Parallel Algorithm Of A Kind Of Elliptical PDEs Via PETSc

Poisson equation and Laplace equation were proposed in Newtonian mechanics to describe potential and its bias,which belong to typical partial differential equations.It is well known that simple homogeneous Poisson equation and Laplace equation are easy to solve numerically by some methods such as combined operator method,extrapolation method in multiple dimension boundary value.The discretization of Laplace equation by the five-point difference method is a traditional issue.However,Laplace equation is a basic partial differential equation,and its algorithm can be extended to the general Poisson equation.On this basis,this paper first generalizes the algorithm to solve a class of partial differential equations with inhomogeneous terms.Next,given that the existing studies all consider the case of constant coefficients,this paper will further propose a numerical algorithm suitable for equations with variable coefficients,which will be a great improvement.In addition,the algorithm proposed in this paper is a parallel numerical algorithm based on the domain decomposition technology,which is realized by PETSC,and uses a supercomputer to improve the operation speed.In terms of theory,for Helmholtz equation,based on Prof.Qian Tao who proposed an Adaptive Fourier Decomposition(AFD)technique in Reproducing Kernel Hilbert Space(RKHS),we proved that the solution space of Helmholtz equation is the Reproducing Kernel Hilbert Space(RKHS).And we also proposed the theoretical steps of the Weak Pre-orthogonal Adaptive Fourier Decomposition(WPOAFD)of this function space on the Riemann manifold given,and the convergence of the algorithm is proved.This kind of equation has a great application space in describing and controlling electromagnetic waves,and further work can develop its numerical algorithm around the theory proposed in this paper.