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A sparse hierarchical $hp$-finite element method on disks and annuli (2402.12831v2)

Published 20 Feb 2024 in math.NA and cs.NA

Abstract: We develop a sparse hierarchical $hp$-finite element method ($hp$-FEM) for the Helmholtz equation with variable coefficients posed on a two-dimensional disk or annulus. The mesh is an inner disk cell (omitted if on an annulus domain) and concentric annuli cells. The discretization preserves the Fourier mode decoupling of rotationally invariant operators, such as the Laplacian, which manifests as block diagonal mass and stiffness matrices. Moreover, the matrices have a sparsity pattern independent of the order of the discretization and admit an optimal complexity factorization. The sparse $hp$-FEM can handle radial discontinuities in the right-hand side and in rotationally invariant Helmholtz coefficients. Rotationally anisotropic coefficients that are approximated by low-degree polynomials in Cartesian coordinates also result in sparse linear systems. We consider examples such as a high-frequency Helmholtz equation with radial discontinuities and rotationally anisotropic coefficients, singular source terms, the time-dependent Schr\"odinger equation, and an extension to a three-dimensional cylinder domain, with a quasi-optimal solve, via the Alternating Direction Implicit (ADI) algorithm.

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References (59)
  1. Robert A Adams and John JF Fournier “Sobolev spaces” Elsevier, 2003
  2. “Unlocking the secrets of locking: Finite element analysis in planar linear elasticity” In Computer Methods in Applied Mechanics and Engineering 395 Elsevier, 2022, pp. 115034 DOI: 10.1016/j.cma.2022.115034
  3. Kendall Atkinson, David Chien and Olaf Hansen “Spectral Methods Using Multivariate Polynomials On The Unit Ball” CRC Press, 2019 DOI: 10.1201/9780429344374
  4. Jared Lee Aurentz and Richard Mikael Slevinsky “On symmetrizing the ultraspherical spectral method for self-adjoint problems” In Journal of Computational Physics 410 Elsevier, 2020, pp. 109383 DOI: 10.1016/j.jcp.2020.109383
  5. “Efficient preconditioning for the p𝑝pitalic_p-version finite element method in two dimensions” In SIAM Journal on Numerical Analysis 28.3 SIAM, 1991, pp. 624–661 DOI: 10.1137/0728034
  6. Ivo Babuška and Milo R Dorr “Error estimates for the combined hℎhitalic_h and p𝑝pitalic_p versions of the finite element method” In Numerische Mathematik 37 Springer, 1981, pp. 257–277 DOI: 10.1007/BF01398256
  7. Ivo Babuška and Barna A. Szabó “Lecture notes on finite element analysis”, 1983–1985
  8. Ivo Babuska, Barna A Szabo and I Norman Katz “The p𝑝pitalic_p-version of the finite element method” In SIAM Journal on Numerical Analysis 18.3 SIAM, 1981, pp. 515–545 DOI: 10.1137/0718033
  9. “Efficient methods for linear Schrödinger equation in the semiclassical regime with time-dependent potential” In Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 472.2193 The Royal Society Publishing, 2016, pp. 20150733 DOI: 10.1098/rspa.2015.0733
  10. Sven Beuchler, Clemens Pechstein and Daniel Wachsmuth “Boundary concentrated finite elements for optimal boundary control problems of elliptic PDEs” In Computational Optimization and Applications 51.2 Springer, 2012, pp. 883–908 DOI: 10.1007/s10589-010-9370-2
  11. “Sparse shape functions for tetrahedral p𝑝pitalic_p-FEM using integrated Jacobi polynomials” In Computing 80.4 Springer, 2007, pp. 345–375 DOI: 10.1007/s00607-007-0236-0
  12. “Sparsity optimized high order finite element functions on simplices” Springer, 2012 DOI: 10.1007/978-3-7091-0794-2˙2
  13. “New shape functions for triangular p𝑝pitalic_p-FEM using integrated Jacobi polynomials” In Numerische Mathematik 103.3 Springer, 2006, pp. 339–366 DOI: 10.1007/s00211-006-0681-2
  14. “Computing with functions in the ball” In SIAM Journal on Scientific Computing 42.4 SIAM, 2020, pp. C169–C191 DOI: 10.1137/19M1297063
