Optimal numerical solution of PDEs using the variational Germano identity

The variational Germano identity (VGI) is a condition that must be satisfied if the variational approximation to a partial differential equation is optimal in a user-defined metric. The VGI is useful as it can be utilized to dynamically determine unknown parameters in the variational approximation without any a-priori knowledge of the exact solution. With these values it yields solutions that are close to the optimal solution. In this manuscript we examine the relation between the definition of an optimal solution and the parameters computed using the VGI. We demonstrate that this relation is made precise by a so-called truncated restriction operator that appears in the final expression for the model parameters and is sensitive to the criterion used for selecting the optimal solution. In order to facilitate the use of the VGI for determining model parameters, we derive explicit expressions for this operator when three distinct approaches are used for defining the optimal solution. Through numerical examples that make use of these expressions, we confirm that the VGI does indeed yield model parameters and numerical solutions that are consistent with the choice of the optimal solution. We also observe that the numerical solutions obtained using the VGI are close to the best numerical solution that can be obtained from a given variational approximation, indicating that it yields nearly optimal values for the parameters.