I want to find the best aproximation to a numerical solution. For that I want to use two aproximated functions (that I already know). If I plot them I see that one of them underestimates the original solution and the other overestimates it.
It would be nice to combine this two "bad" approximations to find a better one.This way I think the error will be more or less compesated, but I don't know how I could do this properly.
Thanks for your ideas!
It is important to know in what sense you want the best approximations, and whether you are looking only for linear combinations of the functions or not. As an example, you can take a linear combination $\tilde{f} = \gamma g + \delta h$ of the 2 approximations $g$ and $h$, choose a set of grid points, use the values of the numerical solution and $\tilde{f}$ at these grid points and calculate simple OLS (Ordinary Least Squares) estimates for $\gamma$ and $\delta$ to get a "best" discrete approximation in $L^2$.