Domain and boundary type meshless methods based on the Direct Multi-Elliptic Interpolation Method are presented. The approach is equivalent to a special RBF-method but completely avoids the solution of large, full and ill-conditioned systems, thus, the computational cost is significantly reduced. The method is illustrated through the example of the usual Poisson problem. Both Dirichlet and Neumann boundary conditions are investigated. The domain version of the method results in particular solutions, while the boundary version can be applied to solve homogeneous problems. Along Neumann boundaries, either off-boundary points can be introduced or a boundary reconstruction technique based on boundary interpolation can be applied. Some further possible applications are also outlined.