We design an adaptive procedure for approximating a selected eigenvalue and its eigen-space for a second-order elliptic boundary-value problem, using an hp finite element method. Such iterative procedure judiciously alternates between a stage in which a near-optimal hp-mesh for the current level of accuracy is generated, and a stage in which such mesh is sufficiently refined to produce a new, enhanced approximation of the eigenfunctions. We identify conditions on the initial mesh and the operator coefficients under which the procedure yields approximations that converge at a geometric rate independent of any discretization parameter, using a number of degrees of freedom comparable to the smallest number needed to get the achieved accuracy. We detail the second stage for a single eigenvalue, relying on a p-robust saturation property.

中文翻译：

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自适应$$ \ mathbf {hp} $$ hp -FEM用于特征值计算

我们使用hp有限元方法设计了一种自适应程序，用于近似求解二阶椭圆形边值问题的选定特征值及其特征空间。此类迭代过程明智地在接近最佳hp的阶段之间交替生成当前精度级别的-mesh，并在此阶段中充分细化此类网格，以生成特征函数的新的增强近似值。我们确定初始网格上的条件和算子系数，在此条件下，该过程将产生近似值，这些近似值以独立于任何离散化参数的几何速率收敛，使用的自由度可与获得所需精度所需的最小数目相媲美。我们依靠p-鲁棒饱和特性，详细描述了单个特征值的第二阶段。