Løvgren, Alf Emil; Maday, Yvon; Rønquist, Einar M. Global \(C^1\) maps on general domains. (English) Zbl 1196.35030 Math. Models Methods Appl. Sci. 19, No. 5, 803-832 (2009). 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)]. Cited in 7 Documents 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 Keywords:lifting of trace; \(C^1\) extension; arbitrary Lagrangian Eulerian (ALE) method; reduced basis element method; transfinite interpolation; pseudo-harmonic extension; mean-valued extension Citations:Zbl 0254.65072; Zbl 0292.41001; Zbl 1069.65553 PDFBibTeX XMLCite \textit{A. E. Løvgren} et al., Math. Models Methods Appl. Sci. 19, No. 5, 803--832 (2009; Zbl 1196.35030) Full Text: DOI References: [1] DOI: 10.1016/0021-9045(73)90020-8 · Zbl 0271.41002 · doi:10.1016/0021-9045(73)90020-8 [2] A. Belyaev, Eurographics Symposium on Geometry Processing 2006 () pp. 89–99. [3] DOI: 10.1080/00036819108840031 · Zbl 0701.41009 · doi:10.1080/00036819108840031 [4] DOI: 10.1016/0021-9045(68)90024-5 · Zbl 0189.40902 · doi:10.1016/0021-9045(68)90024-5 [5] DOI: 10.1007/978-1-4612-3172-1 · Zbl 0788.73002 · doi:10.1007/978-1-4612-3172-1 [6] DOI: 10.1016/S0168-9274(03)00061-8 · Zbl 1031.65109 · doi:10.1016/S0168-9274(03)00061-8 [7] DOI: 10.1016/S0167-8396(03)00002-5 · Zbl 1069.65553 · doi:10.1016/S0167-8396(03)00002-5 [8] Formaggia L., East–West J. Numer. Math. 7 pp 105– [9] Gordon W. J., J. Math. Mech. 18 pp 931– [10] DOI: 10.1137/0708019 · Zbl 0237.41008 · doi:10.1137/0708019 [11] DOI: 10.1007/BF01436298 · Zbl 0254.65072 · doi:10.1007/BF01436298 [12] DOI: 10.1137/0711072 · Zbl 0292.41001 · doi:10.1137/0711072 [13] Grisvard P., Monographs and Studies in Mathematics 24, in: Elliptic Problems in Nonsmooth Domains (1985) · Zbl 0695.35060 [14] DOI: 10.1145/1183287.1183295 · Zbl 05457730 · doi:10.1145/1183287.1183295 [15] Løvgren A. E., Proceedings of ECCOMAS [16] Y. Maday and A. T. Patera, State of the Art Surveys in Computational Mechanics, ed. A. Noor (ASME, 1989) pp. 71–143. [17] DOI: 10.1016/0045-7825(90)90016-F · Zbl 0728.65078 · doi:10.1016/0045-7825(90)90016-F [18] Marsden J. E., Elementary Classical Analysis (1974) · Zbl 0285.26005 [19] DOI: 10.1016/j.cagd.2006.06.005 · Zbl 1171.65312 · doi:10.1016/j.cagd.2006.06.005 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.