A high order modified nodal bi‐cubic spline collocation method is proposed for numerical solution of second‐order elliptic partial differential equation subject to Dirichlet boundary conditions. The approximation is defined on a square mesh stencil using nine grid points. The solution of the method exists and is unique. Convergence analysis has been presented. Moreover, the superconvergent phenomena can be seen in proposed one step method. The numerical results clearly exhibit the superiority of the new approximation, in terms of both accuracy and computational efficiency.

中文翻译：

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椭圆型偏微分方程的高阶收敛修正节点双三次样条配点法

针对Dirichlet边界条件下的二阶椭圆型偏微分方程的数值解，提出了一种高阶改进的节点双三次样条配点方法。近似值是在使用9个网格点的正方形网格模具上定义的。该方法的解决方案存在并且是唯一的。已经提出了收敛分析。此外，超收敛现象可以在提出的一步法中看到。数值结果清楚地显示了新近似方法在准确性和计算效率方面的优越性。