The Research Of Wavelet Boundary Element Method For Neumann Boundary Problem In The Angle Domain

In recent years, wavelet boundary element method has developed a new numericalmethod. Although there is few works about this method, it still shows unique advantagesand strong vitality at the beginning. To overcome the shortcoming of singular integraldifficulty existed in the natural boundary element method, in this paper, we mainly solvethe Neumann boundary value problem of Laplace equation in the angle domain to studythe application of the wavelet base function in the natural boundary element method, it notonly simplify the calculation process, but also improve the computation accuracy. Thebasic idea is that introducing conformal mapping to obtain the natural integral equation,transforming the differential equations into its equivalent variational problem by thenatural boundary naturalization, and then discreting it by the Galerkin-wavelet method orwavelet interpolation to get the corresponding stiffness matrix with unique advantage, itsubstantially reducing computation. Meanwhile we make the further analysis to the errorestimate of the numerical solution. The given example indicates the validity of each kindof wavelet boundary element method.Firstly, we retrospect the development history and research situation of the boundaryelement method, the boundary naturalization theory, conformal mapping and the waveletanalysis theory, and analyze the significance of the research projects. At the same time, weintroduce the natural boundary integral equation of Laplace equation and the Poissonintegral formula by the natural boundary naturalization on the typical domain, and thenatural boundary naturalization theory in the simply-connected domain and the applicationof the above theory in the angle domain, sector domain and rectangle domain are alsogiven.Secondly, since there is difficulty in solving the Neumann boundary value problem ofthe Laplace equation in the angle domain encounters hypersingular integral. In order tosimplify the problem and get more precise numerical solution, we take full advantages ofthe Shannon wavelet limited bandwidth in frequency domain and use Shannon waveletboundary element method to solve the problem. In the meantime we combine the properties of smooth, fast weaken and the interpolation of the wavelet base with thenatural boundary element method, it introduces the conformal mapping. And it effectivelyresolves the trouble of the singular integral when solving the boundary value problem ofLaplace equation in angle domain.Lastly, we apply Hermite Cubic Spline Multi-wavelet Natural Boundary ElementMethod to the Neumann boundary value problem of the Laplace equation in the angledomain and make further research. Moreover, the given example proves that the method iseffective and flexible.