On the robust solution of an isogeometric discretization of bilaplacian equation by using multigrid methods

Abstract

In this paper, we propose a multigrid method for the solution of the biharmonic problem. Isogeometric Analysis (IGA) is considered in order to easily obtain H2-conforming discretizations of the bilaplacian equation, which are difficult to get by means of standard finite element methods. Typically, the design of solvers for isogeometric discretizations that are robust with respect to the polynomial degree is a challenging task. Here, we achieve such robustness by using multiplicative Schwarz methods as smoothers within the multigrid algorithm. The design of the proposed solver is also supported by a local Fourier analysis (LFA), which allows us to choose appropriately the size of the block in the smoother depending on the polynomial degree of the discretization. The robustness and efficiency of the proposed multigrid method is demonstrated through numerical experiments in one- and two-dimensional cases.