We 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.

中文翻译：

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抛物线问题的自适应非层次 Galerkin 方法在移动网格和虚拟单元方法中的应用

我们提出后验的用于线性抛物线问题的不一致和非分层 Galerkin 方法的误差估计，允许它们首次与非常一般的网格修改结合使用。我们处理非分层方案，因为时间步长之间的空间 Galerkin 空间可能彼此完全不相关。通过将我们的结果应用于移动网格的有限元方法并使用估计器来驱动基于任意多边形网格的虚拟元方法的自适应算法，证明了该设置的实际意义。这后验的误差估计，用于测量的误差大号2(H1)和大号∞(大号2)规范，是在一个抽象框架中使用椭圆重建技术推导出来的，该框架旨在精确地封装我们的不一致性和非层次性的概念，并且不需要在连续时间步长上使用的计算网格之间有特殊的兼容性，从而显着放宽了这个基于先前估计的基本假设.