Fractional Elliptic Quasi-Variational Inequalities: Theory and Numerics
Fractional Elliptic Quasi-Variational Inequalities: Theory and Numerics
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分数椭圆拟变分不等式：理论与数值
DOI：
10.4171/ifb/395
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发表时间：
2017
期刊：
影响因子：
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通讯作者：
C. N. Rautenberg
中科院分区：
文献类型：
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作者：
Harbir Antil;C. N. Rautenberg
介绍了一类具有s阶分数扩散的椭圆型拟变分不等式（QVI）问题，研究了其解的存在唯一性，并给出了一种求解算法。由于分数扩散禁止使用标准工具来近似QVI，因此我们将其实现为半无限圆柱体上问题的Dirichlet-to-Neumann映射。我们首先研究了该扩展QVI解的存在唯一性，然后将结果转移到分数阶QVI上，这为分数阶QVI领域引入了一个新的范式。进一步，我们截断了半无限圆柱，并证明了截断问题的解收敛于扩展问题的解，在相当温和的假设下，截断参数$\tau$趋于无穷。由于约束集随解的变化而变化，我们利用Mosco收敛性提出了一个论证。给出了一种求解截断问题的算法，并证明了截断问题在函数空间中的收敛性。最后，给出了几个数值算例。
This paper introduces an elliptic quasi-variational inequality (QVI) problem class with fractional diffusion of order $s \in (0,1)$, studies existence and uniqueness of solutions and develops a solution algorithm. As the fractional diffusion prohibits the use of standard tools to approximate the QVI, instead we realize it as a Dirichlet-to-Neumann map for a problem posed on a semi-infinite cylinder. We first study existence and uniqueness of solutions for this extended QVI and then transfer the results to the fractional QVI: This introduces a new paradigm in the field of fractional QVIs. Further, we truncate the semi-infinite cylinder and show that the solution to the truncated problem converges to the solution of the extended problem, under fairly mild assumptions, as the truncation parameter $\tau$ tends to infinity. Since the constraint set changes with the solution, we develop an argument using Mosco convergence. We state an algorithm to solve the truncated problem and show its convergence in function space. Finally, we conclude with several illustrative numerical examples.