Two-sided approximations of inverses, square roots and Cholesky factors
This article presents an idea in the finite element methods (FEMs) for obtaining two-sided bounds of exact eigenvalues. This approach is based on the combination of nonconforming methods giving lower bounds of the eigenvalues and a postprocessing technique using conforming finite elements. Our results hold for the second and fourth-order problems defined on two-dimensional domains. First, we list analytic and experimental results concerning triangular and rectangular nonconforming elements which...
We derive an optimal lower bound of the interpolation error for linear finite elements on a bounded two-dimensional domain. Using the supercloseness between the linear interpolant of the true solution of an elliptic problem and its finite element solution on uniform partitions, we further obtain two-sided a priori bounds of the discretization error by means of the interpolation error. Two-sided bounds for bilinear finite elements are given as well. Numerical tests illustrate our theoretical analysis....
We derive an optimal lower bound of the interpolation error for linear finite elements on a bounded two-dimensional domain. Using the supercloseness between the linear interpolant of the true solution of an elliptic problem and its finite element solution on uniform partitions, we further obtain two-sided a priori bounds of the discretization error by means of the interpolation error. Two-sided bounds for bilinear finite elements are given as well. Numerical tests illustrate our theoretical analysis. ...
Mathematical treatment of a planar magnetic field excited by permanent magnets is presented. A special two-sided condition for differential magnetic reluctivity is introduced to prove the unique existence of both the weak and the approximate solutions and also a certain error estimate. Notes to numerical algorithm and practical applications are given.
We study the convergence of two-step Ulm-Chebyshev-like method for solving the inverse singular value problems. We focus on the case when the given singular values are positive and multiple. This work extends the result of W. Ma (2022). We show that the new method is cubically convergent. Moreover, numerical experiments are given in the last section, which show that the proposed method is practical and efficient.