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A parametrix method for elliptic surface PDEs (2401.12501v1)

Published 23 Jan 2024 in math.NA and cs.NA

Abstract: Elliptic problems along smooth surfaces embedded in three dimensions occur in thin-membrane mechanics, electromagnetics (harmonic vector fields), and computational geometry. In this work, we present a parametrix-based integral equation method applicable to several forms of variable coefficient surface elliptic problems. Via the use of an approximate Green's function, the surface PDEs are transformed into well-conditioned integral equations. We demonstrate high-order numerical examples of this method applied to problems on general surfaces using a variant of the fast multipole method based on smooth interpolation properties of the kernel. Lastly, we discuss extensions of the method to surfaces with boundaries.

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References (75)
  1. FMM-accelerated solvers for the Laplace-Beltrami problem on complex surfaces in three dimensions. J. Sci. Comput., 97:25, 2023.
  2. The method of fundamental solutions applied to boundary value problems on the surface of a sphere. Comp. Math. Appl., 75(7):2365–2373, 2018.
  3. Anisotropic Laplace-Beltrami Operators for Shape Analysis. In L. Agapito, M. M. Bronstein, and C. Rother, editors, Computer Vision - ECCV 2014 Workshops, pages 299–312, Cham, 2015. Springer International Publishing.
  4. On the Laplace-Beltrami Operator and Brain Surface Flattening. IEEE Trans. Med. Imag., 18(8):700–711, 1999.
  5. A finite element method for surface diffusion: The parametric case. J. Comput. Phys., 203(1):321–343, 2005.
  6. The hitchhiker’s guide to the virtual element method. Math. Models and Meth. Appl. Sci., 24(08):1541–1573, 2014.
  7. Variational problems and partial differential equations on implicit surfaces. J. Comput. Phys., 174(2):759–780, 2001.
  8. An integral equation method for the numerical solution of a Dirichlet problem for second-order elliptic equations with variable coefficients. J. Eng. Math., 112(1):63–73, 2018.
  9. Finite element methods for the Laplace–Beltrami operator, volume 21. Elsevier B.V., 1 edition, 2020.
  10. A Priori Error Estimates for Finite Element Approximations to Eigenvalues and Eigenfunctions of the Laplace–Beltrami Operator. SIAM J. Num. Anal., 56(5):2963–2988, 2018.
  11. J. Bremer and Z. Gimbutas. A Nyström method for weakly singular integral operators on surfaces. J. Comput. Phys., 231:4885–4903, 2012.
  12. H. Brezis. Functional analysis, Sobolev spaces and partial differential equations. Universitext. Springer, New York
  13. A cut discontinuous Galerkin method for the Laplace-Beltrami operator. IMA J. Num. Anal., 37(1):138–169, 2017.
  14. A stable cut finite element method for partial differential equations on surfaces: The Helmholtz–Beltrami operator. Comput. Meth. Appl. Mech. Eng., 362:112803, 2020.
  15. Y. Chen and C. B. Macdonald. The closest point method and multigrid solvers for elliptic equations on surfaces. SIAM J. Sci. Comput., 37(1):A134–A155, 2015.
  16. Analysis of direct boundary-domain integral equations for a mixed BVP with variable coefficient, I: Equivalence and invertibility. J. Integral Equ. Appl., 21(4):499–543, 2009.
  17. Analysis of direct boundary-domain integral equations for a mixed BVP with variable coefficient, II: Solution regularity and asymptotics. J. Integral Equ. Appl., 22(1):19–37, 2010.
  18. Localized Boundary-Domain Singular Integral Equations Based on Harmonic Parametrix for Divergence-Form Elliptic PDEs with Variable Matrix Coefficients. Integral Equ. Oper. Theory, 76(4):509–547, 2013.
  19. Singular localised boundary-domain integral equations of acoustic scattering by inhomogeneous anisotropic obstacle. Math. Meth. Appl. Sci., 41(17):8033–8058, 2018.
  20. M. Chung and J. Taylor. Diffusion smoothing on brain surface via finite element method. In 2004 2nd IEEE International Symposium on Biomedical Imaging: Nano to Macro (IEEE Cat No. 04EX821), pages 432–435 Vol. 1, 2004.
  21. A. Demlow and G. Dziuk. An adaptive finite element method for the Laplace–Beltrami operator on implicitly defined surfaces. SIAM J. Num. Anal., 45(1):421–442, 2007.
