The solution in sense of Prager&Synge is the alternative to the commonly used notion of the numerical solution, which is considered as a limit of grid functions at mesh refinement. Prager&Synge solution is defined as a hypersphere containing the projection of the true solution of the system of partial differentiation equations (PDE) onto the computational grid and does not use any asymptotics. In the original variant it is determined using orthogonal properties specific for certain equations. In the proposed variant, the center and radius of the hypersphere is estimated using the ensemble of numerical solutions obtained by independent algorithms. This approach may be easily expanded for solutions of an arbitrary system of partial differentiation equations that significantly expands the domain of its applicability. Several options for the computation of the Prager&Synge solution are considered and compared herein. The first one is based on the search for the orthogonal truncation errors and their transformation. The second is based on the orthogonalization of approximation errors obtained using the defect correction method and applies a superposition of numerical solutions. These options are intrusive. In third option (nonintrusive) the information regarding orthogonality of errors, which is crucial for the Prager&Synge approach method, is replaced by information that stems from the properties of the ensemble of numerical solutions, obtained by independent numerical algorithms. The values of the angle between the truncation errors on such ensemble or the distances between elements of the ensemble may be used to replace the orthogonality. The variant based on the width of the ensemble of independent numerical solutions does not require any additional a priori information and is the approximate nonintrusive version of the method based on the orthogonalization of approximation errors.

翻译：Prager&Synge意义下的数值解是常用数值解概念的替代方案，后者被视为网格细化时网格函数的极限。Prager&Synge解定义为包含偏微分方程真解在计算网格上投影的超球体，且不依赖任何渐近性质。其原始形式利用特定方程的正交特性进行确定。在提出的变体中，超球体的中心与半径通过独立算法获得的数值解集合进行估计。该方法可轻松推广至任意偏微分方程系统的求解，显著扩展了其适用范围。本文考察并比较了计算Prager&Synge解的几种方案：第一种基于寻找正交截断误差及其变换；第二种采用缺陷修正法获得近似误差的正交化，并应用数值解的叠加。这两种方案具有侵入性。第三种方案（非侵入性）中，Prager&Synge方法的关键误差正交性信息被来自独立数值算法所得数值解集合性质的信息所取代，可利用该集合中截断误差之间的夹角或集合元素间的距离替代正交性。基于独立数值解集合宽度的变体无需任何额外先验信息，是采用近似误差正交化方法的近似非侵入式版本。