An Algebraic Treatment of Essential Boundary Conditions in the Particle–Partition of Unity Method

Abstract

This paper is concerned with the treatment of essential boundary conditions in meshfree methods. In particular, we focus on the particle-partition of unity method (PPUM). However, the proposed technique is applicable to any partition of unity-based approach. We present an efficient scheme for the automatic construction of a direct splitting of a PPUM function space into the degrees of freedom suitable for the approximation of the Dirichlet data and the degrees of freedom that remain for the approximation of the PDE by simple linear algebra. Notably, our approach requires no restrictions on the distribution of the discretization points nor on the employed (local) approximation spaces. We attain the splitting of the global function space from the respective direct splittings of the employed local approximation spaces. Hence, the global splitting can be computed with (sub)linear complexity. Due to this direct splitting of the meshfree PPUM function space, we can implement a conforming local treatment of essential boundary data so that the realization of Dirichlet boundary values in the meshfree PPUM is straightforward. The presented approach yields an optimally convergent scheme, which is demonstrated by the presented numerical results.