A class of difference schemes with flexible local approximation
Journal of Computational Physics
Journal of Computational and Applied Mathematics
Parallel generalized finite element method for magnetic multiparticle problems
VECPAR'04 Proceedings of the 6th international conference on High Performance Computing for Computational Science
A fast object-oriented Matlab implementation of the Reproducing Kernel Particle Method
Computational Mechanics
Quasi-optimal rates of convergence for the Generalized Finite Element Method in polygonal domains
Journal of Computational and Applied Mathematics
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In this paper we present a meshfree discretization technique based only on a set of irregularly spaced points $x_i \in \mathbb R^d$ and the partition of unity approach. In this sequel to [M. Griebel and M. A. Schweitzer, SIAM J. Sci. Comput., 22 (2000), pp. 853--890] we focus on the cover construction and its interplay with the integration problem arising in a Galerkin discretization. We present a hierarchical cover construction algorithm and a reliable decomposition quadrature scheme. Here, we decompose the integration domains into disjoint cells on which we employ local sparse grid quadrature rules to improve computational efficiency. The use of these two schemes already reduces the operation count for the assembly of the stiffness matrix significantly. Now the overall computational costs are dominated by the number of the integration cells. We present a regularized version of the hierarchical cover construction algorithm which reduces the number of integration cells even further and subsequently improves the computational efficiency. In fact, the computational costs during the integration of the nonzeros of the stiffness matrix are comparable to that of a finite element method, yet the presented method is completely independent of a mesh. Moreover, our method is applicable to general domains and allows for the construction of approximations of any order and regularity.