The Laplace eigenvalue problem on circular sectors has eigenfunctions with corner singularities. Standard methods may produce suboptimal approximation results. To address this issue, a novel numerical algorithm that enhances standard isogeometric analysis is proposed in this paper by using a single-patch graded mesh refinement scheme. Numerical tests demonstrate optimal convergence rates for both the eigenvalues and eigenfunctions. Furthermore, the results show that smooth splines possess a superior approximation constant compared to their $C^0$-continuous counterparts for the lower part of the Laplace spectrum. This is an extension of previous findings about excellent spectral approximation properties of smooth splines on rectangular domains to circular sectors. In addition, graded meshes prove to be particularly advantageous for an accurate approximation of a limited number of eigenvalues. The novel algorithm applied here has a drawback in the singularity of the isogeometric parameterization. It results in some basis functions not belonging to the solution space of the corresponding weak problem, which is considered a variational crime. However, the approach proves to be robust. Finally, a hierarchical mesh structure is presented to avoid anisotropic elements, omit redundant degrees of freedom and keep the number of basis functions contributing to the variational crime constant, independent of the mesh size. Numerical results validate the effectiveness of hierarchical mesh grading for the simulation of eigenfunctions with and without corner singularities.

翻译：圆扇形域上的拉普拉斯特征值问题存在具有角点奇异性的特征函数。标准方法可能产生次优的逼近结果。为解决该问题，本文提出一种增强标准等几何分析的新型数值算法，采用单面分级网格细化方案。数值测试表明，特征值与特征函数均达到最优收敛率。此外，结果显示对于拉普拉斯谱的低频部分，光滑样条相比其$C^0$连续对应物具有更优的逼近常数。这是将光滑样条在矩形域上优异的谱逼近性质推广至圆扇形域。同时，分级网格在精确逼近有限数量特征值方面展现出显著优势。本文算法存在等几何参数化奇异性的缺陷，导致部分基函数不属于相应弱问题解空间，这被视为一种变分犯罪。但该方法的鲁棒性已得到验证。最后，引入层次化网格结构以避免各向异性单元、消除冗余自由度，并使参与变分犯罪的基函数数量保持恒定，与网格尺寸无关。数值结果验证了层次网格分级对于模拟含角点与非角点奇异性的特征函数的有效性。