GLOBAL C1 MAPS ON GENERAL DOMAINS
GLOBAL C1 MAPS ON GENERAL DOMAINS
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一般领域的全球 C1 地图
DOI：
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发表时间：
2009
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通讯作者：
Einar M. Rønquist
中科院分区：
文献类型：
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作者：
A. E. Løvgren;Y. Maday;Einar M. Rønquist
在许多情况下，需要构造C1映射，从给定的参考域到一组变形域。在我们的例子中，动机来自于任意拉格朗日欧拉（ALE）方法和基元简化方法的应用。在这些方法中，使用映射来构造变形域上所需的网格点，并使用映射的相应雅可比矩阵将向量场从一个域映射到另一个域。为了保持映射向量场的连续性，雅可比矩阵必须是连续的，因此映射必须是C1。此外，在变形域上构建的网格应该是质量网格，即对于在任意变形域上定义的给定偏微分方程，其解应该是准确的。由于我们感兴趣的是一组变形区域，我们认为偏微分方程的解是由区域的几何形状控制的一组解。讨论并比较了不同的映射策略：Gordon和Hall提出的超限插值，Gordon和Wixom提出的伪调和扩展，Gordon - Hall方法的新推广（如二维一般多边形），调和扩展和floter提出的中值扩展
In many contexts, there is a need to construct C1 maps from a given reference domain to a family of deformed domains. In our case, the motivation comes from the application of the Arbitrary Lagrangian Eulerian (ALE) method and also the reduced basis element method. In these methods, the maps are used to construct the grid-points needed on the deformed domains, and the corresponding Jacobian of the map is used to map vector fields from one domain to another. In order to keep the continuity of the mapped vector fields, the Jacobian must be continuous, and thus the maps need to be C1. In addition, the constructed grids on the deformed domains should be quality grids in the sense that, for a given partial differential equation defined on any of the deformed domains, the solution should be accurate. Since we are interested in a family of deformed domains, we consider the solutions of the partial differential equation to be a family of solutions governed by the geometry of the domains. Different mapping strategies are discussed and compared: the transfinite interpolation proposed by Gordon and Hall,12 the pseudo-harmonic extension proposed by Gordon and Wixom,13 a new generalization of the Gordon–Hall method (e.g., to general polygons in two dimensions), the harmonic extension, and the mean-valued extension proposed by Floater.8