论文标题
基于局部剩余最小化的自适应超授权有限元方法
An adaptive superconvergent finite element method based on local residual minimization
论文作者
论文摘要
我们引入了一种自适应超授权有限元方法，用于一类混合配方，以求解涉及扩散项的部分微分方程。它通过残留最小化结合了原始变量的超对结合后处理技术和自适应有限元方法。这种残留最小化过程是在局部后处理方案上执行的，该方案通常用于混合有限元方法的背景下。鉴于该方法的局部性质，可以通过最小的计算工作来解决与残留最小化相关的基本鞍点问题。我们提出和研究后验误差估计器，包括与残留最小化方案相关的内置残留代表；一方面，一个改进的估计器，它添加了一个残留项，该术语量化了离散通量之间的不匹配，另一方面是固定溶液的元素跳跃。我们使用brezzi-douglas-marini元素作为我们方法的输入，在两个维度上进行了数值实验。这些实验非常适合我们的关键理论发现，并表明我们的估计值很清晰。
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.
