A novel Green element method by mixing the idea of the finite difference method

This paper proposes a novel green element method (GEM) by mixing the idea of finite difference method (FDM). In the novel method, We come to the original formula when boundary integral equation is applied to an element, and use difference quotient of the central nodal value on two sides of the shared edge of adjoining elements to approximate the boundary integration ∫ΓG∇p · nds. This treatment is similar to FDM, and the integral operator relevant to element size controls the estimated error. The novel GEM makes the numerical solution correspond to the actual physical meaning, and the coefficient matrix of the global matrix is a banded sparse matrix with larger bandwidth than previous GEMs. Meanwhile, the instability of the original GEM is illuminated. We have proven it by theoretical error analysis and five numerical examples that, the accuracy of the novel GEM is three-order higher than the original GEM, and the novel GEM has a good convergence and stability, which is the property that the original GEM does not have. Indeed, the novel GEM proposed in this paper is essentially a new numerical method mixed with the idea of boundary element method (BEM), finite element method (FEM), and FDM. In contrast with BEM, FEM, FDM and previous GEM, the characteristics of our novel GEM include: (i) Compared with FEM and FDM, the novel GEM has the accuracy of BEM and can better accord with material balance. (ii) Compared with BEM, the novel GEM can solve nonlinear problems with heterogeneous media, which are hard to be handled by BEM. (iii) Compared with previous GEMs, the novel GEM has a three-order accuracy, and has a better convergence that the calculation error can be well controlled by the element size.