Programming of Finite Element Methods in MATLAB

We discuss how to implement the linear finite element method for solving the Poisson equation. We begin with the data structure to represent the triangulation and boundary conditions, introduce the sparse matrix, and then discuss the assembling process. We pay special attention to an efficient programming style using sparse matrices in MATLAB

A prescription for projectors to compute helicity amplitudes in D dimensions

This article discusses a prescription to compute polarized dimensionally regularized amplitudes, providing a recipe for constructing simple and general polarized amplitude projectors in D dimensions that avoids conventional Lorentz tensor decomposition and avoids also dimensional splitting. Because of the latter, commutation between Lorentz index contraction and loop integration is preserved within this prescription, which entails certain technical advantages. The usage of these D-dimensional polarized amplitude projectors results in helicity amplitudes that can be expressed solely in terms of external momenta, but different from those defined in the existing dimensional regularization schemes. Furthermore, we argue that despite being different from the conventional dimensional regularization scheme (CDR), owing to the amplitude-level factorization of ultraviolet and infrared singularities, our prescription can be used, within an infrared subtraction framework, in a hybrid way without re-calculating the (process-independent) integrated subtraction coefficients, many of which are available in CDR. This hybrid CDR-compatible prescription is shown to be unitary. We include two examples to demonstrate this explicitly and also to illustrate its usage in practice.Comment: Matched to the version to be published in Eur. Phys. J.

Real-time Correlators and Hidden Conformal Symmetry in Kerr/CFT Correspondence

In this paper, we study the real-time correlators in Kerr/CFT, in the low frequency limit of generic non-extremal Kerr(-Newman) black holes. From the low frequency scattering of Kerr-Newman black holes, we show that for the uncharged scalar scattering, there exists hidden conformal symmetry on the solution space. Similar to Kerr case, this suggests that the Kerr-Newman black hole is dual to a two-dimensional CFT with central charges $c_L=c_R=12J$ and temperatures $T_L=\frac{(r_++r_-)-Q^2/M}{4\pi a}, T_R=\frac{r_+-r_-}{4\pi a}$. Using the Minkowski prescription, we compute the real-time correlators of charged scalar and find perfect match with CFT prediction. We further discuss the low-frequency scattering of photons and gravitons by Kerr black hole and find that their retarded Green's functions are in good agreement with CFT prediction. Our study supports the idea that the hidden conformal symmetry in the solution space is essential to Kerr/CFT correspondence.Comment: 15 pages, Latex; typos corrected, references updated; minor correction, published versio

Nonconforming Virtual Element Method for 2m2m-th Order Partial Differential Equations in Rn\mathbb R^n

A unified construction of the $H^m$-nonconforming virtual elements of any order $k$ is developed on any shape of polytope in $\mathbb R^n$ with constraints $m\leq n$ and $k\geq m$. As a vital tool in the construction, a generalized Green's identity for $H^m$ inner product is derived. The $H^m$-nonconforming virtual element methods are then used to approximate solutions of the $m$-harmonic equation. After establishing a bound on the jump related to the weak continuity, the optimal error estimate of the canonical interpolation, and the norm equivalence of the stabilization term, the optimal error estimates are derived for the $H^m$-nonconforming virtual element methods.Comment: 33page