Papers
Topics
Authors
Recent
Gemini 2.5 Flash
Gemini 2.5 Flash
97 tokens/sec
GPT-4o
53 tokens/sec
Gemini 2.5 Pro Pro
44 tokens/sec
o3 Pro
5 tokens/sec
GPT-4.1 Pro
47 tokens/sec
DeepSeek R1 via Azure Pro
28 tokens/sec
2000 character limit reached

Finite element analysis of a spectral problem on curved meshes occurring in diffusion with high order boundary conditions (2404.13994v1)

Published 22 Apr 2024 in math.NA and cs.NA

Abstract: In this work is considered a spectral problem, involving a second order term on the domain boundary: the Laplace-Beltrami operator. A variational formulation is presented, leading to a finite element discretization. For the Laplace-Beltrami operator to make sense on the boundary, the domain is smooth: consequently the computational domain (classically a polygonal domain) will not match the physical one. Thus, the physical domain is discretized using high order curved meshes so as to reduce the \textit{geometric error}. The \textit{lift operator}, which is aimed to transform a function defined on the mesh domain into a function defined on the physical one, is recalled. This \textit{lift} is a key ingredient in estimating errors on eigenvalues and eigenfunctions. A bootstrap method is used to prove the error estimates, which are expressed both in terms of \textit{finite element approximation error} and of \textit{geometric error}, respectively associated to the finite element degree $k\ge 1$ and to the mesh order~$r\ge 1$. Numerical experiments are led on various smooth domains in 2D and 3D, which allow us to validate the presented theoretical results.

