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：基于有限元的计算方案被提出并用于评估一种对偶变分方法，该方法旨在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（边）值问题转化为（空间）时间域中的退化椭圆边值问题，这些椭圆边值问题分别对应上述各问题中恰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（该策略可导出变分公式）在近似求解涵盖能量耗散与守恒的抛物型、双曲型及常微分原始问题方面取得了合理成功。该方案自然地将一族对偶解与唯一的原始解关联起来；这种“规范不变性”在我们对热传导方程和输运方程的计算解中得到了验证，包括热传导方程稳态原始解对应的瞬态对偶解的情形。研究表明，具有因果性的原始演化问题可通过非因果的对偶问题进行正确近似。