This work is motivated by the difficulty in assembling the Galerkin matrix when solving Partial Differential Equations (PDEs) with Isogeometric Analysis (IGA) using B-splines of moderate-to-high polynomial degree. To mitigate this problem, we propose a novel methodology named CossIGA (COmpreSSive IsoGeometric Analysis), which combines the IGA principle with CORSING, a recently introduced sparse recovery approach for PDEs based on compressive sensing. CossIGA assembles only a small portion of a suitable IGA Petrov–Galerkin discretization and is effective whenever the PDE solution is sufficiently sparse or compressible, i.e., when most of its coefficients are zero or negligible. The sparsity of the solution is promoted by employing a multilevel dictionary of B-splines as opposed to a basis. Thanks to sparsity and the fact that only a fraction of the full discretization matrix is assembled, the proposed technique has the potential to lead to significant computational savings. We show the effectiveness of CossIGA for the solution of the 2D and 3D Poisson equation over nontrivial geometries by means of an extensive numerical investigation.

这项工作的动机是，在使用中等至高多项式的B样条通过等几何分析（IGA）求解偏微分方程（PDE）时，难以组装Galerkin矩阵。为了缓解此问题，我们提出了一种名为CossIGA（COmpreSSive等距几何分析）的新颖方法，该方法将IGA原理与CORSING相结合，CORSING是一种新近引入的基于压缩感测的PDE稀疏恢复方法。CossIGA仅组装适当IGA Petrov-Galerkin离散化的一小部分，并且在PDE解决方案足够稀疏或可压缩时（即当其大多数系数为零或可忽略时）有效。通过使用B样条的多级字典（而不是基础）来提高解决方案的稀疏性。由于稀疏性和整个离散化矩阵只有一部分被组装的事实，所提出的技术有可能导致大量的计算节省。通过广泛的数值研究，我们证明了CossIGA在非平凡几何上求解2D和3D泊松方程的有效性。