Showing papers by "Susanne C. Brenner published in 1999"
TL;DR: It is proved that there is a bound (< 1) for the contraction number of the W-cycle algorithm which is independent of mesh level, provided that the number of smoothing steps is sufficiently large.
Abstract: We consider nonconforming multigrid methods for symmetric positive definite second and fourth order elliptic boundary value problems which do not have full elliptic regularity. We prove that there is a bound (< 1) for the contraction number of the W-cycle algorithm which is independent of mesh level, provided that the number of smoothing steps is sufficiently large. We also show that the symmetric variable V-cycle algorithm is an optimal preconditioner.
103 citations
TL;DR: The Poisson equation −Δu=f with homogeneous Dirichlet boundary condition on a two-dimensional polygonal domain Ω with cracks is considered and multigrid methods for the computation of singular solutions and stress intensity factors using piecewise linear functions are analyzed.
Abstract: We consider the Poisson equation −Δu=f with homogeneous Dirichlet boundary condition on a two-dimensional polygonal domain Ω with cracks. Multigrid methods for the computation of singular solutions and stress intensity factors using piecewise linear functions are analyzed. The convergence rate for the stress intensity factors is\(\mathcal{O}(h^{(3/2) - \in } )\) whenfeL2(Ω) and\(\mathcal{O}(h^{(2 - \in )} )\) whenfeH1(Ω). The convergence rate in the energy norm is\(\mathcal{O}(h^{(1 - \in )} )\) in the first case and\(\mathcal{O}(h)\) in the second case. The costs of these multigrid methods are proportional to the number of elements in the triangulation. The general case wherefeHm(Ω) is also discussed.
81 citations
TL;DR: It is shown that for elliptic boundary value problems of order 2m the condition number of the Schur complement matrix that appears in nonoverlapping domain decomposition methods is of order d-1 h-2m+1, where d measures the diameters of the subdomains and h is the mesh size of the triangulation.
Abstract: It is shown that for elliptic boundary value problems of order 2m the condition number of the Schur complement matrix that appears in nonoverlapping domain decomposition methods is of order \(d^{-1}h^{-2m+1}\), where d measures the diameters of the subdomains and h is the mesh size of the triangulation. The result holds for both conforming and nonconforming finite elements.
67 citations
TL;DR: The balancing domain decomposition method is extended to nonconforming plate elements and the condition number of the preconditioned system is shown to be bounded by C, where H measures the diameters of the subdomains, h is the mesh size of the triangulation.
Abstract: In this paper the balancing domain decomposition method is extended to nonconforming plate elements. The condition number of the preconditioned system is shown to be bounded by
$C\big[1+\ln(H/h)\big]^2$
, where H measures the diameters of the subdomains, h is the mesh size of the triangulation, and the constant C is independent of H, h and the number of subdomains.
39 citations
TL;DR: Lower bounds for the condition numbers of the preconditioned systems are obtained for two-level additive Schwarz preconditionsers and are shown that the known upper bounds for both second order and fourth order problems are sharp in the case of a small overlap.
Abstract: Lower bounds for the condition numbers of the preconditioned systems are obtained for two-level additive Schwarz preconditioners. They show that the known upper bounds for both second order and fourth order problems are sharp in the case of a small overlap.
26 citations
TL;DR: In this article, it was shown that ∥u 1 ∥H 1+ ∈(Ω) ≤ C∥u∥H ∈ (Ω), where Ω is a bounded polygonal domain in R 2, 0 < ∈ < (1/2) and u I is the piecewise linear nodal interpolant of u with respect to a triangulation of Ω.
Abstract: We show that ∥u 1 ∥H 1+ ∈(Ω) ≤ C∥u∥H 1+ ∈(Ω), where Ω is a bounded polygonal domain in R 2 , 0 < ∈ < (1/2), u I is the piecewise linear nodal interpolant of u with respect to a triangulation of Ω, and C depends only on E and the minimum angle of the triangulation. Extensions to other finite element nodal interpolations are also discussed.
6 citations