NUMERICAL HOMOGENIZATION OF H(CURL)-PROBLEMS
NUMERICAL HOMOGENIZATION OF H(CURL)-PROBLEMS
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DOI：
10.1137/17m1133932
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发表时间：
2018-01-01
影响因子：
2.9
通讯作者：
Verfuerth, Barbara
中科院分区：
文献类型：
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作者：
Gallistl, Dietmar;Henning, Patrick;Verfuerth, Barbara
如果与 H (curl)- 问题相关的椭圆微分算子涉及粗糙（快速变化）系数，则相应 H (curl)- 问题的解通常具有非常低的正则性，这会导致传统数值方案的收敛速度非常差。本文的目的是证明缺失的规律性可以通过校正算子来补偿。更准确地说，我们考虑最低阶 Nedelec 有限元空间，并证明存在具有四个中心属性的线性校正算子：它是可计算的、H (curl) 稳定的和准局部的，并允许校正粗略有限元函数，以便在右侧属于 H (div) 的情况下获得 H (curl) 范数中的一阶估计（就粗略网格大小而言）。利用这四个性质，实际应用是构造可直接用于伽辽金方法的广义有限元空间。特别是，这表征了均质解和一阶校正器，包括相应的定量误差估计，而不需要尺度分离。所构造的广义有限元方法属于局部正交分解方法，迄今为止尚未针对H(curl)-问题进行研究。
If an elliptic differential operator associated with an H (curl)- problem involves rough (rapidly varying) coefficients, then solutions to the corresponding H (curl)- problem admit typically very low regularity, which leads to arbitrarily bad convergence rates for conventional numerical schemes. The goal of this paper is to show that the missing regularity can be compensated through a corrector operator. More precisely, we consider the lowest-order Nedelec finite element space and show the existence of a linear corrector operator with four central properties: it is computable, H (curl)- stable, and quasi-local and allows for a correction of coarse finite element functions so that first-order estimates (in terms of the coarse mesh size) in the H (curl) norm are obtained provided the right-hand side belongs to H (div). With these four properties, a practical application is to construct generalized finite element spaces which can be straightforwardly used in a Galerkin method. In particular, this characterizes a homogenized solution and a first-order corrector, including corresponding quantitative error estimates without the requirement of scale separation. The constructed generalized finite element method falls into the class of localized orthogonal decomposition methods, which have not been studied for H (curl)- problems so far.