High-Order AFEM for the Laplace–Beltrami Operator: Convergence Rates

• Published in: Foundations of computational mathematics Vol. 16; no. 6; pp. 1473 - 1539
• Main Authors: Bonito, Andrea, Cascón, J. Manuel, Mekchay, Khamron, Morin, Pedro, Nochetto, Ricardo H.
• Format: Journal Article
• Language: English
• Published: New York Springer US 01.12.2016; Springer; Springer Nature B.V
• Subjects: Analysis; Applications of Mathematics; Approximation; Computer Science; Convergence; Convergence (Mathematics); Decay rate; Economics; Errors; Finite element method; Laplacian operator; Linear and Multilinear Algebras; Math Applications in Computer Science; Mathematical analysis; Mathematical models; Mathematics; Mathematics and Statistics; Matrix Theory; Numerical Analysis; Operators (mathematics); Polynomials; Texts; United States; 65N15; 65N12; Laplace–Beltrami operator; 41A25; Convergence rates; A posteriori error estimates; Parametric surfaces; Higher order; Adaptive finite element method; 65N30
• ISSN: 1615-3375; 1615-3383
• DOI: 10.1007/s10208-016-9335-7

We present a new AFEM for the Laplace–Beltrami operator with arbitrary polynomial degree on parametric surfaces, which are globally W ∞ 1 and piecewise in a suitable Besov class embedded in C 1 , α with α ∈ ( 0 , 1 ] . The idea is to have the surface sufficiently well resolved in W ∞ 1 relative to the current resolution of the PDE in H 1 . This gives rise to a conditional contraction property of the PDE module. We present a suitable approximation class and discuss its relation to Besov regularity of the surface, solution, and forcing. We prove optimal convergence rates for AFEM which are dictated by the worst decay rate of the surface error in W ∞ 1 and PDE error in H 1 .