The Numerical Solution in the Sense of Prager&Synge

Abstract

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.