Approximation Of Green's Function Research Articles

Local ultraconvergence of linear and bilinear finite element method for second order elliptic problems

In this article, local ultraconvergence in gradient of the bilinear and linear finite element solutions to two-dimensional second order elliptic problems has been investigated. Special interpolation postprocessing algorithms of numerical solutions by Richardson extrapolation have been developed. The local symmetry theory and estimates of discrete Green functions are used in the analysis. Numerical experiments are provided to confirm the theoretical findings.

Optimal decay to the non-isentropic compressible micropolar fluids

In this paper, we are concerned with the large-time behavior of solutions to the Cauchy problem on the non-isentropic compressible micropolar fluid. For the initial data near the given equilibrium we prove the global well-posedness of classical solutions and obtain the optimal algebraic rate of convergence in the three-dimensional whole space. Moreover, it turns out that the density, the velocity and the temperature tend to the corresponding equilibrium state with rate $ (1+t)^{-3 / 4} $ in $ L^{2} $ norm and the micro-rotational velocity tends to the equilibrium state with the faster rate $ (1+t)^{-5 / 4} $ in $ L^{2} $ norm. The proof is based on the detailed analysis of the Green function and time-weighted energy estimates.

Dirichlet-to-Neumann and elliptic operators on C1+κ-domains: Poisson and Gaussian bounds

We prove Poisson upper bounds for the heat kernel of the Dirichlet-to-Neumann operator with variable Hölder coefficients when the underlying domain is bounded and has a C1+κ-boundary for some κ>0. We also prove a number of other results such as gradient estimates for heat kernels and Green functions G of elliptic operators with possibly complex-valued coefficients. We establish Hölder continuity of ∇x∇yG up to the boundary. These results are used to prove Lp-estimates for commutators of Dirichlet-to-Neumann operators on the boundary of C1+κ-domains. Such estimates are the keystone in our approach for the Poisson bounds.

Decomposition and pointwise estimates of periodic Green functions of some elliptic equations with periodic oscillatory coefficients

This article is about the $\mathbb{Z}^d$-periodic Green function $G_n(x,y)$ of the multiscale elliptic operator $Lu=-{\rm div}\left( A(n\cdot) \cdot \nabla u \right)$, where $A(x)$ is a $\mathbb{Z}^d$-periodic, coercive, and H\older continuous matrix, and $n$ is a large integer. We prove here pointwise estimates on $G_n(x,y)$, $\nabla_x G_n(x,y)$, $\nabla_y G_n(x,y)$ and $\nabla_x \nabla_y G_n(x,y)$ in dimensions $d \geq 2$. Moreover, we derive an explicit decomposition of this Green function, which is of independent interest. These results also apply for systems.

Estimates of Dirichlet heat kernel for symmetric Markov processes

We consider a large class of symmetric pure jump Markov processes dominated by isotropic unimodal Lévy processes with weak scaling conditions. First, we establish sharp two-sided heat kernel estimates for these processes in C1,1 open sets. As corollaries of our main results, we obtain sharp two-sided Green function estimates and a scale invariant boundary Harnack inequality with explicit decay rates in C1,1 open sets.

Poisson equation on complete manifolds

We develop heat kernel and Green's function estimates for manifolds with positive bottom spectrum. The results are then used to establish existence and sharp estimates of the solution to the Poisson equation on such manifolds with Ricci curvature bounded below. As an application, we show that the curvature of a steady gradient Ricci soliton must decay exponentially if it decays faster than linear and the potential function is bounded above.

Near‐field scattering evaluation based on improved PO and EECs

An algorithm for the improvement of the physical optics (PO) and equivalent edge currents (EECs) integral calculating near-field scattering is presented in this work. The solution is obtained by a more accurate representation of the Green function with an expansion centre that lies in the neighbourhood of the sources of interest. Phase and amplitude improvements of the approximate Green function overcome the shortcomings of the conventional PO and EEC for scattering evaluation in near-field regions. The improved algorithm preserves the simplicity and computational efficiency of the classical one and applies directly to near-field scattering. Under the consideration of the antenna radiation pattern, the formula of near-field scattering is derived. The applicability and accuracy of the proposed algorithm is demonstrated by numerical simulations.

