  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.
