A Numerical Method Of Boundary Value Problem For Laplace Equation

Laplace equation,also known as harmonic equation,potential equation,first proposed by French mathematician Laplace.Solving the Laplace equation is an important mathematical problem in electromagnetics,astronomy and fluid dynamics.In this paper,we consider the boundary value problem of Laplace equation on the two-dimensional single-connected region.It appears in many scientific and engineering applications,such as plasma physics,geophysics,nondestructive testing,and cardiology.In recent years,some numerical methods have been proposed to solve this problem,such as difference method,finite element method,Monte Carlo method and so on.The main purpose of this paper is to provide a simple and effective numerical method to solve the boundary value problem of Laplace equation.The main idea isto refine the exact solution by linear combination of harmonic polynomial approximation of the boundary value problem,and the problem is to determine the unknown coefficients of the linear combinations,and then take an appropriate regularization method to solve this problem.The main ideas are as follows:The hypothetical region D???R² is a bounded region with a smooth boundary.Consider the following boundary value problemWhere v is the unit outside the normal vector of ?D,f ? L²??D?.From the theory of complex variable function,the real and imaginary parts of an analytic function satisfy the two-dimensional Laplace equation,which is called the harmonic function,which has the second order continuous partial derivative of the independent variable and satisfies the continuous function of the above equation.Known as k =0,1,……The harmonic polynomial is as follows:Define the harmonic polynomialswhere the constant The main idea of the harmonic polynomial method is to approximate boundary value problem?1?-?2?by a linear combination of harmonic polynomial method of the form where c₀ and c_k,j?k=1.……,N,j ?1,2?are real constants.To determine the parameters c₀ and ck,j,through the boundary conditions,we derive and solve the following equation:where the trace operator defined byThe operator defined by?5?is compact and injective.From lemma 3.4,we knowThe operator AN is compact and thus solve the problem that the operator equation is not suitable.Due to the ill posedness of the equation?5?,we consider the perturbation equation???Where the perturbation data f??L²????D?is satisfied???A regularized solution to?9?is a linear combination of harmonic polynomial???Where the coefficients the following equation:c0?,?,ck,j?,? are determined by solving:???There theorem 3.2 shows that exist positive constants C = C?D,rin,rex?such that???where ?0=rex/rin.If take ?>0,k0>1 and ?=?₀?1+??,N=k₀ ln |ln ?| and choose regularized parameters ?=?2/3 ?0-2N,there are???Where k₁=k₀ln?₀.In the first chapter,we introduce the boundary value problem of Laplace equation.In the second chapter,we introduce several methods for solving the boundary value problem of Laplace equation,such as difference method,boundary element method?BEM?.In the third chapter,we study a new method for solving the boundary value problem of Laplace equation,which is called harmonic polynomial method.and give the stability analysis with regularization.Then,numerical experiments are carried out.The fourth chapter gives the conclusion.