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This article is devoted to the construction and study of the finite element method for solving a two-dimensional nonlinear equation of elliptic type. Equations of this type arise in solving many applied problems, including problems of the theory of multiphase filtering, the theory of semiconductor devices, and many others. The relevance of the study of this problem is associated with the need to develop effective parallel methods for solving this problem. To discretize the equation, a mixed finite element method with Brezzi-Douglas-Marini elements is used. The issue of the convergence of the finite element method is investigated. To linearize the equation, the Picard iterative method is constructed. Two classes of basis functions of finite elements are used in the paper. A comparative analysis of the effectiveness of several direct and iterative methods for solving the resulting system of linear algebraic equations is carried out, including the method based on the Bunch-Kaufman LDLt factorization, the method of minimal residuals, the symmetric LQ method, the stabilized biconjugate gradient method, and a number of other iterative Krylov subspace algorithms with preconditioners based on incomplete LU decomposition. The method has been tested on several model problems by comparing an approximate solution with a known exact solution. The results of the analysis of the method error in various norms depending on the diameter of the mesh are presented.