  15. John P Boyd and Fu Yu “Comparing seven spectral methods for interpolation and for solving the Poisson equation in a disk: Zernike polynomials, Logan–Shepp ridge polynomials, Chebyshev–Fourier series, cylindrical Robert functions, Bessel–Fourier expansions, square-to-disk conformal mapping and radial basis functions” In Journal of Computational Physics 230.4 Elsevier, 2011, pp. 1408–1438 DOI: 10.1016/j.jcp.2010.11.011
  16. Susanne C. Brenner and L. Ridgway Scott “The Mathematical Theory of Finite Element Methods” 15, Texts in Applied Mathematics New York, NY: Springer New York, 2008 DOI: 10.1007/978-0-387-75934-0
  17. Pablo D Brubeck and Patrick E Farrell “A Scalable and Robust Vertex-Star Relaxation for High-Order FEM” In SIAM Journal on Scientific Computing 44.5 SIAM, 2022, pp. A2991–A3017 DOI: 10.1137/21M1444187
  18. “Dedalus: A flexible framework for numerical simulations with spectral methods” In Physical Review Research 2.2 APS, 2020, pp. 023068 DOI: 10.1103/PhysRevResearch.2.023068
  19. Andrea Cangiani, Zhaonan Dong and Emmanuil Georgoulis “h⁢pℎ𝑝hpitalic_h italic_p-version discontinuous Galerkin methods on essentially arbitrarily-shaped elements” In Mathematics of Computation 91.333, 2022, pp. 1–35 DOI: 10.1090/mcom/3667
  20. “ClassicalOrthogonalPolynomials.jl”, 2024 URL: https://github.com/JuliaApproximation/ClassicalOrthogonalPolynomials.jl
  21. Moshe Dubiner “Spectral methods on triangles and other domains” In Journal of Scientific Computing 6 Springer, 1991, pp. 345–390 DOI: 10.1007/BF01060030
  22. Abram C Ellison and Keith Julien “Gyroscopic polynomials” In Journal of Computational Physics Elsevier, 2023, pp. 112268 DOI: 10.1016/j.jcp.2023.112268
  23. Abram C Ellison, Keith Julien and Geoffrey M Vasil “A gyroscopic polynomial basis in the sphere” In Journal of Computational Physics 460 Elsevier, 2022, pp. 111170 DOI: 10.1016/j.jcp.2022.111170
  24. Lawrence C Evans “Partial Differential Equations” American Mathematical Society, 2010
  25. “FastTransforms.jl”, 2023 URL: https://github.com/JuliaApproximation/FastTransforms.jl
  26. Daniel Fortunato, Nicholas Hale and Alex Townsend “The ultraspherical spectral element method” In Journal of Computational Physics 436 Elsevier, 2021, pp. 110087 DOI: 10.1016/j.jcp.2020.110087
  27. “Fast Poisson solvers for spectral methods” In IMA Journal of Numerical Analysis 40.3 Oxford University Press, 2020, pp. 1994–2018 DOI: 10.1093/imanum/drz034
  28. Emilio Gagliardo “Caratterizzazioni delle tracce sulla frontiera relative ad alcune classi di funzioni in n𝑛nitalic_n variabili” In Rendiconti del seminario matematico della universita di Padova 27, 1957, pp. 284–305
  29. Timon S. Gutleb, Sheehan Olver and Richard Mikael Slevinsky “Polynomial and rational measure modifications of orthogonal polynomials via infinite-dimensional banded matrix factorizations”, 2023 arXiv:2302.08448 [math.NA]
  30. “Exponential integrators” In Acta Numerica 19 Cambridge University Press, 2010, pp. 209–286 DOI: 10.1017/S0962492910000048
  31. Lueling Jia, Huiyuan Li and Zhimin Zhang “Sparse spectral-Galerkin method on an arbitrary tetrahedron using generalized Koornwinder polynomials” In Journal of Scientific Computing 91.1 Springer, 2022, pp. 22 DOI: 10.1007/s10915-022-01778-y
  32. George E Karniadakis and Spencer Sherwin “Spectral/h⁢pℎ𝑝hpitalic_h italic_p element methods for computational fluid dynamics” Oxford University Press on Demand, 2005 DOI: 10.1093/acprof:oso/9780198528692.001.0001
  33. Kars Knook, Sheehan Olver and Ioannis P. A. Papadopoulos “Quasi-optimal complexity h⁢pℎ𝑝hpitalic_h italic_p-FEM for Poisson on a rectangle”, 2024 arXiv:2402.11299 [math.NA]
  34. “Spectral approximation on the unit ball” In SIAM Journal on Numerical Analysis 52.6 SIAM, 2014, pp. 2647–2675 DOI: 10.1137/130940591
  35. Alphonse P Magnus “Painlevé-type differential equations for the recurrence coefficients of semi-classical orthogonal polynomials” In Journal of Computational and Applied Mathematics 57.1-2 Elsevier, 1995, pp. 215–237 DOI: 10.1016/0377-0427(93)E0247-J
  36. Virendra N Mahajan “Zernike annular polynomials for imaging systems with annular pupils” In JOSA 71.1 Optica Publishing Group, 1981, pp. 75–85 DOI: 10.1364/JOSA.71.000075
