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Local tangential lifting virtual element method for the diffusion–reaction equation on the non-flat Voronoi discretized surface

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Abstract

In this paper, we propose the surface virtual element method (SVEM) combining with the local tangential lifting technique (LTL) to solve the diffusion–reaction (DR) equation on the non-flat Voronoi discretized surface embedded in \({\mathbb {R}}^3\). It has been a challenge on how to design the efficient numerical method to treat the non-flat discretized surface in comparison with the easy construction of flat discretized surface. Limited to the linear virtual element space, we derive the computable virtual element form of the non-flat Voronoi discretized surface by lifting the Voronoi element into the tangential plane. We demonstrate that this method developed here presents a good numerical simulation on a wide variety of polygonal discretized surfaces. Finally, numerical experiments are carried out to show the efficiency of the proposed method.

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References

  1. Fitzhugh R (1961) Impulses and physiological states in theoretical models of nerve membrane. Biophys J 1(6):445–466

    Google Scholar 

  2. Nagumo J, Arimoto S, Yoshizawa S (1962) An active pulse transmission line simulating nerve axon. Proc IRE 50(10):2061–2070

    Google Scholar 

  3. Sun M, Wang K, Feng X (2020) Numerical simulation of binary fluid-surfactant phase field model coupled with geometric curvature on the curved surface. Comput Methods Appl Mech Eng 367:113123

    MathSciNet  MATH  Google Scholar 

  4. Varea C, Aragon JL, Barrio RA (1999) Turing patterns on a sphere. Phys Rev E 60(4):4588–4592

    Google Scholar 

  5. Xiao X, Feng X, He Y (2019) Numerical simulations for the chemotaxis models on surfaces via a novel characteristic finite element method. Comput Math Appl 78(1):20–34

    MathSciNet  MATH  Google Scholar 

  6. Fornberg B, Flyer N (2015) Solving PDEs with radial basis functions. Acta Numer 24:215–258

    MathSciNet  MATH  Google Scholar 

  7. Frittelli M (2018) Virtual element method for the Laplace–Beltrami equation on surfaces. ESAIM Math Model Numer Anal 52(3):965–993

    MathSciNet  MATH  Google Scholar 

  8. Lehto E, Shankar V, Wright GB (2017) A radial basis function (RBF) compact finite difference (FD) scheme for reaction–diffusion equations on surfaces. SIAM J Sci Comput 39(5):A2129–A2151

    MathSciNet  MATH  Google Scholar 

  9. Li J, Gao Z, Dai Z, Feng X (2020) Divergence-free radial kernel for surface Stokes equations based on the surface Helmholtz decomposition. Comput Phys Commun 256:107408

    MathSciNet  Google Scholar 

  10. Shankar V, Wright GB, Kirby RM, Fogelson AL (2015) A radial basis function (RBF)-finite difference (FD) method for diffusion and reaction–diffusion equations on surfaces. J Sci Comput 63(3):745–768

    MathSciNet  MATH  Google Scholar 

  11. Zhao F, Li J, Xiao X, Feng X (2019) The characteristic RBF-FD method for the convection–diffusion–reaction equation on implicit surfaces. Numer Heat Transf Part A Appl 75(8):548–559

    Google Scholar 

  12. Macdonald CB, Ruuth SJ (2009) The implicit closest point method for the numerical solution of partial differential equations on surfaces. SIAM J Sci Comput 31(6):4330–4350

    MathSciNet  MATH  Google Scholar 

  13. Du Q, Gunzburger MD, Ju L (2003) Voronoi-based finite volume methods, optimal Voronoi meshes, and PDEs on the sphere. Comput Methods Appl Mecha Eng 192(35–36):3933–3957

    MathSciNet  MATH  Google Scholar 

  14. Ju L (2007) Conforming centroidal Voronoi Delaunay triangulation for quality mesh generation. Int J Numer Anal Model 4:531–547

    MathSciNet  MATH  Google Scholar 

  15. Ju L, Du Q (2009) A finite volume method on general surfaces and its error estimates. J Math Anal Appl 352(2):645–668

