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Full-Text Articles in Physical Sciences and Mathematics
Epi-Convergent Discretization Of The Generalizaed Bolza Problem In Dynamic Optimization, Boris S. Mordukhovich, Teemu Pennanen
Epi-Convergent Discretization Of The Generalizaed Bolza Problem In Dynamic Optimization, Boris S. Mordukhovich, Teemu Pennanen
Mathematics Research Reports
The paper is devoted to well-posed discrete approximations of the so-called generalized Bolza problem of minimizing variational functionals defined via extended-real-valued functions. This problem covers more conventional Bolza-type problems in the calculus of variations and optimal control of differential inclusions as well of parameterized differential equations. Our main goal is find efficient conditions ensuring an appropriate epi-convergence of discrete approximations, which plays a significant role in both the qualitative theory and numerical algorithms of optimization and optimal control. The paper seems to be the first attempt to study epi-convergent discretizations of the generalized Bolza problem; it establishes several rather general …
Can We Have Superconvergent Gradient Recovery Under Adaptive Meshes?, Haijun Wu, Zhimin Zhang
Can We Have Superconvergent Gradient Recovery Under Adaptive Meshes?, Haijun Wu, Zhimin Zhang
Mathematics Research Reports
We study adaptive finite element methods for elliptic problems with domain corner singularities. Our model problem is the two dimensional Poisson equation. Results of this paper are two folds. First, we prove that there exists an adaptive mesh (gauged by a discrete mesh density function) under which the recovered.gradient by the Polynomial Preserving Recovery (PPR) is superconvergent. Secondly, we demonstrate by numerical examples that an adaptive procedure with a posteriori error estimator based on PPR does produce adaptive meshes satisfy our mesh density assumption, and the recovered gradient by PPR is indeed supercoveregent in the adaptive process.