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.
翻译:在本文中,我们描述一个稳定的有限元素配方,用于应对消化-扩散-反应问题,从而能够容易地实施稳健的自动适应战略。我们考虑到局部消散、异性和异异性异性异性,以及以对流为主的传播问题。一般稳定的有限元素框架在ArXiv:1907.12605v3中描述和分析,用于一般的线性问题,并测试纯的消化问题。该方法寻求通过在适当稳定的连续加勒金(dG)双重规范上进行残余最小化进程,实现离散的解决方案。这一技术导致一个支撑点问题,提供稳定的离散解决方案,并产生强有力的错误估计,能够驱动中位适应性。在这项工作中,我们展示了极端情况下的方法效率,提供了稳定的解决方案。解决方案的质量和性能与每个网格的离散空间规范中典型的不连续加勒金配方相近。同时,这一技术使我们能够解决粗糙的色色色色色色色色色色色色色色色的解决方案,并调整解决方案以达到用户指定解决方案的质量。