A high-order staggered meshless method for elliptic problems

TL;DR

This paper introduces a novel meshless method for elliptic problems that uses local primal-dual grid complexes and achieves high-order convergence without requiring a mesh.

Contribution

The paper proposes a truly meshless primal-dual discretization approach for elliptic equations that maintains polynomial reproduction and high-order convergence.

Findings

Achieves $O(h^{m})$ convergence in $L^2$ and $H^1$ norms.

Maintains polynomial reproduction to arbitrary orders.

Produces solutions similar to compatible mesh-based methods for discontinuous coefficients.

Abstract

We present a new meshless method for scalar diffusion equations which is motivated by their compatible discretizations on primal-dual grids. Unlike the latter though, our approach is truly meshless because it only requires the graph of nearby neighbor connectivity of the discretization points $\bm{x}_i$. This graph defines a local primal-dual grid complex with a \emph{virtual} dual grid, in the sense that specification of the dual metric attributes is implicit in the method's construction. Our method combines a topological gradient operator on the local primal grid with a Generalized Moving Least Squares approximation of the divergence on the local dual grid. We show that the resulting approximation of the div-grad operator maintains polynomial reproduction to arbitrary orders and yields a meshless method, which attains $O(h^{m})$ convergence in both $L^2$ and $H^1$ norms, similar to mixed finite element methods. We demonstrate this convergence on curvilinear domains using manufactured solutions. Application of the new method to problems with discontinuous coefficients reveals solutions that are qualitatively similar to those of compatible mesh-based discretizations.