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Global \(C^1\) maps on general domains. (English) Zbl 1196.35030

Summary: In many contexts, there is a need to construct \(C^1\) 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 \(C^1\). 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 W. J. Gordon and C. A. Hall [Numer. Math. 21, 109–129 (1973; ; Zbl 0254.65072)], the pseudo-harmonic extension proposed by W. J. Gordon and J. A. Wixom [SIAM J. Numer. Anal. 11, 909–933 (1974; Zbl 0292.41001)], 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 M. S. Floater [Comput. Aided Geom. Des. 20, No. 1, 19–27 (2003; Zbl 1069.65553)].

MSC:

35A35 Theoretical approximation in context of PDEs
65N35 Spectral, collocation and related methods for boundary value problems involving PDEs
65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
65N50 Mesh generation, refinement, and adaptive methods for boundary value problems involving PDEs
35B30 Dependence of solutions to PDEs on initial and/or boundary data and/or on parameters of PDEs
35B65 Smoothness and regularity of solutions to PDEs
37C05 Dynamical systems involving smooth mappings and diffeomorphisms
54C05 Continuous maps
74S05 Finite element methods applied to problems in solid mechanics
74S25 Spectral and related methods applied to problems in solid mechanics
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