Verifications of primal energy identities for variational problems with obstacles

Error identities for variational problems with obstacles

The paper is concerned with a class of nonlinear free boundary problems, which are usually solved by variational methods based on primal (or primal-dual) variational settings. We deduce and investigate special relations (error identities). They show that a certain nonlinear measure of the distance to the exact solution (specific for each problem) is equivalent to the respective duality gap, which minimization is a keystone of all variational numerical methods. Therefore, the identity defines the measure that contains maximal quantitative information on the quality of a numerical solution avail…

基于Brézis-Ekeland-Nayroles原理的次梯度流的基于对偶性的后验误差恒等式

Duality-Based $\textit{A Posteriori}$ Error Identities for Subgradient Flows Based on the Brézis-Ekeland-Nayroles Principle

本文基于Brézis-Ekeland-Nayroles原理推导次梯度流的基于对偶性的后验误差恒等式，将其应用于多个物理相关非稳态变分问题，为次梯度流的误差分析提供了新框架。

具有一般增长的自治障碍问题解的极值性的一个必要条件

A necessary condition for extremality of solutions to autonomous obstacle problems with general growth

该研究针对满足凸性与无穷远超线性条件的自治障碍问题，利用原始-对偶表述刻画其存在且唯一的解，证明采用凸分析经典论证与变分法技巧，给出了解的极值性的必要条件。

Biharmonic obstacle problem: guaranteed and computable error bounds for approximate solutions

The paper is concerned with a free boundary problem generated by the biharmonic operator and an obstacle. The main goal is to deduce a fully guaranteed upper bound of the difference between the exact minimizer u and any function (approximation) from the corresponding energy class (which consists of the functions in $H^2$ satisfying the prescribed boundary conditions and the restrictions stipulated by the obstacle). For this purpose we use the duality method of the calculus of variations and general type error identities earlier derived for a wide class of convex variational problems. By this m…

Optimal Control of a Semidiscrete Cahn-Hilliard-Navier-Stokes System with Non-Matched Fluid Densities

This paper is concerned with the distributed optimal control of a time-discrete Cahn--Hilliard/Navier--Stokes system with variable densities. It focuses on the double-obstacle potential which yields an optimal control problem for a family of coupled systems in each time instance of a variational inequality of fourth order and the Navier--Stokes equation. By proposing a suitable time-discretization, energy estimates are proved and the existence of solutions to the primal system and of optimal controls is established for the original problem as well as for a family of regularized problems. The l…

Acoustic transmission problems: wavenumber-explicit bounds and resonance-free regions

We consider the Helmholtz transmission problem with one penetrable star-shaped Lipschitz obstacle. Under a natural assumption about the ratio of the wavenumbers, we prove bounds on the solution in terms of the data, with these bounds explicit in all parameters. In particular, the (weighted) $H^1$ norm of the solution is bounded by the $L^2$ norm of the source term, independently of the wavenumber. These bounds then imply the existence of a resonance-free strip beneath the real axis. The main novelty is that the only comparable results currently in the literature are for smooth, convex obstacle…