Rate Optimality of Adaptive Finite Elements for Parabolic Partial Differential Equations

• Applicant: Professor Dr. Kunibert G. Siebert
• Subject Area: Mathematics
• Term: from 2015 to 2019
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 273218570

• Final Report Year: 2019

Final Report Abstract

The most important advances made during the project were without a doubt the the design and analysis of the TAFEM algorithm and the a posteriori analysis for dG(s) in time discretizations, as well as a simplified algorithm and the extension of the result to isogeometric discretizations which we expect to be extensible to more general problems. The H1-stability result for the L2-projection also helped extend the type of refinement and polynomial degree for which the quasi-best approximation result holds true, as well as many other applications.

Publications

• A Weak Compatibility Condition for Newest Vertex Bisection in Any Dimension. SIAM J. Sci. Comput., 40(6):A3853–A3872, 2018
  M. Alkämper, F. D. Gaspoz, and R. Klöfkorn
  (See online at https://doi.org/10.1137/17M1156137)
• A convergent time-space adaptive dG(s) finite element method for parabolic problems motivated by equal error distribution. IMA J. of Numer. Anal., 39(2):650–686, 2019
  F. D. Gaspoz, C. Kreuzer, K. G. Siebert, and D. Ziegler
  (See online at https://doi.org/10.1093/imanum/dry005)
• An Alternative Proof of the H1-Stability of the L2 -Projection on Graded Meshes. Stuttgarter Mathematische Berichte 2019-001, Universität Stuttgart, ISSN 1613-8309
  F. D. Gaspoz, C.-J. Heine and K. G. Siebert