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We present a posteriori error estimates for inconsistent and non-hierarchicalGalerkin methods for linear parabolic problems, allowing them to be used inconjunction with very general mesh modification for the first time. We treatschemes which are non-hierarchical in the sense that the spatial Galerkinspaces used on consecutive time-steps may be completely unrelated from oneanother. The practical interest of this setting is demonstrated by applying ourresults to finite element methods on moving meshes and using the estimators todrive an adaptive algorithm based on a virtual element method on a mesh ofarbitrary polygons. The a posteriori error estimates, for the error measured inthe $L^2(H^1)$ and $L^{\infty}(L^2)$ norms, are derived using the ellipticreconstruction technique in an abstract framework designed to preciselyencapsulate our notion of inconsistency and non-hierarchicality and requiringno particular compatibility between the computational meshes used onconsecutive time-steps, thereby significantly relaxing this basic assumptionunderlying previous estimates.

Adaptive non-hierarchical Galerkin methods for parabolic problems with application to moving mesh and virtual element methods