Particle and mesh-free methods offer significant computational advantages in settings where quality mesh generation required for many compatible PDE discretizations may be expensive or even intractable. At the same time, the lack of underlying geometric grid structure makes it more difficult to construct mesh-free methods mirroring the discrete vector calculus properties of mesh-based compatible and mimetic discretization methods. In this talk we survey ongoing efforts at Sandia National Laboratories to develop new classes of locally and globally compatible meshfree methods that attempt to recover some of the key properties of mimetic discretization methods.

We will present two examples of recently developed “mimetic”-like meshfree methods. The first one is motivated by classical staggered discretization methods. We use the local connectivity graph of a discretization particle to define locally compatible discrete operators. In particular, the edge-to-vertex connectivity matrix of the local graph provides a topological gradient, whereas a generalized moving least-squares (GMLS) reconstruction from the edge midpoints defines a divergence operator. The second method can be viewed as a meshfree analogue of a finite volume type scheme. In this method, the metric information that would be normally provided by the mesh, such as cell volumes and face areas, is reconstructed algebraically, without a mesh. This reconstruction process effectively creates virtual cells having virtual faces and ensures a local conservation property matching that of mesh-based finite volumes. In contrast to similar recent efforts our approach does not involve a solution of a global optimization problem to find the virtual cell volumes and faces areas. Instead, we determine the necessary metric information by solving a graph Laplacian problem that can be effectively preconditioned by algebraic multigrid.

Several numerical examples will illustrate the mimetic properties of the new meshfree schemes. The talk will also review some of the ongoing work to build a modern software toolkit for mesh-free and particle discretizations that leverages Sandia’s Trillinos library and performance tools such as Kokkos.

Pavel Bochev is a Distinguished Member of the Technical Staff at Sandia National Laboratories in Albuquerque where he works in the Center for Computing Research. He joined Sandia in 2000 after six years of teaching and research at the University of Texas at Arlington.