A more accurate Green element method in two and three spatial dimensions

The Green element method is a powerful technique for solving nonlinear boundary value problems. Derived from the boundary element method, over the meshes of the finite element method, the GEM combines the second-order accuracy of the BEM with the efficiency and versatility of the FEM. The high accuracy of the Green element method, resulting from the direct representation of normal fluxes as unknowns, comes at the price of very large matrices for problems in 2D and 3D domains. The reason for this is a larger number of inter-element boundaries connected to each internal node, yielding the same number of the normal fluxes to be determined. The currently available technique to avoid this problem approximates the normal fluxes by differentiating the potential estimates within each element. Although this approach produces much smaller matrices, the overall accuracy of the GEM is sacrificed. The technique proposed in this work redefines the present approach of approximating fluxes by considering more elements sharing each internal node. Numerical tests on the potential field exp(x + y) show an increase in accuracy by two orders of magnitude.