Adaptive Refinement of the Variational Grid Method

In this paper, we have developed a new approach to the local improvement of an approximate solution which has been obtained with the finite element method. The proposed improvement is adaptive, and it minimizes the energy functional. The discussed algorithm can be implemented analytically as well as numerically. The proposed approach can be applied to the adaptive refinement of the previously calculated variational grid approximation. The proposed method allows you to expand the approximate space by adding new coordinate functions. The implementation of a local approximation requires a small amount of arithmetic operations. The number of arithmetic operations does not depend on the number of nodes of the previously calculated approximation. Examples of the implementation of the method are considered for a problem with strong degeneracy and for a problem without degeneracy.