A Particle-Partition of Unity Method--Part II: Efficient Cover Construction and Reliable Integration

Abstract

Abstract. In this sequel to [15,16] we focus on the efficient solution of the linear block-systems arising from a Galerkin discretization of an elliptic partial differential equation of second order with the partition of unity method (PUM). We present a cheap multilevel solver for partition of unity discretizations of any order. The shape functions of a PUM are products of piecewise rational partition of unity (PU) functions ϕ i with supp(ϕ i ) = ω i and higher order local approximation functions ψ n i (usually a local polynomial of degree ≤ p i ). Furthermore, they are non-interpolatory. In a multilevel approach we not only have to cope with non-interpolatory basis functions but also with a sequence of nonnested spaces due to the meshfree construction. Hence, injection or interpolatory interlevel transfer operators are not available for our multilevel PUM. Therefore, the remaining natural choice for the prolongation operators are L 2 -projections. Here, we exploit the partition of unity construction of the function spaces and a hierarchical construction of the PU itself to localize the corresponding projection problem. This significantly reduces the computational costs associated with the setup and the application of the interlevel transfer operators. The second main ingredient for our multilevel solver is the use of a block-smoother to treat the local approximation functions ψ n i for all n simultaneously. The results of our numerical experiments in two and three dimensions show that the convergence rate of the proposed multilevel solver is independent of the number of patches card({ω i }). The convergence rate is slightly dependent on the local approximation orders p i .

A Particle-Partition of Unity Method Part V: Boundary Conditions

“…In the following, we shortly review the construction partition of unity spaces and the meshfree Galerkin discretization of an elliptic PDE, see [12,13] for details. Furthermore, we give a summary of the efficient multilevel solution of the arising linear block-system, see [14] for details.…”

“…Here, the functions ϕ i form a partition of unity (PU). They are used to splice the local approximations u i together in such a way that the global approximation u PU benefits from the local approximation orders p i yet it still fulfills global regularity conditions, see [12]. Hence, the global approximation space on Ω is defined as The starting point for any meshfree method is a collection of N independent points P := {x i ∈ R d | x i ∈ Ω, i = 1, .…”

“…This makes the meshfree Galerkin discretization of a partial differential equation (PDE) more challenging than its discretization with a FEM. See [12,13,27,28] for details on the discretization process with the partition of unity method (PUM). Furthermore, the shape functions are in general non-interpolatory which makes the design of optimal linear solvers [14] as well as the implementation of essential boundary conditions not an easy task.…”

“…Many different approaches toward the treatment of Dirichlet problems have been proposed over the years [4,10,12,13,16,17,19,22,27]. Most of them, however, suffer from one or several drawbacks (restrictions on the distribution of the discretization points, need for an additional boundary function space, saddle-point structure, non-optimal convergence rates, etc.).…”

“…

Abstract In this sequel to [12, 13, 14, 15] we focus on the implementation of Dirichlet boundary conditions in our partition of unity method. The treatment of essential boundary conditions with meshfree Galerkin methods is not an easy task due to the non-interpolatory character of the shape functions.

…”