In this paper, we present a new approach to obtain the asymptotic expansion and superconvergence for the linear element on Union Jack mesh. First, we construct a generalized interpolation function and its discrete harmonic extension by using the energy embedding method and the method of separation of variables. Then, we present elaborate estimates for the generalized interpolation function and ...

We present an interesting result regarding the implication of truncating the wavepacket of the harmonic oscillator. We show that disregarding the non-significant tails of a function which is the superposition of eigenfunctions of the harmonic oscillator has a remarkable consequence. Namely, there exit infinitely many different superpositions giving rise to the same function on the interval. Uni...

The circle packing theorem is used to show that on any bounded valence transient planar graph there exists a non constant, harmonic, bounded, Dirichlet function. If P is a bounded circle packing in R 2 whose contacts graph is a bounded valence triangulation of a disk, then, with probability 1, the simple random walk on P converges to a limit point. Moreover, in this situation any continuous fun...

Braneworlds, understood here as double domain wall spacetimes, can be described in terms of a linear harmonic function, with kinks at the locations of the boundary branes. In a dynamical setting, there is therefore the risk that the boundary brane of negative tension, at whose location the value of the harmonic function is always lowest, can encounter a zero of this harmonic function, correspon...

We develop a solver for nonseparable, self adjoint elliptic equations with a variable coefficient. If the coefficient is the square of a harmonic function,a transformation of the dependent variable, results in a constant coefficient Poisson equation. A highly accurate, fast, Fourier-spectral algorithm can solve this equation. When the square root of the coefficient is not harmonic, we approxima...

Ringing, the high-frequency response of a structure to relatively low-frequency waves, has been a great concern in offshore engineering. To investigate this phenomenon, we solve the third-harmonic diffraction problem using a three-dimensional higher-order panel method in the low-frequency range. In this thesis, the boundary integral equations are derived from two different theories , based on c...

We develop a theoretical model that describes the second harmonic generation of light from the surface of the topological insulatorBi2Se3 and experimentally demonstrate that the technique is sensitive to the surface electrons. By performing a crystal symmetry analysis of Bi2Se3(111) we determine the nonlinear electric susceptibility tensor elements that give rise to second harmonic generation. ...