343 is grown in most of the regions of India, and it is one of the renewable sources most suitable for papermaking. Anatomical studies illustrate that vascular bundles near the periphery contain a strong sheath of sclerenchyma cells, which constitutes about 80% of the fibers. The total fibers in wheat straw are about 39.20%, and parenchyma and epidermal cells account for 3...