01591nas a2200145   4500008004100000245013000041210006900171300001200240490000700252520107500259100002101334700002101355700002301376856004601399       2021                        eng d00aAdaptive non-hierarchical Galerkin methods for parabolic problems with application to moving mesh and virtual element methods0 aAdaptive nonhierarchical Galerkin methods for parabolic problems  a711-7510 v313 aWe present a posteriori error estimates for inconsistent and non-hierarchical Galerkin methods for linear parabolic problems, allowing them to be used in conjunction with very general mesh modification for the first time. We treat schemes which are non-hierarchical in the sense that the spatial Galerkin spaces between time-steps may be completely unrelated from one another. The practical interest of this setting is demonstrated by applying our results to finite element methods on moving meshes and using the estimators to drive an adaptive algorithm based on a virtual element method on a mesh of arbitrary polygons. The a posteriori error estimates, for the error measured in the L2(H1) and L∞(L2) norms, are derived using the elliptic reconstruction technique in an abstract framework designed to precisely encapsulate our notion of inconsistency and non-hierarchicality and requiring no particular compatibility between the computational meshes used on consecutive time-steps, thereby significantly relaxing this basic assumption underlying previous estimates.1 aCangiani, Andrea1 aGeorgoulis, E.H.1 aSutton, Oliver, J.  uhttps://doi.org/10.1142/S0218202521500172