Brownian motion on some spaces with varying dimension

In this paper, we introduce and study Brownian motion on a class of state spaces with varying dimension. Starting with a concrete case of such state spaces that models a big square with a flag pole, we construct a Brownian motion on it and study how heat propagates on such a space. We derive sharp two-sided global estimates on its transition density function (also called heat kernel). These two-sided estimates are of Gaussian type, but the measure on the underlying state space does not satisfy volume doubling property. Parabolic Harnack inequality fails for such a process. Nevertheless, we show Hölder regularity holds for its parabolic functions. We also derive the Green function estimates for this process on bounded smooth domains. Brownian motion on some other state spaces with varying dimension are also constructed and studied in this paper.

The Study on Near-Field Scattering of a Target Under Antenna Irradiation by TDSBR Method

An efficient time domain shooting and bouncing ray (TDSBR) method is proposed to analyze the transient near-field scattering from an electrically large complex object illuminated by a far-field antenna source. The time-domain far-field incident sources can be derived by a convolution of pulsed excitation with the inverse Fourier transform of frequency domain (FD) far-fields radiated from an antenna. In order to obtain accurate near-field scattering results, time domain physical optics (TDPO) near-field integral representations are proposed and reduced to closed-form expressions to improve computing efficiency by applying locally expanded Green function approximation. Different from the plane wave incident situation, in the case of antenna radiation, the incident fields of each individual facet in the ray path are represented by the radiation fields from the equivalent mirror antenna derived in this paper. Since the radiation fields naturally contains the time delay term, the complicated processing of the time delay problem can be avoided by solving the equivalent mirror antenna radiation fields. It is simpler and more accurate than the traditional time domain geometrical optics (TDGO) method. Numerical examples are presented to demonstrate the efficiency and accuracy of the proposed method.

On Estimates of Transition Density for Subordinate Brownian Motions with Gaussian Components in C1,1-open Sets

We consider a subordinate Brownian motion $X$ with Gaussian components when the scaling order of purely discontinuous part is between $0$ and $2$ including $2$. In this paper we establish sharp two-sided bounds for transition density of $X$ in ${\mathbb R}^d$ and $C^{1,1}$-open sets. As a corollary, we obtain a sharp Green function estimates.

A neural network approach for determining spatial and geometry dependent Green's functions for thermal stress approximation in power plant header components

The trend in power generation to operate plant with a greater frequency of on/partial/off load conditions creates several concerns for the long term structural integrity of many high temperature components. The Green's function method has been used for many years to estimate the thermal stresses in components such as steam headers by attempting to solve the un-coupled thermal stress problem for a unit temperature step. Once a Green's function for a unit temperature step has been determined, realistic or actual component temperature profiles can be discretised and the time dependent stress profile reconstructed using Duhamel's theorem. Stress fluctuations can therefore be estimated and damage due to fatigue mechanisms can be quantified. A potential difficulty with this method is that Green's function approximations are determined for a single analysis point in a structure. This is because Green's functions are approximated by fitting a trial function to the results of finite element (FE) simulations. While a user can make some judgement on which point in a structure will give the “worst case” (or life limiting) conditions, it is foreseeable that points of interest will be dependent on the specific analysis conditions, such as the stub penetration geometry and the loading condition considered. The neural network approach described in this paper provides a means where transient thermal stress models of complex components (here taken to be steam headers) can be generated relatively quickly and used pro-actively to assess and modify plant operation. A range of header geometries have been considered to make the network applicable over an industry relevant envelope. Coefficients of determination (R2) are typically above 0.92 when reconstructed (from neural network results) unit temperature step stress profiles are compared against “true” FEA results. Mean errors in the stress profiles are, for the majority of cases, less than 10%. Suggestions are also made on possible future improvements to the method through the use of additional constraints on the reconstructed stress profiles.

