Energy methods for numerical analysis (Q819269)

In a quite inaccurate formulation of a class of linear boundary value problems the author introduces an elliptic type inequality for the adjoint operator of the original problem which is termed the ``energy inequality''. Accordingly he proves the existence, uniqueness and regularity results for solutions. Then he adds a nonlinear term and tries to prove the same result using some successive approximation schemes. However, it seems that he is not aware of the classical theory concerning elliptic boundary value problems.