Frame theory, which contains wavelet analysis and Gabor analysis, has become a powerful tool for many applications of mathematics, engineering and quantum mechanics. The study of extension principles of Bessel sequences to frames is important in frame theory. This paper studies transformations on Bessel sequences to generate frames and Riesz bases in terms of operators and scalability. Some characterizations of operators that mapping Bessel sequences to frames and Riesz bases are given. We introduce the definitions of F-scalable and P-scalable Bessel sequences. F-scalability and P-scalability of Bessel sequences are discussed in this paper, then characterizations of scalings of F-scalable or P-scalable Bessel sequences are established. Finally, a perturbation result on F-scalable Bessel sequences is derived.

In this paper, we revisit a priori error analysis of nonconforming finite element methods for the Poisson problem. Based on some techniques developed in the context of the a posteriori error analysis, under two reasonable assumptions on the nonconforming finite element spaces, we prove that, up to some oscillation terms, the consistency error can be bounded by the approximation error. We check these two assumptions for the most used lower order nonconforming finite element methods. Compared with the classical error analysis of the nonconforming finite element method, the a priori analysis herein only needs the H1 regularity of the exact solution.

In this work, we develop an adaptive algorithm for solving elliptic optimal control problems with simultaneously appearing state and control constraints. The algorithm combines a Moreau-Yosida technique for handling state constraints with a semi-smooth Newton method for solving the optimality systems of the regularized sub-problems. The state and adjoint variables are discretized using continuous piecewise linear finite elements while a variational discretization concept is applied for the control. To perform the adaptive mesh refinements cycle we derive local error estimators which extend the goal-oriented error approach to our setting. The performance of the overall adaptive solver is assessed by numerical examples.