论文标题
一种自动适应性稳定的有限元方法，可通过残留最小化，用于异质，各向异性对流扩散反应问题
An automatic-adaptivity stabilized finite element method via residual minimization for heterogeneous, anisotropic advection-diffusion-reaction problems
论文作者
论文摘要
在本文中，我们描述了一个稳定的有限元公式，用于对流扩散反应问题，该问题允许轻松实施强大的自动自适应策略。我们认为局部消失，异质和各向异性扩散性以及以对流为主的扩散问题。在ARXIV：1907.12605V3中，对一般的线性问题进行了总体稳定的有限元框架，并测试了纯对流问题。该方法通过在适当的稳定不连续的Galerkin（DG）双重规范上的残留最小化过程寻求离散解决方案。该技术导致了一个鞍点问题，该问题提供了稳定的离散解决方案和可驱动网格适应性的强大误差估计。在这项工作中，我们在极端情况下演示了该方法的效率，并提供了稳定的解决方案。解决方案的质量和性能可与每个网格的各个离散空间规范中的经典不连续的彩手素公式相媲美。同时，这项技术使我们能够在粗网格上求解并调整解决方案以达到用户指定的解决方案质量。
In this paper, we describe a stable finite element formulation for advection-diffusion-reaction problems that allows for robust automatic adaptive strategies to be easily implemented. We consider locally vanishing, heterogeneous, and anisotropic diffusivities, as well as advection-dominated diffusion problems. The general stabilized finite element framework was described and analyzed in arXiv:1907.12605v3 for linear problems in general, and tested for pure advection problems. The method seeks for the discrete solution through a residual minimization process on a proper stable discontinuous Galerkin (dG) dual norm. This technique leads to a saddle-point problem that delivers a stable discrete solution and a robust error estimate that can drive mesh adaptivity. In this work, we demonstrate the efficiency of the method in extreme scenarios, delivering stable solutions. The quality and performance of the solutions are comparable to classical discontinuous Galerkin formulations in the respective discrete space norm on each mesh. Meanwhile, this technique allows us to solve on coarse meshes and adapt the solution to achieve a user-specified solution quality.
