Wavelet Solutions To A Class Of Elliptic Boundary Value Problems

Partial diferential equations (PDEs) appear shortly after calculus, which comefrom practical problems. Elliptic diferential equations is an important class of PDEs.Since the complicated boundary conditions or other reasons, it is very difcult or evenimpossible to find an analytical solution. Therefore, numerical solutions are needed.Based on the work of Urban and Jia, we shall construct a class of spline wavelet functionswith homogeneous boundary conditions on the interval [0,1], and apply them to a kindof the elliptic boundary value problems of second order with variable coefcients in thisthesis.Firstly, we shall give the concepts of weak derivative, Sobolev space, and describe anelliptic boundary value problem model with homogeneous Dirichlet boundary conditions.After verifying that the problem well-defined, we introduce a numerical method—Galerkin method. Secondly, it is constructed that a class of refinable spline functionsand the corresponding wavelets with homogeneous boundary conditions on [0,1] by thefinite diference method. In addition, we discuss the approximation properties and thecharacterization theorem of wavelet basis. Finally, with pointing out a shortcomingof the multiresolution Galerkin method, we provide a preconditioned wavelet Galerkinmethod. Numerical experiments show efectiveness of our method.