A Robust Multigrid Method for Isogeometric Analysis using Boundary Correction

Abstract

We consider geometric multigrid methods for the solution of linear systems arising from isogeometric discretizations of elliptic partial differential equations. For classical finite elements, such methods are well known to be fast solvers showing optimal convergence behavior. However, the naive application of multigrid to the isogeometric case results in significant deterioration of the convergence rates if the spline degree is increased. Recently, a robust approximation error estimate and a corresponding inverse inequality for B-splines of maximum smoothness have been shown, both with constants independent of the spline degree. We use these results to construct multigrid solvers for discretizations based on B-splines with maximum smoothness which exhibit robust convergence rates.

Cited by in corpus (20)

• Matrix-free weighted quadrature for a computationally efficient isogeometric $k$-method
• Robust Multigrid for Isogeometric Analysis Based on Stable Splittings of Spline Spaces
• Scalable multigrid methods for immersed finite element methods and immersed isogeometric analysis
• Dual-Primal Isogeometric Tearing and Interconnecting solvers for multipatch dG-IgA equations
• Robust multigrid solvers for the biharmonic problem in isogeometric analysis
• A p-multigrid method enhanced with an ILUT smoother and its comparison to h-multigrid methods within Isogeometric Analysis
• Robust approximation error estimates and multigrid solvers for isogeometric multi-patch discretizations
• BPX preconditioners for isogeometric analysis using analysis-suitable T-splines
• Towards Accuracy and Scalability: Combining Isogeometric Analysis with Deflation to Obtain Scalable Convergence for the Helmholtz Equation
• Optimal additive Schwarz preconditioning for adaptive 2D IGA boundary element methods
• An efficient multigrid solver for isogeometric analysis
• A sparse-grid isogeometric solver
• Fast multigrid solvers for conforming and non-conforming multi-patch Isogeometric Analysis
• A parallel multigrid solver for multi-patch Isogeometric Analysis
• Robust Preconditioning for Space-Time Isogeometric Analysis of Parabolic Evolution Problems
• A quasi-robust discretization error estimate for discontinuous Galerkin Isogeometric Analysis
• Robust multigrid methods for isogeometric discretizations of the Stokes equations
• A two level method for isogeometric discretizations
• Compressive Isogeometric Analysis
• Combining p-multigrid and multigrid reduced in time methods to obtain a scalable solver for Isogeometric Analysis