Multigrid methods for isogeometric analysis with THB-splines

作者：Hofreither Clemens; Juettler Bert; Kiss Gabor; Zulehner Walter

来源：Computer Methods in Applied Mechanics and Engineering, 2016, 308: 96-112.

DOI：10.1016/j.cma.2016.05.005

• Isogeometric analysis
• Multigrid
• Hierarchical splines
• Truncated hierarchical B-splines
• Adaptive refinement

We propose a geometric multigrid algorithm for the solution of the linear systems which arise when using hierarchical spline spaces for isogeometric discretizations of elliptic partial differential equations. In particular, we focus on the use of hierarchical B-spline (HB) and truncated hierarchical B-spline (THB) bases to implement adaptively refined isogeometric discretizations. We describe an approach for constructing the nested sequences of hierarchical spline spaces which are required for our geometric multigrid solver. Furthermore, we give an algorithm for the computation of the so-called prolongation matrices which map coarser-space functions to their finer-space representations for both HB- and THB-spline bases.
In several numerical experiments, we study two-dimensional Poisson problems with singularities and employ a posteriori error estimators for the adaptive refinement of the hierarchical spline spaces. Using standard multigrid smoothers, we observe that using THB-spline bases consistently confers a dramatic advantage in terms of iteration numbers over the use of HB-spline bases, in particular for higher spline degrees. When used as a preconditioner for Conjugate Gradient iteration, the proposed multigrid method exhibits practical iteration numbers for two-dimensional problems up to the tested case of fourth-degree splines.

• 出版日期2016-8-15