Hidden convexity in the heat, linear transport, and Euler's rigid body equations: A computational approach

TL;DR

This paper introduces a finite element computational scheme that employs a duality-based variational approach to solve heat, transport, and Euler's rigid body equations, effectively handling diverse PDEs and ODEs with energy conservation and dissipation.

Contribution

It develops a unified dual variational framework for primal problems across parabolic, hyperbolic, and ODE systems, demonstrating gauge invariance and accurate approximation of solutions.

Findings

Successfully approximates solutions of heat, transport, and Euler equations.

Demonstrates gauge invariance with dual and primal solutions.

Handles energy dissipation and conservation effectively.

Abstract

A finite element based computational scheme is developed and employed to assess a duality based variational approach to the solution of the linear heat and transport PDE in one space dimension and time, and the nonlinear system of ODEs of Euler for the rotation of a rigid body about a fixed point. The formulation turns initial-(boundary) value problems into degenerate elliptic boundary value problems in (space)-time domains representing the Euler-Lagrange equations of suitably designed dual functionals in each of the above problems. We demonstrate reasonable success in approximating solutions of this range of parabolic, hyperbolic, and ODE primal problems, which includes energy dissipation as well as conservation, by a unified dual strategy lending itself to a variational formulation. The scheme naturally associates a family of dual solutions to a unique primal solution; such `gauge invariance' is demonstrated in our computed solutions of the heat and transport equations, including the case of a transient dual solution corresponding to a steady primal solution of the heat equation. Primal evolution problems with causality are shown to be correctly approximated by non-causal dual problems.