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Partial differential equation solver based on optimization methods

The numerical solution methods for partial differential equation (PDE) solution allow obtaining a discrete field that converges towards the solution if the method is applied to the correct problem. Nevertheless, the numerical methods usually have the restricted class of the equations, on which the convergence is proved. Only a small amount of "cheap and dirty" numerical methods converge on a wide class of equations with the lower approximation order price. In the article, we present a method that uses an optimization algorithm to obtain a solution that could be used as the initial guess for the wide class of equations.