  22. G. Dziuk. Finite elements for the Beltrami operator on arbitrary surfaces. In Partial differential equations and calculus of variations, pages 142–155. Springer, Berlin, Heidelberg, 1988.
  23. G. Dziuk and C. M. Elliott. Finite element methods for surface PDEs. Acta Numer., 22:289–396, 2013.
  24. C. L. Epstein and L. Greengard. Debye sources and the numerical solution of the time harmonic Maxwell equations. Comm. Pure Appl. Math., dec 2009.
  25. Debye Sources and the Numerical Solution of the Time Harmonic Maxwell Equations II. Comm. Pure Appl. Math., 66(5):753–789, 2013.
  26. A high-order wideband direct solver for electromagnetic scattering from bodies of revolution. J. Comput. Phys., 387:205–229, 2019.
  27. The surface diffusion flow for immersed hypersurfaces. SIAM J. Math. Anal., 29(6):1419–1433, 1998.
  28. L. C. Evans. Partial Differential Equations. American Mathematical Society, Providence, Rhode Island, 2nd edition, 2010.
  29. G. B. Folland. Introduction to Partial Differential Equations. Princeton University Press, Princeton, New Jersey, 2nd edition, 1995.
  30. D. Fortunato. A high-order fast direct solver for surface PDEs. SIAM J. Sci. Comput., 2023.
  31. C. Fresneda-Portillo and Z. Woldemicheal. A new family of boundary-domain integral equations for the Dirichlet problem of the diffusion equation in inhomogeneous media with H-1(ΩΩ\Omegaroman_Ω) source term on Lipschitz domains. Math. Meth. Appl. Sci., 44(12):9817–9830, 2021.
  32. C. Fresneda-Portillo and Z. W. Woldemicheal. On the existence of solution of the boundary-domain integral equation system derived from the 2D Dirichlet problem for the diffusion equation with variable coefficient using a novel parametrix. Complex Var. Elliptic Equ., 65(12):2056–2070, 2020.
  33. M. Frittelli and I. Sgura. Virtual element method for the Laplace-Beltrami equation on surfaces. ESAIM: Math. Modell. Num. Anal., 52(3):965–993, 2018.
  34. P. Garabedian. Partial Differential Equations. Chelsea Publish Company, New York, NY, 2nd edition, 1986.
  35. Fast multipole methods for the evaluation of layer potentials with locally-corrected quadratures. J. Comput. Phys.: X, 10:100092, 2021.
  36. L. Greengard and V. Rokhlin. A fast algorithm for particle simulations. J. Comput. Phys., 73(2):325–348, 1987.
  37. P. Grinfeld. Hamiltonian Dynamic Equations for Fluid Films. Studies in Applied Mathematics, 125(3):223–264, 2010.
  38. P. Grinfeld. Small oscillations of a soap bubble. Stud. Appl. Math., 128(1):30–39, 2012.
  39. J. Hadamard. Le problème de cauchy et les équations aux dérivées partielles linéaires hyperboliques. Paris, 11:243–264, 1932.
  40. E. Hebey. Nonlinear analysis on manifolds : Sobolev spaces and inequalities, volume 5 of Courant Lecture Notes in Mathematics. American Mathematical Society, New York, NY, 2000.
  41. K. Ho. FLAM: Fast Linear Algebra in MATLAB - Algorithms for Hierarchical Matrices. J. Open Source Softw., 5(51):1906, 2020.
  42. L. Hormander. The Analysis of Linear Partial Differential Operators III. Springer-Verlag, 1994.
  43. Boundary integral equations. Applied mathematical sciences (Springer-Verlag New York Inc.) ; v. 164. Springer, 2nd edition, 2021.
  44. L. M. Imbert-Gérard and L. Greengard. Pseudo-Spectral Methods for the Laplace-Beltrami Equation and the Hodge Decomposition on Surfaces of Genus One. Numerical Methods for Partial Differential Equations, 33(3):941–955, 2017.
  45. F. John. Partial Differential Equations. Springer-Verlag, New York, NY, 4th edition, 1982.
  46. A closest point method for surface pdes with interior boundary conditions for geometry processing. arXiv:2305.04711, 2023.
  47. J. Kromer and D. Bothe. Highly accurate numerical computation of implicitly defined volumes using the Laplace-Beltrami operator. arXiv:1805.03136, pages 1–25, 2018.
