Divergence-preserving reconstructions on polygons and a really pressure-robust virtual element method for the Stokes problem
Divergence-preserving reconstructions on polygons and a really pressure-robust virtual element method for the Stokes problem
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DOI：
10.1093/imanum/draa073
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发表时间：
2020-02
期刊：
影响因子：
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通讯作者：
Derk Frerichs;C. Merdon
中科院分区：
文献类型：
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作者：
Derk Frerichs;C. Merdon
不可压缩Stokes问题的无散度离散可能缺乏压力稳健性，其特征是由于动量平衡中的无旋转力而产生较大的离散误差。本文还认为，只要不仔细地对右侧进行离散，多边形网格上的无散度虚拟单元方法就不是真正的压力稳健的。为了能够计算测试函数的右侧，需要对虚拟测试函数进行一些显式内插，该内插可以在任何地方逐点地计算。通过$L^2$-最佳近似的标准离散化不保持散度，因此破坏了无散度测试函数和右侧可能显著的梯度力之间的正交性。为了修复这种正交性和恢复压力稳健性，提出了另一种基于多边形子三角剖分的Raviart-Thomas近似的保散度重建方法。所有的发现都在理论上得到了证明，并在两个维度上得到了数值证明。对于多边形或多面体网格上的混合高阶方法，这种构造也很有趣。
Nondivergence-free discretizations for the incompressible Stokes problem may suffer from a lack of pressure-robustness characterized by large discretizations errors due to irrotational forces in the momentum balance. This paper argues that also divergence-free virtual element methods on polygonal meshes are not really pressure-robust as long as the right-hand side is not discretized in a careful manner. To be able to evaluate the right-hand side for the test functions, some explicit interpolation of the virtual test functions is needed that can be evaluated pointwise everywhere. The standard discretization via an $L^2$-best approximation does not preserve the divergence, and so destroys the orthogonality between divergence-free test functions and possibly eminent gradient forces in the right-hand side. To repair this orthogonality and restore pressure-robustness, another divergence-preserving reconstruction is suggested based on Raviart–Thomas approximations on local subtriangulations of the polygons. All findings are proven theoretically and are demonstrated numerically in two dimensions. The construction is also interesting for hybrid high-order methods on polygonal or polyhedral meshes.