Stability and <inline-formula><tex-math id="M1">\begin{document}$ L^{p}$\end{document}</tex-math></inline-formula> convergence rates of planar diffusion waves for three-dimensional bipolar Euler-Poisson systems

In the paper, we consider a three-dimensional bipolar hydrodynamic model from semiconductor devices and plasmas. This system takes the form of Euler-Poisson with electric field and relaxation term added to the momentum equations. We first construct the planar diffusion waves. Next we show the global existence of smooth solutions for the initial value problem of three-dimensional bipolar Euler-Poisson systems when the initial data are near the planar diffusive waves. Finally, we also establish the \begin{document}$ L^p(p∈[2,+∞])$\end{document} convergence rates of the solutions toward the planar diffusion waves. A frequency decomposition, approximate Green function and delicate energy method are used to prove our results.

Estimates for Green functions of Stokes systems in two dimensional domains

We prove the existence and pointwise bounds of the Green functions for stationary Stokes systems with measurable coefficients in two dimensional domains. We also establish pointwise bounds of the derivatives of the Green functions under a regularity assumption on the $L_1$-mean oscillations of the coefficients.

Pointwise estimates and [formula omitted] convergence rates to diffusion waves for a one-dimensional bipolar hydrodynamic model

In this paper, we study the stability of the diffusion wave to the one-dimensional hydrodynamic model, which takes the bipolar Euler–Poisson system with relaxation effect. The pointwise estimate of the smooth solutions is obtained by the weighted energy method and the approximate Green function when the initial perturbations are sufficiently small. Based on it, we further achieve the optimal time decay rate of the solutions in Lp(1≤p≤+∞). It coincides with the time decay rate of the solution in Gasser et al. (2003).

Efficient approximation of free-surface Green function and OpenMP parallelization in frequency-domain wave–body interactions

The evaluation of Green function without translation has always been the most time-consuming part of the hydrodynamic analysis based on potential flow theory. This research investigates the new implementations of efficient approximations of Green function and the code parallelization with OpenMP library. First, algorithms of infinite-depth case are developed with economized Chebyshev polynomials based on modified region and subdomain partitions. Second, an improved Gauss–Laguerre method for the Cauchy integral is presented to evaluate the finite-depth case with the convergence acceleration by the reduction of fraction and singularity elimination by the infinite-depth case together with the exponential integral. For the analyzed container vessel, on the premise of good numerical accuracy, we conclude that the present method has nearly obtain the same efficiency as Chen (Hydrodynamics in offshore and naval applications—Part I, Bureau Veritas, Paris, 2004) for the infinite-depth case and almost 30% higher efficiency for the finite-depth case. Moreover, we also point out that the numerical efficiency is significantly enhanced again due to the OpenMP parallelization of the serial code.

Retrieval of Elastodynamic Reflections From Passive Double‐Couple Recordings

AbstractVirtual Green's functions obtained by seismic interferometry (SI) can provide valuable reflectivity data that can complement tomographic inversion schemes. However, virtual reflections are affected by illumination irregularities that are typical of earthquake‐induced wavefields recorded by the receiver array. As a consequence, irregular source distributions, scattering (in case of suboptimal illumination), and complex source mechanisms can significantly disturb the retrieval of Green's function approximations by conventional SI methods. We introduce SI by full‐field multidimensional deconvolution (MDD) for elastodynamic wavefields as an alternative method to obtain body wave Green's functions under those typical circumstances. The advantage of this method compared to other MDD methods is that the kernel of its governing equation is exact. This alternative formulation of the kernel pertains to several advantages: the solution is less sensitive to artifacts and utilizes the free‐surface multiples in the data to estimate primary reflections. Moreover, the point spread function of the full‐field MDD method corrects more affectively for irregular illumination because it also addresses irregularities caused by scattering inside the medium. In order to compare full‐field MDD to existing SI methods, we model synthetic earthquake recordings in a subduction zone setting using an elastodynamic finite‐difference scheme with double couples of different orientations and peak frequencies. Our results show that SI by cross correlation suffers most under these circumstances. Higher‐quality reflections are obtained by the MDD methods, of which full‐field MDD involves the most stable inversion, and its results are least contaminated by artifacts.

