Abstract:
Adaptive finite element methods rely on a stable discretisation. For singularly perturbed and indefinite problems, the Galerkin residual no longer tracks the error, and marking may refine the wrong elements. Residual minimisation restores stability, but usually at the expense of broken test spaces, stabilisation parameters, or an ill-conditioned ambient solve. We present a minimum-residual realisation on three nested scales (coarse, fine, ambient) of a single continuous Galerkin space. Only the coarse and fine problems are solved, and no broken space or tuning parameter is needed. The coarse-to-fine correction is computable, and its energy equals the drop in the residual. We define markers as monotone set functions, with D\"orfler marking as the additive case, and show that the objective (the energy) decides which candidates are selected. A chain of computable constants measures how much of that decision the discrete solve realises. A three-branch algorithm, monitored by saturation and reliability indicators, contracts the error. We illustrate the theory on one- and two-dimensional finite element studies.

Bio:
Dr Victor Calo is a Professor of Applied Mathematics and a John Curtin Distinguished Professor at Curtin University. He holds the Professorial Chair in Computational Geoscience at Curtin University. Professor Calo is a highly cited researcher actively involved in disseminating knowledge: Victor has authored over 300 peer-reviewed publications, a book, and two patents. Prof Calo's research interests include adaptive finite element analysis of non-linear dynamics and high-performance computing driven by applications in geomechanics, fluid dynamics, flow in porous media, phase separation, fluid-structure interaction, and solid mechanics.

组织者：邓权灵