A Posteriori Error Estimates for Hypersingular Integral Equation on Spheres with Spherical Splines

Abstract

A posteriori residual and hierarchical upper bounds for the error estimates are proved when solving the hypersingular integral equation on the unit sphere by using the Galerkin method with spherical splines. Based on these a posteriori error estimates, adaptive mesh refining procedures are used to reduce complexity and computational cost of the discrete problems. Numerical experiments illustrate our theoretical results.

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Funding

This research is funded by the Vietnam National Foundation for Science and Technology Development (NAFOSTED) under grant no. 101.99–2016.13. A part of this research was conducted during the first author’s 2-month visit to the Vietnam Insitute for Advanced Study in Mathematics (VIASM). The authors would like to thank VIASM for the supports.

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Correspondence to Duong Thanh Pham.

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Pham, D.T., Le, T. A Posteriori Error Estimates for Hypersingular Integral Equation on Spheres with Spherical Splines. Acta Math Vietnam (2019). https://doi.org/10.1007/s40306-019-00331-8

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Keywords

  • Hypersingular integral equation
  • Spherical spline
  • A posteriori error estimate
  • Adaptivity

Mathematics Subject Classification (2010)

  • 65N30
  • 65N38
  • 65N15
  • 65N50