Clément-type interpolation on spherical domains - interpolation error estimates and application to a posteriori error estimation

• Main Authors: Apel, Thomas, Pester, Cornelia
• Other Authors: TU Chemnitz, SFB 393
• Format: Others
• Language: English
• Published: Universitätsbibliothek Chemnitz 2006
• Subjects: Clément-type interpolation; spherical domains; ddc:510; A-posteriori-Abschätzung; Finite-Elemente-Methode; Laplace-Beltrami-Operator
• Online Access: http://nbn-resolving.de/urn:nbn:de:swb:ch1-200601335; http://nbn-resolving.de/urn:nbn:de:swb:ch1-200601335; http://www.qucosa.de/fileadmin/data/qucosa/documents/5246/data/index.html; http://www.qucosa.de/fileadmin/data/qucosa/documents/5246/data/.htaccess; http://www.qucosa.de/fileadmin/data/qucosa/documents/5246/data/SFBcov.png; http://www.qucosa.de/fileadmin/data/qucosa/documents/5246/data/pdf.gif; http://www.qucosa.de/fileadmin/data/qucosa/documents/5246/data/pre.css; http://www.qucosa.de/fileadmin/data/qucosa/documents/5246/data/ps.gif; http://www.qucosa.de/fileadmin/data/qucosa/documents/5246/data/sfb03-13.pdf; http://www.qucosa.de/fileadmin/data/qucosa/documents/5246/data/sfb03-13.ps.gz; http://www.qucosa.de/fileadmin/data/qucosa/documents/5246/20060133.txt

In this paper, a mixed boundary value problem for the Laplace-Beltrami operator is considered for spherical domains in $R^3$, i.e. for domains on the unit sphere. These domains are parametrized by spherical coordinates (\varphi, \theta), such that functions on the unit sphere are considered as functions in these coordinates. Careful investigation leads to the introduction of a proper finite element space corresponding to an isotropic triangulation of the underlying domain on the unit sphere. Error estimates are proven for a Clément-type interpolation operator, where appropriate, weighted norms are used. The estimates are applied to the deduction of a reliable and efficient residual error estimator for the Laplace-Beltrami operator.