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：本文开发并采用基于有限元的计算方案，评估一种对偶变分方法在求解以下问题中的应用：一维空间与时间域中的线性热传导与输运偏微分方程，以及描述刚体绕定点旋转的欧拉非线性常微分方程组。该公式将初（边）值问题转化为（时空域中的）退化椭圆边值问题，这些边值问题分别表示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（该策略适合变分公式化），证明了在近似求解这一系列抛物型、双曲型及常微分原始问题（包括能量耗散与守恒情况）方面取得了合理成功。该方案自然地将一族对偶解与一个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与输运方程解中得到验证，包括热传导方程稳态原始解对应的瞬态对偶解情形。研究表明，具有因果性的原始演化问题可通过非因果的对偶问题得到正确近似。