Condition number bounds for IETI-DP methods that are explicit in h and p

@article{Schneckenleitner2019ConditionNB,
  title={Condition number bounds for IETI-DP methods that are explicit in h and p},
  author={Rainer Schneckenleitner and Stefan Takacs},
  journal={ArXiv},
  year={2019},
  volume={abs/1912.07909}
}
We study the convergence behavior of Dual-Primal Isogeometric Tearing and Interconnecting (IETI-DP) methods for solving large-scale algebraic systems arising from multi-patch Isogeometric Analysis. We focus on the Poisson problem on two dimensional computational domains. We provide a convergence analysis that covers several choices of the primal degrees of freedom: the vertex values, the edge averages, and the combination of both. We derive condition number bounds that show the expected… 
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Convergence Theory for IETI-DP Solvers for Discontinuous Galerkin Isogeometric Analysis that is Explicit in ℎ and 𝑝

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A convergence theory for Dual-Primal Isogeometric Tearing and Interconnecting (IETI-DP) solvers for isogeometric multi-patch discretizations of the Poisson problem, where the patches are coupled using discontinuous Galerkin is developed.

Inexact IETI-DP for conforming isogeometric multi-patch discretizations

TLDR
The sparse LU factorizations are replaced by fast diagonalization based preconditioners to get a faster IETI-DP method while maintaining the same explicit condition number bound.

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This paper considers dual-primal isogeometric tearing and interconnection (IETI-DP) solvers for multi-patch geometries in Isogeometric Analysis and provides numerical experiments that indicate that similar results may hold for three dimensional domains.

A IETI-DP method for discontinuous Galerkin discretizations in Isogeometric Analysis with inexact local solvers

TLDR
A convergence theory is presented that shows that the condition number of the preconditioned system only grows poly-logarithmically with the grid size, and that the convergence of the overall solver only mildly depends on the spline degree.

Stable discretizations and IETI-DP solvers for the Stokes system in multi-patch Isogeometric Analysis

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It is shown how stability results for single-patch domains can be carried over to multi- patch domains, and how stability strongly depends on the shape of the geometry.

IETI-DP methods for discontinuous Galerkin multi-patch Isogeometric Analysis with T-junctions

Dual-Primal Isogeometric Tearing and Interconnecting methods for the Stokes problem

TLDR
This work uses Dual-Primal Isogeometric Tearing and Interconnecting methods to test out two different scaled Dirichlet preconditioners with different choices of primal degrees of freedom for a fast solver for linear systems obtained by discretizing the Stokes problem with multi-patch I sogeometric Analysis.

Towards a IETI-DP solver on non-matching multi-patch domains

TLDR
This paper presents a generalization of isogeometric tearing and interconnecting solvers that means that the patches can meet in T-junctions, which increases the flexibility of the geometric model significantly.

Convergence theory for IETI-DP solvers for discontinuous Galerkin Isogeometric Analysis that is explicit in h and p

TLDR
A convergence theory for Dual-Primal Isogeometric Tearing and Interconnecting (IETI-DP) solvers for isogeometric multi-patch discretizations of the Poisson problem, where the patches are coupled using discontinuous Galerkin.

References

SHOWING 1-10 OF 37 REFERENCES

Dual-Primal Isogeometric Tearing and Interconnecting Methods

  • C. HoferU. Langer
  • Computer Science, Mathematics
    Computational Methods in Applied Sciences
  • 2018
TLDR
This paper generalizes the Dual-Primal Finite Element Tearing and Interconnecting method to linear algebraic systems arising from the Isogemetric Analysis of elliptic diffusion problems with heterogeneous diffusion coefficients in two- and three-dimensional multipatch domains with \(C^0\) smoothness across the patch interfaces.

Dual-primal isogeometric tearing and interconnecting solvers for multipatch dG-IgA equations

A Polylogarithmic Bound for an Iterative Substructuring Method for Spectral Elements in Three Dimensions

TLDR
An iterative method is designed for which the condition number of the relevant operator grows only in proportion to $(1+\log p)^2 .$ this bound is independent of jumps in the coefficient of the elliptic problem across the interfaces between the subregions.

Optimal extensions on tensor-product meshes

An algebraic theory for primal and dual substructuring methods by constraints

Optimal Multilevel Extension Operators

TLDR
The norm-preserving explicit operator for the extension of finite-element functions from boundaries of domains into the inside is suggested, based on the multilevel decomposition of functions on the boundaries and on the equivalent norm for this decomposition.

Symbol-Based Multigrid Methods for Galerkin B-Spline Isogeometric Analysis

TLDR
This work considers the stiffness matrices arising from the Galerkin B-spline isogeometric analysis discretization of classical elliptic problems and designs an efficient multigrid method for the fast solution of the related linear systems.

Isogeometric Preconditioners Based on Fast Solvers for the Sylvester Equation

TLDR
It is shown that a preconditioning strategy which is based on the solution of a Sylvester-like equation at each step of an iterative solver is robust with respect to both mesh size and spline degree, although it may suffer from the presence of complicated geometry or coefficients.

Robust approximation error estimates and multigrid solvers for isogeometric multi-patch discretizations

  • S. Takacs
  • Computer Science
    Mathematical Models and Methods in Applied Sciences
  • 2018
TLDR
The approximation error estimates and the multigrid solver are extended to this multi-patch case and allow to solve the linear system arising from the discretization of a partial differential equation in Isogeometric Analysis in a single-patch setting with convergence rates that are provably robust both in the grid size and the spline degree.

Extension Operators on Tensor Product Structures in Two and Three Dimensions

  • S. Beuchler
  • Computer Science, Mathematics
    SIAM J. Sci. Comput.
  • 2005
TLDR
In this paper, a uniformly elliptic second order boundary value problem in two dimensions is discretized by the p-version of the finite element method and it is proved that this type of extension is optimal, i.e., the $H^1(\Omega)$-norm of the extended function is bounded by the given function.