Bayesian rupture imaging in a complex medium: The 29 May 2012 Emilia, Northern Italy, earthquake

AbstractWe develop a new approach to image earthquake rupture from strong motion data. We use a large data set of aftershock waveforms, interpolated over the seismic fault to obtain Green's function approximations. Next we deploy a Bayesian inversion method to characterize the slip distribution, the rupture velocity, the slip duration, and their uncertainties induced by errors in the Green's functions. The method is applied to the 29 May 2012 Mw 6 Emilia earthquake, which ruptured a fault buried below the Po Plain sediments (Northern Italy). Despite the particularly complex wave propagation, the near‐field strong motion observations are well reproduced with 15 rupture parameters. The rupture and slip velocities were notably slow (~0.5 Vs and <0.5 m/s, respectively), implying that the fault was difficult to break. This method opens some perspectives for earthquake rupture studies in areas where numerical simulations suffer from imprecise knowledge of the velocity structure.

Application of time reversal to passive acoustic remote sensing of the ocean

This paper investigates a novel approach to processing records of ambient noise in the ocean that are measured concurrently in spatially separated locations. The approach is a synthesis of two well-known phase-coherent signal processing techniques. At the first stage of processing, an approximation to the transient acoustic Green function is found by the method of noise interferometry. At the second stage, the approximate Green function is time reversed and back propagated from the location of one of the receivers, thereby producing a focus in the vicinity of the other receiver. Unlike the earlier work, measurements at just two points (rather than vertical array measurements) are used when the sound-propagation range is large compared to the ocean depth. The requirement for optimal focusing of the back-propagated field is shown to lead to extraction of estimates of the unknown physical parameters of the waveguide and, hence, to passive acoustic remote sensing of the ocean.

Integral propagator method as a kinetic operator to describe discontinuous plasmas

We present a path-integral-like method to numerically solve drift-diffusion equations for plasma physics. The algorithm uses short-time propagators as approximate Green's functions that tend to smooth typical discontinuities arising in plasma dynamics as, for instance, the effects of plasma-wall interaction or localized particle flows. The usual numerical schemes based on differences may fail to represent these abrupt conditions by inducing numerical viscosities and instabilities. However, the robust mesli-free computational integral method has been proved to be unconditionally stable in the simple case of no imposed restrictions by boundary conditions. The extension of the method to deal with boundary value problems is analysed. The advancing scheme is also useful to deal with the merging of natural discontinuities in the system as those induced by the effects of electromagnetic fields generated by charge separation as well as for the existence of two differentiated plasma regimes. In any case, the kinetic equation may have drift and/or diffusion coefficients that are likewise discontinuous. The method works almost being numerically insensitive to these discontinuities, leading to feasible physically meaningful solutions. The scheme works as an effective integral kinetic operator even for Fokker-Planck equations connecting two different dynamical statistics, like Maxwell-Bolt.zmann and Fermi- Dirac ones. It is robust to describe spatially heterogeneous plasmas.

Merging Features from Green’s Functions and Time Dependent Density Functional Theory: A Route to the Description of Correlated Materials out of Equilibrium?

We propose a description of nonequilibrium systems via a simple protocol that combines exchange-correlation potentials from density functional theory with self-energies of many-body perturbation theory. The approach, aimed to avoid double counting of interactions, is tested against exact results in Hubbard-type systems, with respect to interaction strength, perturbation speed and inhomogeneity, and system dimensionality and size. In many regimes, we find significant improvement over adiabatic time dependent density functional theory or second Born nonequilibrium Green's function approximations. We briefly discuss the reasons for the residual discrepancies, and directions for future work.