Application of essential boundary conditions in mesh-free methods: a corrected collocation method

Collocation methods for applying essential boundary conditions are defined as those methods in which conditions are enforced exactly at a discrete set of boundary nodes. In mesh-free methods, this is usually accomplished by replacing rows of the matrix equations which result from discretization of the weak form with equations which ensure the enforcement of boundary conditions. In this paper, an inconsistency in this method is pointed out, and a correction is derived. Numerical test are done on one- and two-dimensional equations; it is shown that convergence rates decrease with the use of the invalid traditional collocation and are restored with the corrected method.