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.

翻译：本文发展并应用一种基于有限元的计算格式，以评估基于对偶变分方法求解一维空间与时间域中的线性热传导方程和输运偏微分方程，以及描述刚体绕定点转动的欧拉非线性常微分方程组。该公式将初始（边）值问题转化为（时空）域上的退化椭圆边值问题，这些边值问题代表了上述各问题中适当设计的对偶泛函的欧拉-拉格朗日方程。我们通过统一的对偶策略——该策略适用于变分公式——在逼近这类抛物型、双曲型及常微分原始问题（包括能量耗散与守恒情形）的解方面取得了合理成功。该格式天然地将一族对偶解与唯一的原始解相关联；这种“规范不变性”在我们计算的热传导方程和输运方程解中得到验证，包括对应于热传导方程稳态原始解的瞬态对偶解情形。研究表明，具有因果性的原始演化问题可由非因果的对偶问题正确逼近。