The energy norm convergence rate of the finite element solution of the heat equation is reduced by the time-regularity of the exact solution. This paper presents an adaptive finite element treatment of time-dependent ...The energy norm convergence rate of the finite element solution of the heat equation is reduced by the time-regularity of the exact solution. This paper presents an adaptive finite element treatment of time-dependent singularities on the one-dimensional heat equation. The method is based on a Fourier decomposition of the solution and an extraction formula of the coefficients of the singularities coupled with a predictor-corrector algorithm. The method recovers the optimal convergence rate of the finite element method on a quasi-uniform mesh refinement. Numerical results are carried out to show the efficiency of the method.展开更多
The evolution of polarization singularities supported in a one-dimensional periodic plasmonic system is studied.The lateral inversion symmetry of the system,which breaks the in-plane inversion symmetry and up-down mir...The evolution of polarization singularities supported in a one-dimensional periodic plasmonic system is studied.The lateral inversion symmetry of the system,which breaks the in-plane inversion symmetry and up-down mirror symmetry simultaneously,yields abundant polarization states.A complete evolution process with geometry for the polarization states is traced.In the evolution,circularly polarized points(C points)can stem from 3 different processes.In addition to the previously reported processes occurring in an isolated band,a new type of C point appearing in two bands simultaneously due to the avoided band crossing,is observed.Unlike the dielectric system with a similar structure which only supports at-Γbound states in the continuum(BICs),accidental BICs off theΓpoint are realized in this plasmonic system.This work provides a new scheme of polarization manipulation for the plasmonic systems.展开更多
It is generally believed that matter inside or once entering a black hole will gravitationally fall into the center and form a size-less singularity, where the density goes to infinity and the spacetime breaks down wi...It is generally believed that matter inside or once entering a black hole will gravitationally fall into the center and form a size-less singularity, where the density goes to infinity and the spacetime breaks down with infinite curvature or gravitation. In accordance to the Unruh effect, one of the most surprizing predictions of quantum field theory, however, it is found from this study that such singularity cannot be actually formed because it violates the law of energy conservation. The total Unruh radiation energy of the size-less singularity is shown to be infinite, much greater than that the collapsing matter can generate. All the energies of the collapsing matter including the gravitational potential energy, deducted, are far below the Unruh radiation energy, increased, for the collapsing matter to form the singularity. The collapsing matter actually formed is shown to be not a size-less singular point but a small sphere with a finite radius, which is found to be dependent of the mass of the singularity sphere, approximately proportional to the square root of the mass. The radius of the singularity sphere cannot be zero, unless the mass also approaches to zero. The result obtained from this study not only provides us a quantum solution to the problem of black hole singularity, but also leads to profound implications to the spacetime and cosmology. The Unruh effect excludes a black hole to form a size-less singularity, which has a finite mass but infinite density, curvature, and Unruh radiation energy. A point-like or size-less singularity can only be massless and naked.展开更多
文摘The energy norm convergence rate of the finite element solution of the heat equation is reduced by the time-regularity of the exact solution. This paper presents an adaptive finite element treatment of time-dependent singularities on the one-dimensional heat equation. The method is based on a Fourier decomposition of the solution and an extraction formula of the coefficients of the singularities coupled with a predictor-corrector algorithm. The method recovers the optimal convergence rate of the finite element method on a quasi-uniform mesh refinement. Numerical results are carried out to show the efficiency of the method.
基金Project supported by the National Natural Science Foundation of China (Grant Nos.12074049 and 12047564)the Fundamental Research Funds for the Central Universities,China (Grant Nos.2020CDJQY-Z006 and 2020CDJQYZ003)the Research Foundation of SWUST (Grant No.21zx7141)。
文摘The evolution of polarization singularities supported in a one-dimensional periodic plasmonic system is studied.The lateral inversion symmetry of the system,which breaks the in-plane inversion symmetry and up-down mirror symmetry simultaneously,yields abundant polarization states.A complete evolution process with geometry for the polarization states is traced.In the evolution,circularly polarized points(C points)can stem from 3 different processes.In addition to the previously reported processes occurring in an isolated band,a new type of C point appearing in two bands simultaneously due to the avoided band crossing,is observed.Unlike the dielectric system with a similar structure which only supports at-Γbound states in the continuum(BICs),accidental BICs off theΓpoint are realized in this plasmonic system.This work provides a new scheme of polarization manipulation for the plasmonic systems.
文摘It is generally believed that matter inside or once entering a black hole will gravitationally fall into the center and form a size-less singularity, where the density goes to infinity and the spacetime breaks down with infinite curvature or gravitation. In accordance to the Unruh effect, one of the most surprizing predictions of quantum field theory, however, it is found from this study that such singularity cannot be actually formed because it violates the law of energy conservation. The total Unruh radiation energy of the size-less singularity is shown to be infinite, much greater than that the collapsing matter can generate. All the energies of the collapsing matter including the gravitational potential energy, deducted, are far below the Unruh radiation energy, increased, for the collapsing matter to form the singularity. The collapsing matter actually formed is shown to be not a size-less singular point but a small sphere with a finite radius, which is found to be dependent of the mass of the singularity sphere, approximately proportional to the square root of the mass. The radius of the singularity sphere cannot be zero, unless the mass also approaches to zero. The result obtained from this study not only provides us a quantum solution to the problem of black hole singularity, but also leads to profound implications to the spacetime and cosmology. The Unruh effect excludes a black hole to form a size-less singularity, which has a finite mass but infinite density, curvature, and Unruh radiation energy. A point-like or size-less singularity can only be massless and naked.
