In this paper, we discuss adaptive approximations of an elliptic eigenvalue optimization problem in a phase-field setting by a conforming finite element method. An adaptive algorithm is proposed and implemented in several two dimensional numerical examples for illustration of efficiency and accuracy. Theoretical findings consist in the vanishing limit of a subsequence of estimators and the convergence of the relevant subsequence of adaptively-generated solutions to a solution to the continuous optimality system.

翻译：本文讨论在相场框架下通过协调有限元方法对椭圆特征值优化问题进行自适应逼近。我们提出了一种自适应算法，并通过多个二维数值算例验证其效率与精度。理论结果包括：估计子序列的消失极限，以及自适应生成解的相关子序列收敛至连续最优性系统解的特性。