Simplicial Bernstein form and positivity certificates for solutions obtained in a stationary digital twin by Bernstein Bubnov-Galerkin method.

In this article, using the simplicial Bernstein form of polynomials on a simplex, we provide positivity certificates for the approximate solution of a linear elliptic PDE obtained by simplicial Bernstein Bubnov-Galerkin method. Particularly, we show how to obtain a simplicial Bernstein certificate of positivity for the approximate solution, although the discretized system does not satisfy a discrete weak maximum principle.