Two-dimensional polynomial splines

{~i} and their derivatives of order k or less converge uniformly to u and its corresponding derivatives. We show that these splines can be used to obtain a numerical method for solving elliptic boundary value problems. The method of construction can be easily extended to higher dimensional Euclidean spaces. For one dimensional splines, see, for example, references [2] and [3JWe denote by G a bounded open domain of class C o (see reference It] for definition) in the Euclidean plane with boundary ~G and closure G. For a function uEC/(G) denote by Illuilli, the maximum norm max max l D~u(~)] and denote by the norm (f ~, I D~u(~)]ZdY)t, Here ~ = (~1, ~)is the multiple G I~l 1 and an integer p >--_ 0. Subdivide the domain ~ by a grid into a finite number of subdomains such that each subdomain is a polygon with infinitely differentiable curvilinear sides. We call any such subdomain a cell. Let {Si} be the set of grid lines forming the sides of cells. We require that the interior of any grid line Si must not contain a vertex. Let sup (Length of S¢) -- h. We denote by {Gt} the disjoint collection of interiors of the cells and let G 0---U Gi. Assume that each grid line, S t, is a non-singular curve, defined by an equation of the form x = fi (~3