This paper considers the application of extrapolation techniques in finding approximate solutions of some optimization problems with constraints defined by the Robin boundary problem for the Laplace equation. When applied extrapolation techniques produce very accurate solutions of the boundary problems on relatively coarse meshes, but this paper demonstrates that this is not a real restriction when dealing with optimization problems. Producing a solution of continuous problem by polynomial extrapolation based on the low-order discrete problem solutions significantly reduces both computational time and memory. The present paper illustrates this approach using finite-difference and finite-element methods, and finally makes a brief remark about some tacit engineering assumptions regarding numerical solutions of conductive media problems by construction of equivalent resistor networks.