Stable enrichment and local preconditioning in the particle-partition of unity method

Abstract

Summary. This paper is concerned with the stability and approximation properties of enriched meshfree methods. In particular we focus on the particle-partition of unity method (PPUM) yet the presented results hold for any partition of unity based enrichment scheme.The goal of our enrichment scheme is to recover the optimal convergence rate of the uniform h-version independent of the regularity of the solution. Hence, we employ enrichment not only for modeling purposes but rather to improve the approximation properties of the numerical scheme. To this end we enrich our PPUM function space in an enrichment zone hierarchically near the singularities of the solution. This initial enrichment however can lead to a severe ill-conditioning and can compromise the stability of the discretization. To overcome the ill-conditioning of the enriched shape functions we present an appropriate local preconditioner which yields a stable and optimally conditioned basis independent of the employed initial enrichment. The construction of this preconditioner is of linear complexity with respect to the number of discretization points.We obtain optimal error bounds for an enriched PPUM discretization with local preconditioning that are independent of the regularity of the solution globally and within the employed enrichment zone we attain a kind of super-convergence.The results of our numerical experiments clearly show that our enriched PPUM with local preconditioning recovers the optimal convergence rate of O(h p ) of the uniform h-version globally. For the considered model problems from linear elastic fracture mechanics we obtain an improved convergence rate of O(h p+δ ) with δ ≥ 1 2 for p = 1. The condition number of the stiffness matrix is independent of the employed enrichment zone.

Multilevel preconditioners for embedded enriched partition of unity approximations

“…Thus, we employ enrichment by a singular function everywhere in ENR , see [24] for details on the construction of a stable basis for V ENR . Moreover, we use so-called block-Gauß-Seidel relaxation in V ENR where we collect all degrees of freedom defined on the same patch into a single block, see [22,23] for details.…”

Multilevel particle-partition of unity method

“…The algebraic refinement of an approximation spaces obviously improves the approximation properties, however, it may also adversely affect the stability of the basis functions and thereby the efficient iterative solution of the arising linear system. In the PPUM we can overcome these stability issues easily via the use of a so-called flat top PU and a local preconditioning technique [35]. Moreover, there is a multilevel solver available for the PPUM [17] that can cope with arbitrary enrichment functions.…”

“…We must rather enrich a vicinity of the crack tip (so-called geometric enrichment) to overcome the pollution effect and recover the optimal convergence behavior of V PU on independently of the regularity of the solution, see also [4,25]. We employ a hierarchical scheme [33,35] which yields such a geometric enrichment in the vicinity of the crack tip automatically during refinement of the cover C . The maximal size of the enrichment zone is determined by the curvature of the crack, since we must assume a straight crack for the validity of (13).…”

“…Such a preconditioner can be constructed efficiently via a Schur complement approach and eigenvalue decompositions of the local mass matrices with respect to the flat-top region ω FT,i on each enriched patch ω i ∩ E = ∅, see [34,35] for details of this construction. Thus, we can assume that the local approximation spaces satisfy (16) with…”

“…near the singularity of the solution [35]. Recall from Theorem 1 that it is sufficient to obtain boundsˆ i and˜ i for the local errors with respect to V i on ω i ∩ to bound the global error with respect to V PU on .…”