More precise trail function in Rayleigh–Ritz method

Question

In order to obtain displacement field of an elasticity problem, say a plate structure, we approximate the solution using trigonometric series with unknown coefficients which satisfy the essential boundary condition. Unknown coefficients can be obtained by minimizing the total potential energy of the system. after that, deflections can be determined at any arbitrary point. We choose two different shape functions, e.g. polynomial and trigonometric. after determining the unknown coefficient for both of them, we have two solutions in hand. one gives larger displacement (say 3cm) for an arbitrary point, the other one gives smaller (say 2.5cm). Without having the exact solution, how can we find out which solution is more precise? smaller one or larger?