We introduce an adaptive superconvergent finite element method for a class of mixed formulations to solve partial differential equations involving a diffusion term. It combines a superconvergent postprocessing technique for the primal variable with an adaptive finite element method via residual minimization. Such a residual minimization procedure is performed on a local postprocessing scheme, commonly used in the context of mixed finite element methods. Given the local nature of that approach, the underlying saddle point problems associated with residual minimizations can be solved with minimal computational effort. We propose and study a posteriori error estimators, including the built-in residual representative associated with residual minimization schemes; and an improved estimator which adds, on the one hand, a residual term quantifying the mismatch between discrete fluxes and, on the other hand, the interelement jumps of the postprocessed solution. We present numerical experiments in two dimensions using Brezzi-Douglas-Marini elements as input for our methodology. The experiments perfectly fit our key theoretical findings and suggest that our estimates are sharp.
翻译:我们为一组混合配方引入了适应性超相容的有限要素方法,以解决包含扩散术语的部分差异方程式。它将原始变量的超相容后处理技术与通过残留最小化的适应性有限元素方法相结合。这种残余最小化程序是在本地的后处理方法上实施的,通常在混合限定元素方法中使用。鉴于这种方法的局部性质,与残余最小化相关的潜在临界点问题可以通过最小的计算努力来解决。我们提议并研究一个事后误差估计器,包括与残留最小化计划相关的内在残留代表;以及一个改进的估算器,它一方面增加了一个剩余术语,用以量化离散通量与后处理解决方案的相互偏差。我们用布雷兹-杜格拉斯-马里尼元素作为我们方法的投入,在两个层面进行数字实验。这些实验完全符合我们的主要理论结论,并表明我们的估计是精确的。