Classical Methods of Approximation

This chapter aims to determine which methods of approximation yield variationally consistent integral forms for which differential operators. It considers various classical methods of approximation based on the integral form constructed using the fundamental lemma of the calculus of variations. It then considers three different classes of differential operators: self-adjoint, nonself-adjoint and nonlinear for this method of approximation. The integral forms resulting from the Galerkin method are always VIC regardless of the nature of the differential operator. The operator is non-self-adjoint, hence all methods of approximation except least-squares processes yield VIC integral forms. It considers the Galerkin method with weak form and least-squares method. The VIC integral forms, on the other hand, yield non-symmetric coefficient matrices which are not always ensured to be positive-definite. a variety of boundary value problems are considered to present the details of various methods of approximation for each of the types of differential operators.