A new hybrid block collocation method for solving elliptic PDEs.

In this article, we introduce a new hybrid block collocation method (NHBCM) using polynomial approximation for solving two-dimensional elliptic partial differential equations (PDEs). The theoretical analysis demonstrates that the NHBCM achieves fifth-order accuracy, highlighting its robustness in stability and convergence. We validate the effectiveness of the NHBCM through a series of test problems, showcasing its practical applications. The comparison results indicate that the NHBCM provides significantly more efficient solutions for both linear and nonlinear elliptic model PDEs than other numerical methods used for comparison, thereby validating the superior performance of the NHBCM.