Piecewise Linear Approximation by the Method of Worst Segment Division

Abstract The finite element method for numerical solving two-dimensional boundary value problem is based on domain triangulation and piecewise linear approximation. The present paper describes how to minimize the number of triangulation vertices without exceeding the given level of the approximation error. The paper proposes a method for constructing piecewise linear approximations for continuous two-dimensional functions by dividing the ‘‘worst’’ segment. The possibilities of applying the proposed method in solving boundary value problems are investigated. The main theorem gives a sufficient condition on the minimized functional so that the best mesh would be the Delaunay triangulation. An example of a numerical solution of the Maxwell equation of the electromagnetic field using the method of the worst segment division with Delaunay triangulation is given.