  In this paper we develop a new approach to the design of direct numerical
methods for multidimensional problems of the calculus of variations. The
approach is based on a transformation of the problem with the use of a new
class of Sobolev-like spaces that is studied in the article. This
transformation allows one to analytically compute the direction of steepest
descent of the main functional of the calculus of variations with respect to a
certain inner product, and, in turn, to construct new direct numerical methods
for multidimensional problems of the calculus of variations. In the end of the
paper we point out how the approach developed in the article can be extended to
the case of problems with more general boundary conditions, problems for
functionals depending on higher order derivatives, and problems with
isoperimetric and/or pointwise constraints.
