Papers
Topics
Authors
Recent
Gemini 2.5 Flash
Gemini 2.5 Flash 79 tok/s
Gemini 2.5 Pro 49 tok/s Pro
GPT-5 Medium 45 tok/s
GPT-5 High 43 tok/s Pro
GPT-4o 103 tok/s
GPT OSS 120B 475 tok/s Pro
Kimi K2 215 tok/s Pro
2000 character limit reached

Reduced basis stabilization and post-processing for the virtual element method (2310.00625v2)

Published 1 Oct 2023 in math.NA and cs.NA

Abstract: We present a reduced basis method for cheaply constructing (possibly rough) approximations to the nodal basis functions of the virtual element space, and propose to use such approximations for the design of the stabilization term in the virtual element method and for the post-processing of the solution.

Definition Search Book Streamline Icon: https://streamlinehq.com
References (62)
  1. A stream virtual element formulation of the Stokes problem on polygonal meshes. SIAM Journal on Numerical Analysis, 52(1):386–404, 2014.
  2. Agglomeration-based geometric multigrid schemes for the virtual element method. SIAM Journal on Numerical Analysis, 61(1):223–249, 2023.
  3. A multigrid algorithm for the p-version of the virtual element method. ESAIM: M2AN, 52(1):337–364, 2018.
  4. Virtual element method for general second-order elliptic problems on polygonal meshes. Mathematical Models and Methods in Applied Sciences, 26(04):729–750, 2016.
  5. Basic principles of virtual element methods. Mathematical Models and Methods in Applied Sciences, 23(01):199–214, 2013.
  6. Virtual elements for linear elasticity problems. SIAM Journal on Numerical Analysis, 51(2):794–812, 2013.
  7. The hitchhiker’s guide to the virtual element method. Mathematical models and methods in applied sciences, 24(08):1541–1573, 2014.
  8. Serendipity nodal VEM spaces. Computers & Fluids, 141:2–12, 2016.
  9. Basic principles of hp virtual elements on quasiuniform meshes. Mathematical Models and Methods in Applied Sciences, 26(08):1567–1598, 2016.
  10. Stability analysis for the virtual element method. Mathematical Models and Methods in Applied Sciences, 27(13):2557–2594, 2017.
  11. Divergence free virtual elements for the Stokes problem on polygonal meshes. ESAIM: Mathematical Modelling and Numerical Analysis, 51(2):509–535, 2017.
  12. A hybrid mortar virtual element method for discrete fracture network simulations. Journal of Computational Physics, 306:148–166, 2016.
  13. Lowest order stabilization free virtual element method for the Poisson equation. arXiv preprint arXiv:2103.16896, 2021.
  14. Comparison of standard and stabilization free virtual elements on anisotropic elliptic problems. Applied Mathematics Letters, 129:107971, 2022.
  15. S. Berrone and M. Busetto. A virtual element method for the two-phase flow of immiscible fluids in porous media. Computational Geosciences, 26(1):195–216, 2022.
  16. Virtual element simulation of two-phase flow of immiscible fluids in discrete fracture networks. Journal of Computational Physics, 473:111735, 2023.
  17. S. Berrone and A. Raeli. Efficient partitioning of conforming virtual element discretizations for large scale discrete fracture network flow parallel solvers. Engineering Geology, 306:106747, 2022.
  18. Stabilization of the nonconforming virtual element method. Computers & Mathematics with Applications, 116:25–47, 2022.
  19. BDDC and FETI-DP for the virtual element method. Calcolo, 54:1565–1593, 2017.
  20. FETI-DP for the three dimensional virtual element method. SIAM Journal on Numerical Analysis, 58(3):1556–1591, 2020.
  21. Interior estimates for the virtual element method. arXiv preprint arXiv:2204.09955, 2022.
  22. Weakly imposed dirichlet boundary conditions for 2d and 3d virtual elements. Computer Methods in Applied Mechanics and Engineering, 400:115454, 2022.
  23. Some estimates for virtual element methods. Computational Methods in Applied Mathematics, 17(4):553–574, 2017.
  24. Basic principles of mixed virtual element methods. ESAIM: Mathematical Modelling and Numerical Analysis, 48(4):1227–1240, 2014.
  25. F. Brezzi and L. D. Marini. Virtual element methods for plate bending problems. Computer Methods in Applied Mechanics and Engineering, 253:455–462, 2013.
  26. hp-version discontinuous galerkin methods on polytopic meshes. to appear, 2017.
  27. A posteriori error estimates for the virtual element method. Numerische mathematik, 137:857–893, 2017.
  28. Hourglass stabilization and the virtual element method. International Journal for Numerical Methods in Engineering, 102(3-4):404–436, 2015.
  29. L. Chen and J. Huang. Some error analysis on virtual element methods. Calcolo, 55:1–23, 2018.
  30. A virtual element method for 3D contact problems with non-conforming meshes. Computer Methods in Applied Mechanics and Engineering, 402:115385, 2022.
  31. The mimetic finite difference method for elliptic problems, volume 11. Springer, 2014.
  32. Bend 3D mixed virtual element method for Darcy problems. Computers & Mathematics with Applications, 119:1–12, 2022.
  33. F. Dassi and S. Scacchi. Parallel block preconditioners for three-dimensional virtual element discretizations of saddle-point problems. Computer Methods in Applied Mechanics and Engineering, 372:113424, 2020.
