2018/02/05 by Andrea Bonito, Bonito, Andrea, Alan Demlow +1
Engineering · #58J32 #65N15 #65N30 #Advanced Numerical Methods in Computational Mathematics #Electromagnetic Simulation and Numerical Methods #FOS: Mathematics #Numerical Analysis (math.NA) #Numerical methods in engineering
paper · pdf · doi:10.48550/arxiv.1802.01625
openalex publication_date 2018/02/05 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We prove new a posteriori error estimates for surface finite element methods\n(SFEM). Surface FEM approximate solutions to PDE posed on surfaces.\nPrototypical examples are elliptic PDE involving the Laplace-Beltrami operator.\nTypically the surface is approximated by a polyhedral or higher-order\npolynomial approximation. The resulting FEM exhibits both a geometric\nconsistency error due to the surface approximation and a standard Galerkin\nerror. A posteriori estimates for SFEM require practical access to geometric\ninformation about the surface in order to computably bound the geometric error.\nIt is thus advantageous to allow for maximum flexibility in representing\nsurfaces in practical codes when proving a posteriori error estimates for SFEM.\nHowever, previous a posteriori estimates using general parametric surface\nrepresentations are suboptimal by one order on C2 surfaces. Proofs of error\nestimates optimally reflecting the geometric error instead employ the closest\npoint projection, which is defined using the signed distance function. Because\nthe closest point projection is often unavailable or inconvenient to use\ncomputationally, a posteriori estimates using the signed distance function have\nnotable practical limitations. We merge these two perspectives by assuming it\npractical access only to a general parametric representation of the surface,\nbut using the distance function as a it theoretical tool. This allows us to\nderive sharper geometric estimators which exhibit improved experimentally\nobserved decay rates when implemented in adaptive surface finite element\nalgorithms.\n