This paper proposes a novel adaptive higher-order finite element ($hp$-FEM) method for solving elliptic eigenvalue problems, where $n$ eigenpairs are calculated simultaneously, but on individual higher-order finite element meshes. The meshes are automatically $hp$-refined independently of each other, with the goal to use an optimal mesh sequence for each eigenfunction. The method and the adaptive algorithm are described in detail. Numerical examples clearly demonstrate the superiority of the novel method over the standard approach where all eigenfunctions are approximated on the same finite element mesh.

中文翻译：

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用自适应多重网格解决椭圆本征问题 $Hp$有限元法

本文提出了一种新颖的自适应高阶有限元（$Hp$-FEM）方法来解决椭圆特征值问题，其中 $ñ$本征对是同时计算的，但要在各个高阶有限元网格上进行。网格是自动的$Hp$-彼此独立地优化，目标是为每个本征函数使用最佳网格序列。详细描述了该方法和自适应算法。数值示例清楚地证明了该新方法优于标准方法的优越性，在标准方法中，所有本征函数都在相同的有限元网格上进行近似。