New Direct Numerical Methods for Some Multidimensional Problems of the Calculus of Variations

TL;DR

This paper introduces a novel numerical approach for multidimensional calculus of variations problems using Sobolev-like spaces, enabling analytical computation of steepest descent directions and extending to complex boundary and constraint conditions.

Contribution

The paper presents a new transformation-based method utilizing Sobolev-like spaces for direct numerical solutions of multidimensional variational problems, including extensions to complex boundary and constraint cases.

Findings

Developed a new transformation approach for variational problems

Constructed direct numerical methods based on steepest descent directions

Extended the approach to problems with complex boundary conditions and constraints

Abstract

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.