  48. M. C. Kropinski and N. Nigam. Fast integral equation methods for the Laplace-Beltrami equation on the sphere. Adv. Comput. Math., 40(2):577–596, 2014.
  49. Integral equation methods for the Yukawa-Beltrami equation on the sphere. Adv. Comput. Math., 42(2):469–488, 2016.
  50. The vibrations of bubbles and balloons. Acoustics Australia, 40(3):183–187, 2012.
  51. P. D. Lax. Functional analysis, volume 55. John Wiley & Sons, 2002.
  52. G. Leoni. A First Course in Sobolev Spaces: Second Edition. American Mathematical Society, Providence, Rhode Island, second edition, 2017.
  53. Level set equations on surfaces via the closest point method. J. Sci. Comput., 35:219–240, 2008.
  54. The implicit closest point method for the numerical solution of partial differential equations on surfaces. SIAM Journal on Scientific Computing, 31(6):4330–4350, 2010.
  55. Taylor States in Stellarators: A Fast High-order Boundary Integral Solver. J. Comput. Phys., Feb 2019.
  56. P.-G. Martinsson. Fast direct solvers for elliptic PDEs. SIAM, 2019.
  57. P.-G. Martinsson and V. Rokhlin. A fast direct solver for boundary integral equations in two dimensions. J. Comput. Phys., 205(1):1–23, 2005.
  58. P.-G. Martinsson and V. Rokhlin. An accelerated kernel-independent fast multipole method in one dimension. SIAM J. Sci. Comput., 29(3):1160–1178, 2007.
  59. A. Naumovets and Y. S. Vedula. Surface diffusion of adsorbates. Surface Science Reports, 4(7-8):365–434, 1985.
  60. J.-C. Nedéléc. Acoustic and Electromagnetic Equations. Springer-Verlag, New York, NY, 2001.
  61. Diffusion on a curved surface coupled to diffusion in the volume: Application to cell biology. J. Comput. Phys., 226(2):1271–1290, 2007.
  62. M. O’Neil. Second-kind integral equations for the Laplace-Beltrami problem on surfaces in three dimensions. Adv. Comput. Math., 44(5):1385–1409, 2018.
  63. M. O’Neil and A. J. Cerfon. An integral equation-based numerical solver for Taylor states in toroidal geometries. J. Comput. Phys., 359:263–282, 2018.
  64. J. Punchin. Weakly singular integral operators as mappings between function spaces. The Journal of Integral Equations and Applications, pages 303–320, 1988.
  65. M. Rachh. Integral equation methods for problems in electrostatics, elastostatics and viscous flow. PhD thesis, New York University, 2015.
  66. Petascale direct numerical simulation of blood flow on 200k cores and heterogeneous architectures. In SC’10: Proceedings of the 2010 ACM/IEEE International Conference for High Performance Computing, Networking, Storage and Analysis, pages 1–11. IEEE, 2010.
  67. S. J. Ruuth and B. Merriman. A simple embedding method for solving partial differential equations on surfaces. J. Comput. Phys., 227(3):1943–1961, 2008.
  68. Y. Saad and M. H. Schultz. GMRES: A generalized minimal residual algorithm for solving nonsymmetric linear systems. SIAM J. Sci. Stat. Comput., 7(3):856–869, 1986.
  69. B. Vallet and B. Lévy. Spectral Geometry Processing with Manifold Harmonics. Computer Graphics Forum, 27(2):251–260, 2008.
  70. A fast algorithm for simulating vesicle flows in three dimensions. J. Comput. Phys., 230(14):5610–5634, 2011.
  71. B. Vioreanu and V. Rokhlin. Spectra of Multiplication Operators as a Numerical Tool. SIAM J. Sci. Comput., 36(1):A267–A288, 2014.
  72. Modified Virtual Grid Difference for Discretizing the Laplace–Beltrami Operator on Point Clouds. SIAM J. Sci. Comput., 40(1):A1–A21, 2018.
  73. F. W. Warner. Foundations of Differentiable Manifolds and Lie Groups. Springer, New York, NY, 2013.
  74. X. Xing and E. Chow. Interpolative decomposition via proxy points for kernel matrices. SIAM J. Matrix Anal. Appl., 41(1):221–243, 2020.
  75. A kernel-independent adaptive fast multipole algorithm in two and three dimensions. J. Comput. Phys., 196(2):591–626, 2004.

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