Selfadaptive Mesh Modification for Parabolic FBPs: Theory and Computation
Selfadaptive Mesh Modification for Parabolic FBPs: Theory and Computation
复制标题
抛物线 FBP 的自适应网格修改：理论与计算
DOI：
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发表时间：
1990
期刊：
影响因子：
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通讯作者：
C. Verdi
中科院分区：
文献类型：
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作者：
R. Nochetto;M. Paolini;C. Verdi
抛物型自由边界问题(FBPs)固定区域格式的有限元近似通常具有次最优收敛速度。例如，对于两相Stefan问题，L 2中关于温度的分段线性逼近的收敛阶永远不会好于线性[13，14]，即使根据内插理论人们可能期望二次收敛。理论结果更加悲观[5，13，16]。这种全球数值污染是由界面上的奇异性产生的，这种奇异性本身就是一个与时间相关的未知数。因此，主要目标是通过使用适当分级的网格来消除这种污染影响。反过来，它们应该用于均匀分布内插误差以及伴随界面运动。设计一种自适应算法，自动修改(或再生)网格以满足某些预定的质量标准，这是一项具有挑战性的任务。这篇文章的目的是阐明几个关键的理论和计算问题，这些问题决定了自适应方法的质量和效率。
Finite element approximations of fixed domain formulations of parabolic free boundary problems (FBPs) typically exhibit a suboptimal rate of convergence. For the two-phase Stefan problem, for instance, the order of convergence in L 2 for piecewise linear approximations of temperature is never better than linear [13,14], even though one might expect quadratic convergence according to the interpolation theory. Theoretical results are even more pessimistic [5,13,16]. This sort of global numerical pollution is produced by the singularity located on the interface, which is a relevant time dependent unknown on its own right. The main goal is thus to eliminate such a pollution effect by using properly graded meshes. They, in turn, should serve to equidistribute interpolation errors as well as accompany the interface motion. The task of designing an adaptive algorithm, that automatically modifies (or regenerates) a mesh to meet certain predetermined quality criteria, is a challenging one. The aim of this paper is to shed light on several crucial theoretical and computational issues that dictate the quality and efficiency of an adaptive method for FBPs.