    MathSciNet  MATH  Google Scholar 

  16. Demlow A, Dziuk G (2007) An adaptive finite element method for the Laplace-Beltrami operator on implicitly defined surfaces. SIAM J Numer Anal 45(1):421–442

    MathSciNet  MATH  Google Scholar 

  17. Demlow A (2009) Higher-order finite element methods and pointwise error estimates for elliptic problems on surfaces. SIAM J Numer Anal 47(2):805–827

    MathSciNet  MATH  Google Scholar 

  18. Dziuk G (1988) Finite elements for the Beltrami operator on arbitrary surfaces. Springer, Berlin, pp 142–155

    MATH  Google Scholar 

  19. Dziuk G, Elliott CM (2013) Finite element methods for surface PDEs. Acta Numer 22(22):289–396

    MathSciNet  MATH  Google Scholar 

  20. Fries TP (2018) Higher-order surface FEM for incompressible Navier–Stokes flows on manifolds. Int J Numer Methods Fluids 88(2):55–78

    MathSciNet  Google Scholar 

  21. Olshanskii MA, Annalisa Q, Arnold R, Vladimir Y (2018) A finite element method for the surface Stokes problem. SIAM J Sci Comput 40(4):A2492–A2518

    MathSciNet  MATH  Google Scholar 

  22. Olshanskii MA, Reusken A, Grande J (2009) A finite element method for elliptic equations on surfaces. SIAM J Numer Anal 47(5):3339–3358

    MathSciNet  MATH  Google Scholar 

  23. Xiao X, Feng X, Yuan J (2018) The lumped mass finite element method for surface parabolic problems: error estimates and maximum principle. Comput Math Appl 76:488–507

    MathSciNet  MATH  Google Scholar 

  24. Xiao X, Feng X, Yuan J (2017) The stabilized semi-implicit finite element method for the surface Allen–Cahn equation. Discret Contin Dyn Syst Ser B 22(7):2857–2877

    MathSciNet  MATH  Google Scholar 

  25. Xiao X, Wang K, Feng X (2018) A lifted local Galerkin method for solving the reaction–diffusion equations on implicit surfaces. Comput Phys Commun 231:107–113

    MathSciNet  Google Scholar 

  26. Xiao X, Dai Z, Feng X (2020) A positivity preserving characteristic finite element method for solving the transport and convection–diffusion–reaction equations on general surfaces. Comput Phys Commun 247:106941

    MathSciNet  Google Scholar 

  27. Xiao X, Feng X, Li Z (2019) A gradient recovery-based adaptive finite element method for convection–diffusion–reaction equations on surfaces. Int J Numer Methods Eng 120(7):901–917

    MathSciNet  Google Scholar 

  28. Xiao X, Zhao J, Feng X (2020) A layers capturing type H-adaptive finite element method for convection–diffusion–reaction equations on surfaces. Comput Methods Appl Mecha Eng 361:112792

    MathSciNet  MATH  Google Scholar 

  29. Zhao S, Xiao X, Tan Z, Feng X (2018) Two types of spurious oscillations at layers diminishing methods for convection–diffusion–reaction equations on surface. Numer Heat Transf Part A Appl 74(7):1387–1404

    Google Scholar 

  30. Antonietti PF, Da Veiga LB, Scacchi S, Verani M (2016) A \(C^1\) virtual element method for the Cahn–Hilliard equation with polygonal meshes. SIAM J Numer Anal 54(1):34–56

    MathSciNet  MATH  Google Scholar 

  31. Cangiani A, Gyrya V, Manzini G (2016) The nonconforming virtual element method for the Stokes equations. SIAM J Numer Anal 54(6):3411–3435

    MathSciNet  MATH  Google Scholar 

  32. De Dios BA, Lipnikov K, Manzini G (2016) The nonconforming virtual element method. ESAIM Math Model Numer Anal 50(3):879–904

    MathSciNet  MATH  Google Scholar 

  33. Li Y, Xie C, Feng X (2020) Streamline diffusion virtual element method for convection-dominated diffusion problems. East Asian J Appl Math 10(1):158–180

