PUMA publications for /group/simtech/adaptivityFri Jul 20 10:54:15 CEST 2018IMA Journal of Numerical Analysis3947-970A Convergence Proof for Adaptive Finite Elements without Lower Bound312011adaptivity convergence elements finite density vorlaeufig We analyse the adaptive finite-element approximation to solutions of partial differential equations in variational formulation. Assuming well-posedness of the continuous problem and requiring only basic properties of the adaptive algorithm, we prove convergence of the sequence of discrete solutions to the true one. The proof is based on the ideas by Morin, Siebert and Veeser but replaces local efficiency of the estimator by a local density property of the adaptively generated finite-element spaces. As a result, estimators without a discrete lower bound are also included in our theory. The assumptions of the presented framework are fulfilled by a large class of important applications, estimators and adaptive strategies.Fri Jul 20 10:54:15 CEST 2018Journal of Computational Physics268-283A fast and accurate adaptive solution strategy for two-scale models with continuous inter-scale dependencies2402013Adaptivity vorlaeufig This article presents a fast and accurate adaptive algorithm that numerically solves a two-scale model with continuous inter-scale dependencies. The examined sample two-scale model describes a phase transition of a binary mixture with the evolution of equiaxed dendritic microstructures. It consists of a macroscopic heat equation and a family of microscopic cell problems that model the phase transition of the mixture. Both scales are coupled: the macroscopic temperature field influences the evolution of the microstructure and the microscopic fields enter to the macroscopic heat equation via averaged coefficients. Adaptivity exploits the constitutive assumption that the evolving microstructure depends in a continuous way on the macroscopic temperature field: macroscopic nodes with similar temperature evolutions use the same microscopic data. A suitable metric compares temperature evolutions and adaptive methods select active macroscopic nodes. Microscopic cell problems are solved for active nodes only; microscopic data in inactive nodes is approximated from microscopic data of active nodes with a similar temperature evolution. The set of active nodes is updated in course of the simulation: active nodes are deactivated until all active nodes have unsimilar temperature evolutions, and inactive nodes are activated until for every inactive node there exists at least one active node with a similar temperature evolution. Numerical examples, in two and in three space dimensions, show that the adaptive solution is only slightly less accurate than the direct solution, but it is computationally much more efficient. Therefore, the adaptive algorithm enables the solution of two-scale models with continuous inter-scale dependencies on large computational macroscopic and microscopic grids within an acceptable period of time for computation.