The purpose of this paper is the determination of the numerical solution of a classical unilateral stationary elliptic obstacle problem. The numerical technique combines Moreau-Yoshida penalty and spectral finite element approximations. The penalized method transforms the obstacle problem into a family of semilinear partial differential equations. The discretization uses a non-overlapping spectral finite element method with Legendre–Gauss–Lobatto nodal basis using a conforming mesh. The strategy is based on approximating the solution using a spectral finite element method. In addition, by coupling the penalty and the discretization parameters, we prove a priori and a posteriori error estimates where reliability and efficiency of the estimators are shown for Legendre spectral finite element method. Such estimators can be used to construct adaptive methods for obstacle problems. Moreover, numerical results are given to corroborate our error estimates.

本文的目的是确定经典的单边固定椭圆障碍问题的数值解。数值技术结合了Moreau-Yoshida罚分和频谱有限元逼近。惩罚方法将障碍问题转换为一族半线性偏微分方程。离散化使用非重叠频谱有限元方法，并使用合格网格划分了Legendre-Gauss-Lobatto节点。该策略基于使用频谱有限元方法近似解的基础。此外，通过结合惩罚和离散化参数，我们证明了先验和后验误差估计，其中勒让德谱有限元方法的估计器的可靠性和效率得到了证明。此类估计器可用于构建障碍问题的自适应方法。此外，给出了数值结果以证实我们的误差估计。