We present a general framework to compute upper and lower bounds for linear-functional outputs of the exact solutions of the Poisson equation based on reconstructions of the field variable and flux for both the primal and adjoint problems. The method is devised from a generalization of the complementary energy principle and the duality theory. Using duality theory, the computation of bounds is reduced to finding independent potential and equilibrated flux reconstructions. A generalization of this result is also introduced, allowing to derive alternative guaranteed bounds from nearly-arbitrary H(div;{\Omega}) flux reconstructions (only zero-order equilibration is required). This approach is applicable to any numerical method used to compute the solution. In this work, the proposed approach is applied to derive bounds for the hybridizable discontinuous Galerkin (HDG) method. An attractive feature of the proposed approach is that superconvergence on the bound gap is achieved, yielding accurate bounds even for very coarse meshes. Numerical experiments are presented to illustrate the performance and convergence of the bounds for the HDG method in both uniform and adaptive mesh refinements.

翻译：我们提出了一个总框架,用以根据对实地变量和通量的重建,对 Poisson 方程式的精度-功能性输出进行线性输出的上下界限的计算。该方法来自对补充能源原则的概括化和双重理论。使用双重理论,界限的计算将缩小到找到独立潜力和平衡通量重建。还采用了这一结果的概括化,从近乎任意的H(div; \Omega}) 通量重建(只要求零级平衡)中得出替代的保证界限。这个方法适用于用于计算解决方案的任何数字方法。在这项工作中,拟议的方法用于为混合的不连续加勒金(HDG)方法划定界限。拟议方法的一个吸引因素是,在封闭的距离上实现超级一致,甚至为非常粗略的中层提供了准确的界限。数量实验展示了在统一和适应性微微微的微微微微微微微微微微微微微微微微微微微微小的微小微小微微微微小微微微微微微微微微微微微微微微微微微微微微微微微微微微微微微微微微微微微微微微微微微微微微微微微微微微微微微微微微微微微微微微微微微微微微微微微微微微微微微微微微微微微微微微微微微微微微微微微微微微微微微微微微微微微微微微微微微微微微微微微微微微微微微微微微微微微微微微微微微微微微微微微微微微微微微微微微微微微微微微微微微微微微微微微微微微微微微微微微微微微微微微微微微微微微微微微微微微微微微微微微微微微微微微微微微微微微微微微微微微微微微微微微微微微微微微微微微微微微微微微微微微微微微微微微微微微微微微微微微微微微微微微微微微微微微微微微微微微微微微微微微微微微微