  34. F. Dassi and S. Scacchi. Parallel solvers for virtual element discretizations of elliptic equations in mixed form. Computers & Mathematics with Applications, 79(7):1972–1989, 2020.
  35. Robust and scalable adaptive BDDC preconditioners for virtual element discretizations of elliptic partial differential equations in mixed form. Computer Methods in Applied Mechanics and Engineering, 391:114620, 2022.
  36. The nonconforming virtual element method. ESAIM: Mathematical Modelling and Numerical Analysis, 50(3):879–904, 2016.
  37. T. DeRose and M. Meyer. Harmonic coordinates. 2006.
  38. D. A. Di Pietro and J. Droniou. The hybrid high-order method for polytopal meshes. Number 19 in Modeling, Simulation and Application, 2020.
  39. M. S. Floater. Generalized barycentric coordinates and applications. Acta Numerica, 24:161–214, 2015.
  40. On the virtual element method for three-dimensional linear elasticity problems on arbitrary polyhedral meshes. Computer Methods in Applied Mechanics and Engineering, 282:132–160, 2014.
  41. Certified reduced basis methods for parametrized partial differential equations, volume 590. Springer, 2016.
  42. K. Hormann and N. Sukumar, editors. Generalized Barycentric Coordinates in Computer Graphics and Computational Mechanics. CRC Press, Boca Raton, FL, 2017.
  43. J. L. Lions and E. Magenes. Non-homogeneous boundary value problems and applications: Vol. 1, volume 181. Springer Science & Business Media, 2012.
  44. New perspectives on polygonal and polyhedral finite element methods. Mathematical Models and Methods in Applied Sciences, 24(08):1665–1699, 2014.
  45. Polyhedral finite elements using harmonic basis functions. In Computer graphics forum, volume 27, pages 1521–1529. Wiley Online Library, 2008.
  46. L. Mascotto. Ill-conditioning in the virtual element method: Stabilizations and bases. Numerical Methods for Partial Differential Equations, 34(4):1258–1281, 2018.
  47. A nonconforming Trefftz virtual element method for the Helmholtz problem. Mathematical Models and Methods in Applied Sciences, 29(09):1619–1656, 2019.
  48. A nonconforming Trefftz virtual element method for the Helmholtz problem: numerical aspects. Computer Methods in Applied Mechanics and Engineering, 347:445–476, 2019.
  49. The nonconforming Trefftz virtual element method: general setting, applications, and dispersion analysis for the Helmholtz equation. In The Virtual Element Method and its Applications, pages 363–410. Springer, 2022.
  50. An engineering perspective to the virtual element method and its interplay with the standard finite element method. Computer Methods in Applied Mechanics and Engineering, 350:995–1023, 2019.
  51. A virtual element method for 2D linear elastic fracture analysis. Computer Methods in Applied Mechanics and Engineering, 340:366–395, 2018.
  52. A Static condensation Reduced Basis Element method : approximation and a posteriori error estimation. ESAIM: Mathematical Modelling and Numerical Analysis, 47(1):213–251, 2013.
  53. FETI-DP preconditioners for the virtual element method on general 2D meshes. In Numerical Mathematics and Advanced Applications ENUMATH 2017, pages 157–164. Springer, 2019.
  54. N. Sukumar and E. Malsch. Recent advances in the construction of polygonal finite element interpolants. Archives of Computational Methods in Engineering, 13:129–163, 2006.
  55. O. J. Sutton. The virtual element method in 50 lines of MATLAB. Numerical Algorithms, 75(4):1141–1159, 2017.
  56. PolyMesher: a general-purpose mesh generator for polygonal elements written in Matlab. Structural and Multidisciplinary Optimization, 45:309–328, 2012.
  57. G. Vacca. An H1superscript𝐻1{H}^{1}italic_H start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT-conforming virtual element for Darcy and Brinkman equations. Mathematical Models and Methods in Applied Sciences, 28(01):159–194, 2018.
  58. P. Valtr. Probability that n random points are in convex position. 1994.
  59. S. Vendoschot. Generating random convex polygons, 2017.
  60. P. Wriggers and B. Hudobivnik. A low order virtual element formulation for finite elasto-plastic deformations. Computer Methods in Applied Mechanics and Engineering, 327:459–477, 2017.
  61. Efficient virtual element formulations for compressible and incompressible finite deformations. Computational Mechanics, 60:253–268, 2017.
  62. Y. Yu. mVEM: A MATLAB software package for the virtual element methods. arXiv preprint arXiv:2204.01339, 2022.
Citations (2)
List To Do Tasks Checklist Streamline Icon: https://streamlinehq.com

Collections

Sign up for free to add this paper to one or more collections.

Summary

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

Dice Question Streamline Icon: https://streamlinehq.com

Follow-up Questions

We haven't generated follow-up questions for this paper yet.

Don't miss out on important new AI/ML research

See which papers are being discussed right now on X, Reddit, and more:

“Emergent Mind helps me see which AI papers have caught fire online.”

Philip

Philip

Creator, AI Explained on YouTube