2020/05/12 by Andrea Cangiani, Cangiani, Andrea, Emmanuil H. Georgoulis +3 · 2 citations
Engineering · Computer Science · Mathematics · #Advanced Numerical Methods in Computational Mathematics #Advanced Mathematical Modeling in Engineering #Differential Equations and Numerical Methods
paper · pdf · doi:10.48550/arxiv.2005.05661
We present a posteriori error estimates for inconsistent and non-hierarchical\nGalerkin methods for linear parabolic problems, allowing them to be used in\nconjunction with very general mesh modification for the first time. We treat\nschemes which are non-hierarchical in the sense that the spatial Galerkin\nspaces used on consecutive time-steps may be completely unrelated from one\nanother. The practical interest of this setting is demonstrated by applying our\nresults to finite element methods on moving meshes and using the estimators to\ndrive an adaptive algorithm based on a virtual element method on a mesh of\narbitrary polygons. The a posteriori error estimates, for the error measured in\nthe L2(H1) and L\∞(L2) norms, are derived using the elliptic\nreconstruction technique in an abstract framework designed to precisely\nencapsulate our notion of inconsistency and non-hierarchicality and requiring\nno particular compatibility between the computational meshes used on\nconsecutive time-steps, thereby significantly relaxing this basic assumption\nunderlying previous estimates.\n