scholarly journals Second-Order Regularity Estimates for Singular Schrödinger Equations on Convex Domains

2014 ◽  
Vol 2014 ◽  
pp. 1-10
Author(s):  
Xiangxing Tao
Keyword(s):  
Green Function ◽  
Singular Potential ◽  
Coefficient Matrix ◽  
Second Order ◽  
Schrodinger Equations ◽  
Schrödinger Equations ◽  
Convex Domains ◽  
Regularity Estimates ◽  
The Green Function ◽  
Atomic Hardy Space

LetΩ⊂ℝnbe a nonsmooth convex domain and letfbe a distribution in the atomic Hardy spaceHatp(Ω); we study the Schrödinger equations-div⁡(A∇u)+Vu=finΩwith the singular potentialVand the nonsmooth coefficient matrixA. We will show the existence of the Green function and establish theLpintegrability of the second-order derivative of the solution to the Schrödinger equation onΩwith the Dirichlet boundary condition forn/(n+1)<p≤2. Some fundamental pointwise estimates for the Green function are also given.

scholarly journals An Extremely Efficient Boundary Element Method for Wave Interaction with Long Cylindrical Structures Based on Free-Surface Green’s Function

2016 ◽  
Author(s):  
Yingyi Liu ◽  
Ying Gou ◽  
Bin Teng
Keyword(s):  
Free Surface ◽  
Green Function ◽  
Water Waves ◽  
Chebyshev Approximation ◽  
Wave Interaction ◽  
Coefficient Matrix ◽  
Influence Coefficient ◽  
Cauchy Type ◽  
Cylindrical Structures ◽  
The Green Function

The present study aims to develop an efficient numerical method for computing the diffraction and radiation of water waves with horizontal long cylindrical structures, such as floating breakwaters. A higher-order scheme is used to discretize geometry of the structure as well as the relevant physical quantities. As the kernel of this method, Wehausen&rsquo;s free-surface Green function is calculated by a newly-developed Gauss-Kronrod adaptive quadrature algorithm after elimination of its Cauchy-type singularities. To improve computational efficiency, a Chebyshev approximation approach is applied to a fast calculation of the Green function that needs evaluation thousands of times. In addition, OpenMP parallel technique is used to the formation of influence coefficient matrix, which significantly reduces CPU time. Finally, computations are performed on wave exciting forces and hydrodynamic coefficients for the long cylindrical structures, either floating or submerged. Comparison with other numerical and analytical methods demonstrates good performance of the present method.

A global random walk on grid algorithm for second order elliptic equations

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Karl K. Sabelfeld ◽  
Dmitrii Smirnov
Keyword(s):  
Differential Equation ◽  
Random Walk ◽  
Green Function ◽  
Elliptic Equations ◽  
Symmetry Property ◽  
Variable Coefficients ◽  
Second Order ◽  
Grid Algorithm ◽  
The Green Function ◽  
Second Order Elliptic Equations

Abstract We suggest in this paper a global random walk on grid (GRWG) method for solving second order elliptic equations. The equation may have constant or variable coefficients. The GRWS method calculates the solution in any desired family of m prescribed points of the gird in contrast to the classical stochastic differential equation based Feynman–Kac formula, and the conventional random walk on spheres (RWS) algorithm as well. The method uses only N trajectories instead of mN trajectories in the RWS algorithm and the Feynman–Kac formula. The idea is based on the symmetry property of the Green function and a double randomization approach.

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