The master's thesis addresses a method for numerically solving elliptic boundary value problems of two variables in weak form using the Bernstein basis. It provides conditions for the existence and uniqueness of the solution and its approximation, along with an error estimate. The method is based on the finite element method, which is also described in the master's thesis. The solution is sought in the space of polynomial splines over triangulations, where each spline over an individual triangle is represented in the Bernstein basis rather than the typical polynomial basis. The thesis describes the properties of this basis, which enable us to seek an approximation as the solution to an optimization problem with additional constraints to ensure the satisfaction of boundary conditions and sufficient smoothness order.
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