论文标题
双旋转方程的不合格的虚拟元素，具有莫利自由度的多边形网格
Nonconforming virtual elements for the biharmonic equation with Morley degrees of freedom on polygonal meshes
论文作者
论文摘要
最低阶的不合格虚拟元素将Morley三角元素扩展到多边形，以使弱解决方案$ u \ in V：= H^2_0（ω）$的近似值延伸到Biharmonic方程。抽象框架允许（甚至是）局部离散空间的两个示例$ v_h（p）$，并且更光滑允许粗糙的源项$ f \ in v^*= h^= h^{ - 2}（ω）$。本文中的先验和后验错误分析通过一些可计算的符合伴侣运算符$ j：v_h \ to v $从不合格的虚拟元素space $ v_h $绕过任何第二个衍生物的痕迹。操作员$ j $是插值操作员的直接右侧，可导致分段Sobolev Norms中的最佳错误估算，而无需对$ U \ in V $的任何其他规则性假设。随着伴侣操作员的更平滑，将修改离散的右侧，然后允许准最佳近似。明确的基于残差的后验误差估计器可靠，有效地对数据振荡。数值示例显示了自适应网格再填充的均匀和最佳收敛速率的预测经验收敛速率。
The lowest-order nonconforming virtual element extends the Morley triangular element to polygons for the approximation of the weak solution $u\in V:=H^2_0(Ω)$ to the biharmonic equation. The abstract framework allows (even a mixture of) two examples of the local discrete spaces $V_h(P)$ and a smoother allows rough source terms $F\in V^*=H^{-2}(Ω)$. The a priori and a posteriori error analysis in this paper circumvents any trace of second derivatives by some computable conforming companion operator $J:V_h\to V$ from the nonconforming virtual element space $V_h$. The operator $J$ is a right-inverse of the interpolation operator and leads to optimal error estimates in piecewise Sobolev norms without any additional regularity assumptions on $u\in V$. As a smoother the companion operator modifies the discrete right-hand side and then allows a quasi-best approximation. An explicit residual-based a posteriori error estimator is reliable and efficient up to data oscillations. Numerical examples display the predicted empirical convergence rates for uniform and optimal convergence rates for adaptive mesh-refinement.
