  One of strengths in the finite element (FE) and Galerkin methods is their
capability to apply weak formulations via integration by parts, which leads to
elements matching at lower degree of continuity and relaxes requirements of
choosing basis functions. However, when applied to nonlinear problems, the
methods of this type require a great amount of computing effort of repeated
numerical integration. It is well known that the method of weighted residual is
the very basis of various popular numerical techniques including the FE and
Galerkin methods. This paper presents a novel methodology of weighted residual
for nonlinear computation with objectives to avoid the above-mentioned
shortcomings. It is shown that the presented nonlinear formulations of the FE
and Galerkin methods can be expressed in the Hadamard product form as in the
collocation and finite difference methods. Therefore, the recently developed
SJT product approach can be applied in the evaluation of the Jacobian matrix of
these nonlinear formulations. This also provides possibility to introduce the
nonlinear uncoupling technique to the FE and Galerkin nonlinear computing.
Furthermore, the present scheme of weighted residuals also greatly eases the
use of the least square and boundary element methods to nonlinear problems
