2022 , Volume 27, № 1, p.39-51

The finite element method is widely used to solve partial differential equations. The process of solving leads to a sparse system of linear algebraic equations with a singular matrix. It is known that the accuracy of solving systems of linear equations depends on the condition number of the system matrix. Previously, it was proposed to estimate the condition number for singular matrices using their non-zero eigenvalues, as for regular matrices. In the article, all eigenvalues of the stiffness matrix that arises when solving a one-dimensional problem using the same linear finite elements and all possible regularizations of the stiffness matrix — matrices obtained by deleting the row and column containing the selected diagonal element are analytically calculated. It is shown that for a stiffness matrix with an odd number of rows, the smallest of the condition numbers of regularization matrices is less than the number previously proposed as the condition number of a singular matrix. However, they are asymptotically equal. An estimate for the condition number of the matrix and its asymptotics are given. It is shown that the generally accepted method to increase the accuracy of calculations by the finite element method, which consists in element refinement, has a theoretical accuracy limit.