Comparison principles are developed for discrete quasilinear elliptic partial differential equations. We consider the analysis of a class of nonmonotone Leray-Lions problems featuring both nonlinear solution and gradient dependence in the principal coefficient, and a solution dependent lower-order term. Sufficient local and global conditions on the discretization are found for piecewise linear finite element solutions to satisfy a comparison principle, which implies uniqueness of the solution. For problems without a lower-order term, our analysis shows the meshsize is only required to be locally controlled, based on the variance of the computed solution over each element. We include a discussion of the simpler semilinear case where a linear algebra argument allows a sharper mesh condition for the lower order term.

中文翻译：

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拟线性椭圆偏微分方程的离散比较原理

比较原理是为离散拟线性椭圆偏微分方程开发的。我们考虑对一类非单调 Leray-Lions 问题的分析，该问题具有主系数的非线性解和梯度相关性，以及与解相关的低阶项。为分段线性有限元解找到了足够的离散化的局部和全局条件，以满足比较原则，这意味着解的唯一性。对于没有低阶项的问题，我们的分析表明网格大小只需要根据每个元素的计算解的方差进行局部控制。我们讨论了更简单的半线性情况，其中线性代数参数允许低阶项的网格条件更清晰。