Definition Search Book Streamline Icon: https://streamlinehq.com
References (34)
  1. G. Allaire. Numerical analysis and optimization. Numerical Mathematics and Scientific Computation. Oxford University Press, Oxford, 2007.
  2. I. Babuška and J. E. Osborn. Estimates for the errors in eigenvalue and eigenvector approximation by Galerkin methods, with particular attention to the case of multiple eigenvalues. SIAM J. Numer. Anal., 24(6):1249–1276, 1987.
  3. I. Babuška and J. E. Osborn. Finite element-Galerkin approximation of the eigenvalues and eigenvectors of selfadjoint problems. Math. Comp., 52(186):275–297, 1989.
  4. U. Banerjee and J. E. Osborn. Estimation of the effect of numerical integration in finite element eigenvalue approximation. Numer. Math., 56(8):735–762, 1990.
  5. C. Bernardi. Optimal finite-element interpolation on curved domains. SIAM J. Numer. Anal., 26(5):1212–1240, 1989.
  6. A priori error estimates for finite element approximations to eigenvalues and eigenfunctions of the Laplace-Beltrami operator. SIAM J. Numer. Anal., 56(5):2963–2988, 2018.
  7. The mathematical theory of finite element methods, volume 15 of Texts in Applied Mathematics. Springer-Verlag, New York, second edition, 2002.
  8. C. Carstensen and J. Gedicke. An oscillation-free adaptive FEM for symmetric eigenvalue problems. Numer. Math., 118(3):401–427, 2011.
  9. Numerical analysis of a diffusion equation with ventcel boundary condition using curved meshes. (submitted), 2023.
  10. Numerical study of a diffusion equation with ventcel boundary condition using curved meshes. Monografías Matemáticas García de Galdeano, 2023.
  11. P. G. Ciarlet. The finite element method for elliptic problems, volume 40 of Classics in Applied Mathematics. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2002. Reprint of the 1978 original [North-Holland, Amsterdam; MR0520174 (58 #25001)].
  12. Interpolation theory over curved elements, with applications to finite element methods. Comput. Methods Appl. Mech. Engrg., 1:217–249, 1972.
  13. C. Dapogny and P. Frey. Computation of the signed distance function to a discrete contour on adapted triangulation. Calcolo, 49(3):193–219, 2012.
  14. A. Demlow. Higher-order finite element methods and pointwise error estimates for elliptic problems on surfaces. SIAM J. Numer. Anal., 47(2):805–827, 2009.
  15. A. Demlow and G. Dziuk. An adaptive finite element method for the Laplace-Beltrami operator on implicitly defined surfaces. SIAM J. Numer. Anal., 45(1):421–442, 2007.
  16. F. Dubois. Discrete vector potential representation of a divergence-free vector field in three-dimensional domains: numerical analysis of a model problem. SIAM J. Numer. Anal., 27(5):1103–1141, 1990.
  17. G. Dziuk. Finite elements for the Beltrami operator on arbitrary surfaces. In Partial differential equations and calculus of variations, volume 1357 of Lecture Notes in Math., pages 142–155. Springer, Berlin, 1988.
  18. G. Dziuk and C. M. Elliott. Finite element methods for surface PDEs. Acta Numer., 22:289–396, 2013.
  19. D. Edelmann. Isoparametric finite element analysis of a generalized Robin boundary value problem on curved domains. SMAI J. Comput. Math., 7:57–73, 2021.
  20. C. M. Elliott and T. Ranner. Finite element analysis for a coupled bulk-surface partial differential equation. IMA J. Numer. Anal., 33(2):377–402, 2013.
  21. A. Ern and J.-L. Guermond. Theory and practice of finite elements, volume 159 of Applied Mathematical Sciences. Springer-Verlag, New York, 2004.
  22. D. Gallistl. An optimal adaptive FEM for eigenvalue clusters. Numer. Math., 130(3):467–496, 2015.
  23. S. Giani and I. G. Graham. A convergent adaptive method for elliptic eigenvalue problems. SIAM J. Numer. Anal., 47(2):1067–1091, 2009.
  24. D. Gilbarg and N. S. Trudinger. Elliptic partial differential equations of second order. Classics in Mathematics. Springer-Verlag, Berlin, 2001. Reprint of the 1998 edition.
  25. G. R. Goldstein. Derivation and physical interpretation of general boundary conditions. Adv. Differential Equations, 11(4):457–480, 2006.
  26. A. Henrot and M. Pierre. Variation et optimisation de formes: une analyse géométrique, volume 48. Springer Science & Business Media, 2006.
  27. Well-posedness, regularity, and convergence analysis of the finite element approximation of a generalized Robin boundary value problem. SIAM J. Numer. Anal., 53(1):105–126, 2015.
  28. New a priori FEM error estimates for eigenvalues. SIAM J. Numer. Anal., 43(6):2647–2667, 2006.
  29. M. Lenoir. Optimal isoparametric finite elements and error estimates for domains involving curved boundaries. SIAM J. Numer. Anal., 23(3):562–580, 1986.
  30. J.-C. Nédélec. Curved finite element methods for the solution of singular integral equations on surfaces in R3superscript𝑅3R^{3}italic_R start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT. Comput. Methods Appl. Mech. Engrg., 8(1):61–80, 1976.
  31. C. Pierre. The finite element library Cumin, curved meshes in numerical simulations. repository: https://plmlab.math.cnrs.fr/cpierre1/cumin, hal-0393713(v1), 2023.
  32. R. Scott. Interpolated boundary conditions in the finite element method. SIAM J. Numer. Anal., 12:404–427, 1975.
  33. A. D. Ventcel. Semigroups of operators that correspond to a generalized differential operator of second order. Dokl. Akad. Nauk SSSR (N.S.), 111:269–272, 1956.
  34. A. D. Ventcel. On boundary conditions for multi-dimensional diffusion processes. Theor. Probability Appl., 4:164–177, 1959.
User Edit Pencil Streamline Icon: https://streamlinehq.com
Authors (3)
  1. Fabien Caubet (12 papers)
  2. Joyce Ghantous (4 papers)
  3. Charles Pierre (15 papers)

Summary

We haven't generated a summary for this paper yet.

Ai Sparkles Streamline Icon: https://streamlinehq.com Summarize Now
X Twitter Logo Streamline Icon: https://streamlinehq.com

Tweets