General finite-volume framework for saddle-point problems of various physics

This article is dedicated to the general finite-volume framework used to discretize and solve saddle-point problems of various physics. The framework applies the Ostrogradsky–Gauss theorem to transform a divergent part of the partial differential equation into a surface integral, approximated by the summation of vector fluxes over interfaces. The interface vector fluxes are reconstructed using the harmonic averaging point concept resulting in the unique vector flux even in a heterogeneous anisotropic medium. The vector flux is modified with the consideration of eigenvalues in matrix coefficients at vector unknowns to address both the hyperbolic and saddle-point problems, causing nonphysical oscillations and an inf-sup stability issue. We apply the framework to several problems of various physics, namely incompressible elasticity problem, incompressible Navier–Stokes, Brinkman–Hazen–Dupuit–Darcy, Biot, and Maxwell equations and explain several nuances of the application. Finally, we test the framework on simple analytical solutions.

Transfer of spin angular momentum to a dielectric particle

We show here that in the sharp focus of a linearly polarized laser beam the spin vector flux has only transverse components (the effect of photonic wheels or photonic helicopter). For a linearly polarized optical vortex, the orbit-spin conversion leads to the appearance of both longitudinal and transverse components of the spin density vector in the focus. Spin-orbit conversion is experimentally demonstrated for a circularly polarized Gaussian beam when a transverse energy flux (orbital angular momentum) arises in the focus, which is transmitted to a microparticle and makes it rotate. Switching the handedness of circular polarization (from left to right) switches the microparticle rotation direction. It is also shown here that an azimuthally polarized vortex beam with an arbitrary integer topological charge generates in the focus a spin density vector that only has an axial component (pure magnetization), while the transverse spin flux is absent.

Error Estimates of H1-Galerkin Mixed Finite Element Methods for Nonlinear Parabolic Problem

In this paper, H1-Galerkin mixed element method is proposed to simulate the nonlinear Parabolic problem. The problem is considered in one dimensional space. and optimal error estimates are also established. In particular, our methods can simultaneously approximate the scalar unknown and the vector flux effectively, without requiring the LBB consistency condition.

Electrical resistivity

The next six chapters describe the transport phenomena associated with the flow of charge, heat, and matter. In each case there is a vector flux that is governed by a vector field. Linear relationships between flux and field include electrical resistivity (Chapter 17), thermal conductivity (Chapter 18), diffusion (Chapter 19), and thermoelectricity (Chapter 21). All are represented by second rank tensors similar to electric permittivity (Chapter 9), but the underlying physics is somewhat different. Transport properties are nonequilibrium phenomena governed by statistical mechanics and the concept of microscopic reversibility, rather than the second law of thermodynamics that applies to equilibrium properties such as specific heat, permittivity, and elasticity. Higher order tensors appear when the transport experiments are carried out in the presence of magnetic fields or mechanical stresses. Galvanomagnetic, thermomagnetic (Chapter 20), and piezoresistance effects (Chapter 22) require third- and fourth-rank tensors. When an electric field is applied to a conductor, an electric current flows through the sample. The field Ei (in V/m) is related to the current density Jj (in A/m2) through Ohm’s Law, where ρij is the electrical resistivity (in Ω m). In tensor form, . . . Ei = ρijJj . . . Ei and Jj are polar vectors (first rank polar tensors) and ρij is a second rank polar tensor property which follows Neumann’s law in the usual way. Sometimes it is more convenient to use the reciprocal relation involving the electrical conductivity σij : . . . Ji = σijEj . . . .