  37. Marcela Molina Meyer and Frank Richard Prieto Medina “Polar differentiation matrices for the Laplace equation in the disk under nonhomogeneous Dirichlet, Neumann and Robin boundary conditions and the biharmonic equation under nonhomogeneous Dirichlet conditions” In Computers & Mathematics with Applications 89 Elsevier, 2021, pp. 1–19 DOI: 10.1016/j.camwa.2021.02.005
  38. Marcela Molina-Meyer and Frank Richard Prieto Medina “A collocation-spectral method to solve the bi-dimensional degenerate diffusive logistic equation with spatial heterogeneities in circular domains” In Editors-in-chief 52, 2020, pp. 311–344 DOI: 10.13137/2464-8728/30917
  39. “MultivariateOrthogonalPolynomials.jl”, 2023 URL: https://github.com/JuliaApproximation/MultivariateOrthogonalPolynomials.jl
  40. “NIST Digital Library of Mathematical Functions”, http://dlmf.nist.gov/, Release 1.1.4 of 2022-01-15, 2022 URL: http://dlmf.nist.gov/
  41. Sheehan Olver, Richard Mikaël Slevinsky and Alex Townsend “Fast algorithms using orthogonal polynomials” In Acta Numerica 29 Cambridge University Press, 2020, pp. 573–699 DOI: 10.1017/S0962492920000045
  42. “A fast and well-conditioned spectral method” In SIAM Review 55.3 SIAM, 2013, pp. 462–489 DOI: 10.1137/120865458
  43. Sheehan Olver, Alex Townsend and Geoffrey Vasil “A sparse spectral method on triangles” In SIAM Journal on Scientific Computing 41.6 SIAM, 2019, pp. A3728–A3756 DOI: 10.1137/19M1245888
  44. Eduardo L Ortiz “The tau method” In SIAM Journal on Numerical Analysis 6.3 SIAM, 1969, pp. 480–492 DOI: 10.1137/0706044
  45. Ioannis P. A. Papadopoulos “ioannisPApapadopoulos/SparseDiskFEM.jl: v0.0.1” Zenodo, 2024 DOI: 10.5281/zenodo.10682797
  46. “Building hierarchies of semiclassical Jacobi polynomials for spectral methods in annuli”, 2023 arXiv:2310.07541 [math.NA]
  47. “PiecewiseOrthogonalPolynomials.jl”, 2024 URL: https://github.com/JuliaApproximation/PiecewiseOrthogonalPolynomials.jl
  48. “RadialPiecewisePolynomials.jl”, 2024 URL: https://github.com/ioannisPApapadopoulos/RadialPiecewisePolynomials.jl
  49. Ch Schwab “p𝑝pitalic_p-and h⁢pℎ𝑝hpitalic_h italic_p-finite element methods: Theory and applications in solid and fluid mechanics” Clarendon Press, 1998
  50. “SemiclassicalOrthogonalPolynomials.jl”, 2024 URL: https://github.com/JuliaApproximation/SemiclassicalOrthogonalPolynomials.jl
  51. Richard Mikaël Slevinsky “Fast and backward stable transforms between spherical harmonic expansions and bivariate Fourier series” In Applied and Computational Harmonic Analysis 47.3 Elsevier, 2019, pp. 585–606 DOI: 10.1016/j.acha.2017.11.001
  52. “Sparse spectral and-finite element methods for partial differential equations on disk slices and trapeziums” In Studies in Applied Mathematics 145.1 Wiley Online Library, 2020, pp. 3–35 DOI: 10.1111/sapm.12303
  53. “Sparse spectral methods for partial differential equations on spherical caps” In Transactions of Mathematics and Its Applications 5.1 Oxford University Press, 2021, pp. tnab001 DOI: 10.1093/imatrm/tnab001
  54. “SparseDiskFEM.jl”, 2024 URL: https://github.com/ioannisPApapadopoulos/SparseDiskFEM.jl
  55. “Introduction to finite element analysis: formulation, verification and validation” John Wiley & Sons, 2011
  56. Berge Tatian “Aberration balancing in rotationally symmetric lenses” In JOSA 64.8 Optica Publishing Group, 1974, pp. 1083–1091 DOI: 10.1364/JOSA.64.001083
  57. “Tensor calculus in polar coordinates using Jacobi polynomials” In Journal of Computational Physics 325 Elsevier, 2016, pp. 53–73 DOI: 10.1016/j.jcp.2016.08.013
  58. Heather Wilber, Alex Townsend and Grady B Wright “Computing with functions in spherical and polar geometries II. The disk” In SIAM Journal on Scientific Computing 39.3 SIAM, 2017, pp. C238–C262 DOI: 10.1137/16M1070207
  59. James E. Wiss “Quantum Mechanics in Multidimensions” In Physics 485 (Fall 2015) Course Notes, accessed: 2023-11-22 Unversity of Illinois, 2015 URL: https://courses.physics.illinois.edu/phys485/fa2015/web/dimensions.pdf
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