Adaptive non-hierarchical Galerkin methods for parabolic problems with application to moving mesh and virtual element methods
Adaptive non-hierarchical Galerkin methods for parabolic problems with application to moving mesh and virtual element methods
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DOI：
10.1142/s0218202521500172
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发表时间：
2020-05
期刊：
影响因子：
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通讯作者：
A. Cangiani;E. Georgoulis;Oliver J. Sutton
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文献类型：
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作者：
A. Cangiani;E. Georgoulis;Oliver J. Sutton
我们提出了线性抛物问题的不一致和非分层Galerkin方法的后验误差估计，允许它们首次与非常一般的网格修改结合使用。我们在时间步之间的空间伽辽金空间可能彼此完全无关的意义上处理非分层的方案。通过将我们的结果应用于移动网格的有限元方法，并使用估计器在任意多边形网格上驱动基于虚拟元素方法的自适应算法，证明了这种设置的实际意义。对于[公式：见文本]和[公式：见文本]规范中测量的误差，后验误差估计是在抽象框架中使用椭圆重建技术推导出来的，该框架旨在精确地概括我们的不一致性和非层次性概念，并且不要求在连续时间步上使用的计算网格之间具有特别的兼容性，从而大大放松了先前估计的基本假设。
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 [Formula: see text] and [Formula: see text] 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.