论文标题
具有平衡通量的自适应虚拟元素方法
Adaptive virtual element methods with equilibrated fluxes
论文作者
论文摘要
我们基于布拉格的HyperCircle方法和Synge提出了HP自适应虚拟元素方法（VEM），以实现解决扩散问题的溶液。我们引入了可靠有效的后验误差估计器，该估计量是通过解决辅助全局混合问题来计算的。我们表明，混合VEM满足离散的INF-SUP条件，INF-SUP常数与离散参数无关。此外，我们为混合VEM构建了一个稳定，就该方法的局部准确性而言，具有明确的界限。理论结果得到了几个数值实验的支持，包括与残留A后验误差估计器的比较。该数字表现出提出的误差估计量的P-固定性。此外，我们还提供了虚拟元素框架中局部通量重建的第一步，该框架导致了一个可靠的后验误差估计器，该估计值是通过解决局部（廉价到解决方案和可行的）混合问题来计算的。我们提供了理论和数值证据，表明所提出的局部误差估计器缺乏效率。
We present an hp-adaptive virtual element method (VEM) based on the hypercircle method of Prager and Synge for the approximation of solutions to diffusion problems. We introduce a reliable and efficient a posteriori error estimator, which is computed by solving an auxiliary global mixed problem. We show that the mixed VEM satisfies a discrete inf-sup condition, with inf-sup constant independent of the discretization parameters. Furthermore, we construct a stabilization for the mixed VEM, with explicit bounds in terms of the local degree of accuracy of the method. The theoretical results are supported by several numerical experiments, including a comparison with the residual a posteriori error estimator. The numerics exhibit the p-robustness of the proposed error estimator. In addition, we provide a first step towards the localized flux reconstruction in the virtual element framework, which leads to an additional reliable a posteriori error estimator that is computed by solving local (cheap-to-solve and parallelizable) mixed problems. We provide theoretical and numerical evidence that the proposed local error estimator suffers from a lack of efficiency.
