Accurate triangulation of the domain plays a pivotal role in computing the numerical approximation of the differential operators. A good triangulation is the one which aids in reducing discretization errors. In a standard collocation technique, the smooth curved domain is typically triangulated with a mesh by taking points on the boundary to approximate them by polygons. However, such an approach often leads to geometrical errors which directly affect the accuracy of the numerical approximation. To restrict such geometrical errors, \textit{isoparametric}, \textit{subparametric}, and \textit{iso-geometric} methods were introduced which allow the approximation of the curved surfaces (or curved line segments). In this paper, we present an efficient finite element method to approximate the solution to the elliptic boundary value problem (BVP), which governs the response of an elastic solid containing a v-notch and inclusions. The algebraically nonlinear constitutive equation along with the balance of linear momentum reduces to second-order quasi-linear elliptic partial differential equation. Our approach allows us to represent the complex curved boundaries by smooth \textit{one-of-its-kind} point transformation. The main idea is to obtain higher-order shape functions which enable us to accurately compute the entries in the finite element matrices and vectors. A Picard-type linearization is utilized to handle the nonlinearities in the governing differential equation. The numerical results for the test cases show considerable improvement in the accuracy.

翻译：域的精确三角剖分在微分算子数值逼近计算中起着关键作用。良好的三角剖分有助于减少离散化误差。在标准配置法中，通常通过在边界上取点用多边形近似光滑曲面域进行网格三角剖分。然而，这种方法往往会导致直接影响数值逼近精度的几何误差。为限制此类几何误差，引入了等参、亚参和等几何方法，这些方法允许对曲面（或曲线段）进行逼近。本文提出一种高效有限元方法，用于逼近控制含V形切口和夹杂物弹性固体响应的椭圆边值问题（BVP）。结合线动量平衡的代数百非线性能量方程简化为二阶拟线性椭圆偏微分方程。我们的方法通过独特的点变换将复杂曲面边界表示为光滑形式。主要思路是获得高阶形函数，从而能够精确计算有限元矩阵和向量中的元素。采用Picard型线性化处理控制微分方程中的非线性项。测试案例的数值结果表明精度显著提升。