arXiv subjects

Morten E. Nielsen

Publications and source records attributed to Morten E. Nielsen.

2 recordsLinked to original sources

arXiv2024

High-order numerical method for solving elliptic partial differential equations on unfitted node sets

In this paper, we present how high-order accurate solutions to elliptic partial differential equations can be achieved in arbitrary spatial domains using radial basis function-generated finite differences (RBF-FD) on unfitted node sets (i.e., not adjusted to the domain boundary). In this novel method, we only collocate on nodes interior to the domain boundary and enforce boundary conditions as constraints by means of Lagrange multipliers. This combination enables full geometric flexibility near boundaries without compromising the high-order accuracy of the RBF-FD method. The high-order accuracy and robustness of two formulations of this approach are illustrated by numerical experiments.

math.NA↗

arXiv2023

Node Subsampling for Multilevel Meshfree Elliptic PDE Solvers

Subsampling of node sets is useful in contexts such as multilevel methods, computer graphics, and machine learning. On uniform grid-based node sets, the process of subsampling is simple. However, on node sets with high density variation, the process of coarsening a node set through node elimination is more interesting. A novel method for the subsampling of variable density node sets is presented here. Additionally, two novel node set quality measures are presented to determine the ability of a subsampling method to preserve the quality of an initial node set. The new subsampling method is demonstrated on the test problems of solving the Poisson and Laplace equations by multilevel radial basis function-generated finite differences (RBF-FD) iterations. High-order solutions with robust convergence are achieved in linear time with respect to node set size.

math.NA↗