Efficient matrix computation for isogeometric discretizations with hierarchical B-splines in any dimension

Abstract

Hierarchical B-splines, which possess the local refinement capability, have been recognized as a useful tool in the context of isogeometric analysis. However, similar as for tensor-product B-splines, isogeometric simulations with hierarchical B-splines face a big computational burden from the perspective of matrix assembly, particularly if the spline degree p is high. To address this issue, we extend the recent work (Pan et al., 2020) – which introduced an efficient assembling approach for tensor-product B-splines – to the case of hierarchical B-splines. In the new approach, the integrand factor is transformed into piecewise polynomials via quasi-interpolation. Subsequently, the resulting elementary integrals are pre-computed and stored in a look-up table. Finally, the sum-factorization technique is adopted to accelerate the assembly process. We present a detailed analysis, which reveals that the presented method achieves the expected complexity of O(pd+1) per degree of freedom (without taking sparse matrix operations into account) under the assumption of mesh admissibility. We verify the efficiency of the new method by applying it to an elliptic problem on the three-dimensional domain and a parabolic problem on the four-dimensional domain in space–time, respectively. A comparison with standard Gaussian quadrature is also provided.