A High-Order Staggered Meshless Method for Elliptic Problems

Abstract

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 x i . This graph defines a local primal-dual grid complex with a 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.

A conservative, consistent, and scalable meshfree mimetic method

“…Generalized moving least squares (GMLS) is a non-parametric regression technique for the approximation of bounded linear functionals from scattered samples of their arguments [14]. GMLS has been used in meshfree collocation schemes to approximate point values of derivatives; see [19], [20], [15] and the references therein. Here we shall apply GMLS to approximate integrals as our goal is to develop a meshfree divergence operator based on the definition (2).…”

“…The virtual boundary in (15) is completely determined by the list of edges connected to the primal vertex v i . Any graph G(V, E) built on the point cloud can provide this information.…”

“…In [15] we used the local connectivity graph of each particle and its formal dual to mimic the staggered arrangement in div-grad stencils on primal-dual grids. The resulting "staggered" meshfree scheme behaves similarly to mesh-based div-compatible discretizations but is not locally conservative because the primal-dual mesh surrogate is defined independently for each particle.…”

“…The resulting "staggered" meshfree scheme behaves similarly to mesh-based div-compatible discretizations but is not locally conservative because the primal-dual mesh surrogate is defined independently for each particle. This paper is a logical extension of [15] aiming to deliver a meshfree scheme mirroring the conservation properties of traditional compatible discretizations.…”

A spatially adaptive high-order meshless method for fluid–structure interactions

“…This is important not only in handling large deformations relevant to FSI, but also in handling the insertion and deletion of nodes in h-adaptive schemes without the need to preserve mesh quality. In this work, we consider the generalized moving least squares (GMLS), a recently developed meshless method that has been shown to provide stable high-order solution to the div-grad [14] and steady Stokes [15] problems. GMLS provides a flexible optimization framework to obtain optimal recovery of linear target functionals over a desired reproduction space directly from scattered samples (GMLS particles), and is built upon a rigorous approximation theory [16,17].…”