Hermite bicubic collocation method in domain with curvilinear boundary

In this paper the collocation method with Hermite bicubic basis functions is considered, applied to the first boundary value problem for an elliptic equation in a domain with a curved boundary. The collocation method has some advantages compared with Galerkin finite element method: no need to compute the integrals for the determination of the coefficients of the stiffness matrix. Hermite bicubic functions belongs to the class C 1. The consistent with the boundary mesh is constructed for solving the problem. The grid is consistent with the border so that two nodes in irregular cells were lying on a curvilinear boundary. This approach allows to reduce the total number of basis functions in the domain. As an internal collocation nodes the points of the set of Gauss are taken. The collocation points are distributed evenly on a curvilinear boundary. Under such arrangement, the total number of collocation points equal the total number of basis functions with the given boundary conditions. The problem is reduced to solution of a linear system Au=f where A is a square matrix. The results of numerical experiments of solving Poisson equation with different right sides show that the algorithm of solution has the convergence of high order.