Boundary-value problems for elliptic equations are the most common problems of mathematical physics. They have many practical applications. They also appear in the reduced forms of parabolic and hyperbolic equations. Therefore it is quite natural to pay particular attention to problems related to elliptic operators. Here we will not focus on well-known facts connected with the statement of the boundary-value problem of elliptic type, and the dependence of the solutions on the properties of the input data. This information can be found in the literature. We will give only the necessary facts from the theory, and focus on the application of the various approaches to improving the difference solutions to such problems. In addition to linear problems a solvable nonlinear problem will also be considered. This nonlinear problem, as well as a simple diffraction problem, have been chosen because we wanted to demonstrate the improvement in the accuracy of the solutions for relatively simple problems. Our aim was also to acquaint the reader with basic algorithmic techniques used.
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