    MathSciNet  MATH  Google Scholar 

  34. Ahmad B, Alsaedi A, Brezzi F, Marini LD, Russo A (2013) Equivalent projectors for virtual element methods. Comput Math Appl 66(3):376–391

    MathSciNet  MATH  Google Scholar 

  35. Da Veiga LB, Brezzi F, Cangiani A, Manzini G, Marini LD, Russo A (2013) Basic principles of virtual element methods. Math Models Methods Appl Sci 23(1):199–214

    MathSciNet  MATH  Google Scholar 

  36. Da Veiga LB, Brezzi F, Marini LD, Russo A (2014) The Hitchhiker’s guide to the virtual element method. Math Models Methods Appl Sci 24(08):1541–1573

    MathSciNet  MATH  Google Scholar 

  37. Vacca G, Da Veiga LB (2015) Virtual element methods for parabolic problems on polygonal meshes. Numer Methods Partial Differ Equ 31(6):2110–2134

    MathSciNet  MATH  Google Scholar 

  38. Chen SG, Wu JY (2013) Discrete conservation laws on curved surfaces. SIAM J Sci Comput 35(2):A719–A739

    MathSciNet  MATH  Google Scholar 

  39. Wu JY, Chi MH, Chen SG (2010) A local tangential lifting differential method for triangular meshes. Math Comput Simul 80(12):2386–2402

    MathSciNet  MATH  Google Scholar 

  40. Voronoi G (1908) Nouvelles applications des paramètres continus à la théorie des formes quadratiques. J Reine Angew Math 134:198–207

    MathSciNet  MATH  Google Scholar 

  41. Thiessen AH (1911) Precipitation averages for large areas. Mon Weather Rev 39(7):1082–1084

    Google Scholar 

  42. Demanet L (2006) Painless highly accurate discretizations of the Laplacian on a smooth manifold. Technical report, Stanford University

  43. Suchde P, Kuhnert J (2019) A meshfree generalized finite difference method for surface PDEs. Comput Math Appl 78(8):2789–2805

    MathSciNet  MATH  Google Scholar 

  44. Sutton OJ (2017) The virtual element method in 50 lines of MATLAB. Numer Algorithms 75(4):1141–1159

    MathSciNet  MATH  Google Scholar 

  45. Fuselier EJ, Wright GB (2013) A high-order kernel method for diffusion and reaction–diffusion equations on surfaces. J Sci Comput 56(3):535–565

    MathSciNet  MATH  Google Scholar 

  46. Chatzipantelidis P, Lazarov RD, Vidar Thome (2012) Some error estimates for the lumped mass finite element method for a parabolic problem. Math Comput 81(277):1–20

    MathSciNet  MATH  Google Scholar 

  47. Chen CM, Thomee V (1985) The lumped mass finite element method for a parabolic problem. ANZIAM J 26(3):329–354

    MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. College of Mathematics and System Science, Xinjiang University, Urumqi, 830046, People’s Republic of China

    Jingwei Li, Xinlong Feng & Yinnian He

  2. Laboratory of Mathematics and Complex Systems and School of Mathematical Sciences, Beijing Normal University, Beijing, 100875, People’s Republic of China

    Jingwei Li

  3. School of Mathematics and Statistics, Xi’an Jiaotong University, Xi’an, 710049, People’s Republic of China

    Yinnian He

Authors
  1. Jingwei Li
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  2. Xinlong Feng
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  3. Yinnian He
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Correspondence to Jingwei Li.

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This work is in part supported by the Excellent Doctor Innovation Program of Xinjiang University (No. XJUBSCX-2017006), China Postdoctoral Science Foundation (No. 2021M700476) and the NSF of China (Nos. 11671345, 11362021, 61962056).

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Li, J., Feng, X. & He, Y. Local tangential lifting virtual element method for the diffusion–reaction equation on the non-flat Voronoi discretized surface. Engineering with Computers 38, 5297–5307 (2022). https://doi.org/10.1007/s00366-021-01595-1

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  • DOI: https://doi.org/10.1007/s00366-021-01595-1

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