In this paper, we consider a quadratic finite volume method (FVM) for solving second-order nonlinear elliptic problems. Under reasonable assumptions, we shall establish the existence and uniqueness of the quadratic FVM approximation and develop the error analysis of the approximation solution. To be specific, without any additional requirements on the underlying triangular meshes, we derive the optimal error estimate by assumption that \(u\in H^{3}\cap W^{2,\infty }\), where u is the solution of the nonlinear elliptic problems. Numerical experiments are presented to confirm the theoretical results.

中文翻译：

---

非线性椭圆问题的二次有限体积方法

在本文中，我们考虑用于解决二阶非线性椭圆问题的二次有限体积方法（FVM）。在合理的假设下，我们将建立二次FVM逼近的存在性和唯一性，并进行逼近解的误差分析。具体而言，在底层三角形网格上没有任何其他要求的情况下，我们通过假设\（u \ in H ^ {3} \ cap W ^ {2，\ infty} \）得出最佳误差估计，其中u是非线性椭圆问题的解。数值实验证实了理论结果。