In this paper we develop a $C^0$-conforming virtual element method (VEM) for a class of second-order quasilinear elliptic PDEs in two dimensions. We present a posteriori error analysis for this problem and derive a residual based error estimator. The estimator is fully computable and we prove upper and lower bounds of the error estimator which are explicit in the local mesh size. We use the estimator to drive an adaptive mesh refinement algorithm. A handful of numerical test problems are carried out to study the performance of the proposed error indicator.

翻译：本文针对二维空间中的一类二阶拟线性椭圆型偏微分方程，发展了一种$C^0$相容的虚拟元方法。我们对该问题进行了后验误差分析，并推导出一种基于残差的误差估计子。该估计子是完全可计算的，我们证明了误差估计子的上下界，这些界限在局部网格尺寸上是显式的。我们利用该估计子驱动自适应网格细化算法。通过若干数值测试问题，对所提出的误差指示子的性能进行了研究。