Robust multigrid solvers for the biharmonic problem in isogeometric analysis
For researchers in isogeometric analysis, this provides efficient solvers for higher-order discretizations, though the methods are incremental extensions of existing multigrid techniques.
The paper develops robust multigrid solvers for the biharmonic problem in isogeometric analysis, proving grid-size robustness for both Gauss-Seidel and mass smoothing methods, with the latter also robust in spline degree. Numerical experiments demonstrate efficiency, especially for a hybrid approach.
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.