Trigonometric Wavelet Methods Based On Natural Boundary Reduction For Some Elliptic Boundary Value Problems

In this paper, trigonometric wavelet methods based on natural boundary reduction for a class of elliptic boundary value problems are investigated. Poission integral equations and natural integral equations of the elliptic boundary value problems in an interior unit disk or an exterior unit one are obtained by the principle of the natural boundary reduction method. Hermite trigonometric wavelets introduced by Quak [Math. Comp., 65:214(1996), 683-722] as trial functions are used to construct the finite element subspace. They are used to the Galerkin discretization with 2^j+1 nodes on the boundary to solve Poission integral equations and natural integral equations numerically. It is proved that the stiffness matrix of the finite element solution is a block diagonal matrix and its elements are some symmetric and block circulant submatrices. The simple computational formulae of the entries in the stiffness matrix are obtained and they are algebraic operations of finite degree. These show that we only need to compute 2(2^j+1) elements of a 2^j+2 × 2^j+2 stiffness matrix. The convergence of the approximate solutions and their error estimates are obtained. Finally, some numerical examples are presented to show that our methods are effective.