Abstract We introduce and study a fully discrete nonconforming finite element approximation for a parabolic variational inequality associated with a general obstacle problem. The method comprises of the Crouzeix–Raviart finite element method for space discretization and implicit backward Euler scheme for time discretization. We derive an error estimate of optimal order 𝒪 ⁢ ( h + Δ ⁢ t ) {\mathcal{O}(h+\Delta t)} in a certain energy norm defined precisely in the article. We only assume the realistic regularity u t ∈ L 2 ⁢ ( 0 , T ; L 2 ⁢ ( Ω ) ) {u_{t}\in L^{2}(0,T;L^{2}(\Omega))} and moreover the analysis is performed without any assumptions on the speed of propagation of the free boundary. We present a numerical experiment to illustrate the theoretical order of convergence derived in the article.

中文翻译：

---

抛物线障碍问题的 Crouzeix-Raviart 有限元近似

摘要 我们介绍并研究了与一般障碍问题相关的抛物线变分不等式的完全离散非一致性有限元近似。该方法包括用于空间离散化的 Crouzeix-Raviart 有限元方法和用于时间离散化的隐式后向欧拉格式。我们在文章中精确定义的某个能量范数中推导出最优阶数 𝒪 ⁢ ( h + Δ ⁢ t ) {\mathcal{O}(h+\Delta t)} 的误差估计。我们只假设现实规律 ut ∈ L 2 ⁢ ( 0 , T ; L 2 ⁢ ( Ω ) ) {u_{t}\in L^{2}(0,T;L^{2}(\Omega)) } 此外，分析是在没有对自由边界的传播速度进行任何假设的情况下进行的。我们提出了一个数值实验来说明文章中推导出的理论收敛顺序。