In this paper, we present a novel approach to attain fourth-order approximate solution of 2D quasi-linear elliptic partial differential equation on an irrational domain. In this approach, we use nine grid points with dissimilar mesh in a single compact cell. We also discuss appropriate fourth-order numerical methods for the solution of the normal derivatives on a dissimilar mesh. The method has been protracted for solving system of quasi-linear elliptic equations. The convergence analysis is discussed to authenticate the proposed numerical approximation. On engineering applications, we solve various test problems, such as linear convection–diffusion equation, Burgers’equation, Poisson equation in singular form, NS equations, bi- and tri-harmonic equations and quasi-linear elliptic equations to show the efficiency and accuracy of the proposed methods. A comprehensive comparative computational experiment shows the accuracy, reliability and credibility of the proposed computational approach.

中文翻译：

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二维拟线性椭圆边值问题系统的高分辨率紧致数值方法及无理域上的法导数解与工程应用

在本文中，我们提出了一种在无理域上获得二维拟线性椭圆偏微分方程四阶近似解的新方法。在这种方法中，我们在单个紧凑单元中使用九个具有不同网格的网格点。我们还讨论了求解不同网格上法向导数的适当四阶数值方法。该方法已被推广用于求解拟线性椭圆方程组。讨论了收敛分析以验证所提出的数值近似。在工程应用方面，我们解决了各种测试问题，如线性对流扩散方程、伯格斯方程、奇异形式的泊松方程、NS方程、双和三谐波方程和准线性椭圆方程来显示所提出方法的效率和准确性。综合比较计算实验表明了所提出的计算方法的准确性、可靠性和可信度。