Enhancement of the accuracy of the Green element method: Application to potential problems

The 2-D formulation of the Green element method (GEM) which approximates the internal normal directional fluxes by difference expressions in terms of the field variable had been recognized to be fraught with errors that comprise its accuracy. However, this approach is computational attractive because there is only one degree of freedom at every node, the system matrix is slender, and it does require additional compatibility relationships. There have been attempts to reduce the numerical errors of this original GEM formulation by the use of flux-based formulations which essentially retain the internal fluxes but at the expense of those attractive numerical features. Here the original GEM is revisited and shown that, with difference approximation of the internal normal fluxes whose error is of the order of the square of the size of the element, its accuracy is greatly enhanced to a level comparable to the flux-based formulations. This approach is demonstrated on regular domains with rectangular elements and irregular domains with triangular elements using six examples that cover steady, transient, linear and nonlinear potential flow and heat transfer problems in homogeneous and heterogeneous media. ► The accuracy of the Green element method (GEM) is improved with a new formulation. ► The formulation uses a second order difference approximation of internal fluxes. ► This new formulation is implemented on rectangular and triangular elements. ► The formulation is applied to six examples of linear and nonlinear potential flows. ► The accuracy of the formulation is comparable to flux-based formulations.