About convergence of a mixed variational-grid method

V. L. Leont'ev

Sibirskij žurnal vyčislitelʹnoj matematiki, Tome 5 (2002) no. 1, pp. 25-34

Résumé

The variational-grid method connected to mixed variational principle and to the approximation of the exact solution by orthogonal finite functions is considered. Such functions differ from other orthogonal finite functions by simplicity of structure and symmetry and consequently simplify the algorithm of a method. The convergence of the approximate solutions in a task of mathematical physics and in a plane task of the theory of elasticity is investigated. The estimations of speed of their convergence are established. The orthogonal finite basic functions define the structure of the system of the grid equations of the variational-grid method, which supposes the exception of a part of unknown nodal quantities. It makes possible the use of classical technique of the research of the convergence of difference schemes and eliminates the basic lack of the mixed variationalgrid methods, which is connected to the increased number of nodal unknown quantities in comparison with methods based on variational principles for convex functionals. Thus all advantages of the mixed methods, caused by independent approximation of the unknown functions and their derivatives are kept.