New formulation of the Green element method to maintain its second-order accuracy in 2D/3D

The Green element method (GEM) is a powerful technique for solving nonlinea r boundary value problems. Derived from the boundary element method (BEM), over the meshes of the finite element method (FEM), the GEM combines the se cond-order accuracy of the BEM with the efficiency and versatility of the F EM. The high accuracy of the GEM, resulting from the direct representation of n ormal fluxes as unknowns, comes at the price of very large matrices for pro blems 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 numb er of the normal fluxes to be determined. The currently available technique to avoid this problem approximates the normal fluxes by differentiating th e potential estimates within each element. Although this approach produces much smaller matrices, the overall accuracy of the GEM is sacrificed. The first of the two techniques 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 i ncrease in accuracy by two orders of magnitude. The second approach is a reformulation of the standard GEM in terms of the flux vector, replacing its normal component. The original accuracy of the G EM is preserved while the number of unknowns is reduced as many as ten-time s in the case of a mesh consisting of tetrahedrons. The additional benefit of this novel technique is the fact that the entire flux field is a mere by product of the basic procedure for determining the unspecified boundary va lues. (C) 2001 Elsevier Science Ltd. All rights reserved.