Robust multigrid solvers for the biharmonic problem in isogeometric analysis

Abstract

In this paper, we develop multigrid solvers for the biharmonic problem in the framework of isogeometric analysis (IgA). In this framework, one typically sets up B-splines on the unit square or cube and transforms them to the domain of interest by a global smooth geometry function. With this approach, it is feasible to set up $H^2$-conforming discretizations. We propose two multigrid methods for such a discretization, one based on Gauss Seidel smoothing and one based on mass smoothing. We prove that both are robust in the grid size, the latter is also robust in the spline degree. Numerical experiments illustrate the convergence theory and indicate the efficiency of the proposed multigrid approaches, particularly of a hybrid approach combining both smoothers.

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• Towards Accuracy and Scalability: Combining Isogeometric Analysis with Deflation to Obtain Scalable Convergence for the Helmholtz Equation
• Fast multigrid solvers for conforming and non-conforming multi-patch Isogeometric Analysis
• Application of optimal spline subspaces for the removal of spurious outliers in isogeometric discretizations
• A quasi-robust discretization error estimate for discontinuous Galerkin Isogeometric Analysis
• Outlier-free isogeometric discretizations for Laplace eigenvalue problems: closed-form eigenvalue and eigenvector expressions
• Combining p-multigrid and multigrid reduced in time methods to obtain a scalable solver for Isogeometric Analysis