Partition of Unity Method

Abstract

In the following, we present a general partition of unity method (PUM) for a meshfree discretization of an elliptic partial differential equation. The approach is roughly as follows: The discretization is stated in terms of points x _i only. To obtain a trial and test space V^PU, a patch or volume ω _i ⊂ ℝ^d is attached to each point x _i such that the union of these patches form an open cover C_Ω= {ω _i } of the domain Ω, i.e. $\bar{\Omega } \subset \cup {\omega _{i}}$. Now, with the help of weight functions W _i : ℝ^d → ℝ with supp $({W_i}) = \overline {{\omega _i}}$ local shape functions φ _i are constructed by Shepard’s method. The functions φ _i form a partition of unity (PU). Then, each partition of unity function φ _i is multiplied with a sequence of local approximation functions ψ ⁿ _i to assemble higher order shape functions. These product functions φ _i ψ ⁿ _i are finally plugged into the weak form to set up a linear system of equations via a Galerkin discretization, which we discuss in the next chapter.