The construction of boundary analogues of variational methods to approximate weak solutions of boundary-value problems in the theory of elasticity

Abstract

Methods of approximating weak solutions of certain boundary-value problems in the theory of elasticity are proposed based on expanding the approximate solution in a finite series in basis functions which identically satisfy a homogeneous differential equation in the domain. The coefficients of the expansion are found by constructing a boundary analogue of the method of least squares (BAMLS). It is proved that the approximate solution thus obtained converges to a weak solution of the problem. Sufficient conditions for the stability of the BAMLS, easily verifiable by computational means, are derived. The construction of a boundary analogue of the collocation method (BACM) is proposed on the basis of the BAMLS, combined with discretization of the scalar product by quadrature formulae. The BACM obtained is convergent and stable and possesses better computational properties than the BAMLS.