  In this paper, a new localized radial basis function (RBF) method based on
partition of unity (PU) is proposed for solving boundary and initial-boundary
value problems. The new method is benefited from a direct discretization
approach and is called the `direct RBF partition of unity (D-RBF-PU)' method.
Thanks to avoiding all derivatives of PU weight functions as well as all lower
derivatives of local approximants, the new method is faster and simpler than
the standard RBF-PU method. Besides, the discontinuous PU weight functions can
now be utilized to develop the method in a more efficient and less expensive
way. Alternatively, the new method is an RBF-generated finite difference
(RBF-FD) method in a PU setting which is much faster and in some situations
more accurate than the original RBF-FD. The polyharmonic splines are used for
local approximations, and the error and stability issues are considered. Some
numerical experiments on irregular 2D and 3D domains, as well as cost
comparison tests, are performed to support the theoretical analysis and to show
the efficiency of the new method.
