Abstract
This paper deals with the numerical approximation of solutions of Stokes and Brinkman systems using meshless methods. The aim is to solve a problem containing a nonzero body force, starting from the well known decomposition in terms of a particular solution and the solution of a homogeneous force problem. We propose two methods for the numerical construction of a particular solution. One method is based on the Neuber-Papkovich potentials, which we extend to nonhomogeneous Brinkman problems. A second method relies on a Helmholtz-type decomposition for the body force and enables the construction of divergence-free basis functions. Such basis functions are obtained from Hänkel functions and justified by new density results for the space H1(Ω). Several 2D numerical experiments are presented in order to discuss the feasibility and accuracy of both methods.
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References
Abramowitz, M., Stegun, I.: Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. Dover, New York (1964)
Alves, C.J.S., Chen, C.S.: A new method of fundamental solutions applied to nonhomogeneous elliptic problems. Adv. Comput. Math. 23, 125–142 (2005)
Alves, C.J.S., Silvestre, A.L.: Density results using Stokeslets and a method of fundamental solutions for the Stokes equations. Eng. Anal. Bound. Elem. 28, 1245–1252 (2004)
Barnett, A.H., Betcke, T.: Stability and convergence of the method of fundamental solutions for Helmholtz problems on analytic domains. J. Comput. Phys. 227(14), 7003–7026 (2008)
Bogomolny, A.: Fundamental solutions method for elliptic boundary value problems. SIAM J. Numer. Anal. 22, 644–669 (1985)
Brinkman, H.C.: A calculation of the viscous force exerted by a flowing fluid in a dense swarm of particles. Appl. Sci. Res. A1, 27–34 (1949)
Cheng, A.H.D., Hong, Y.: An overview of the method of fundamental solutions - solvability, uniqueness, convergence, and stability. Eng. Anal. Bound. Elem. 120, 118–152 (2020)
Evans, L.: Partial differential equations Graduate Studies in Mathematics, vol. 19. American Mathematical Society, Providence (1998)
Girault, V., Raviart, P.-A.: Finite element methods for Navier-Stokes equations. Theory and algorithms. In: Springer Series in Computational Mathematics, vol. 5. Springer, Berlin (1986)
Katsurada, M., Okamoto, H.: A mathematical study of the charge simulation method I. J. Fac. Sci. Univ. Tokyo Sect. IA Math. 35, 507–518 (1988)
Karageorghis, A., Lesnic, D.: The method of fundamental solutions for the Oseen steady-state viscous flow past obstacles of known or unknown shapes. Numer. Methods Partial Differ. Equ. 35(6), 2103–2119 (2019)
Krotkiewski, M., Ligaarden, I.S., Lie, K.-A., Schmid, D.W.: On the importance of the Stokes-Brinkman equations for computing effective permeability in carbonate karst reservoirs. Commun. Comput. Phys. 10(5), 1315–1332 (2011)
Li, X.: On convergence of the method of fundamental solutions for solving the Dirichlet problem of Poisson’s equation. Adv. Comp. Math 23, 265–277 (2005)
Martins, N.F.M., Rebelo, M.: Meshfree methods for nonhomogeneous Brinkman flows. Comput. Math. with Appl. 68(8), 872–886 (2014)
McLean, W.: Strongly Elliptic Systems and Boundary Integral Equations. Cambridge University Press, New-York (2000)
Nath, D., Kalra, M.S., Munshi, P.: One-stage method of fundamental and particular solutions (MFS-MPS) for the steady Navier-Stokes equations in a lid-driven cavity. Eng. Anal. Bound. Elem. 58, 39–47 (2015)
Neuber, H.: Ein neuer Ansatz zur lösung räumlicher Probleme der elastizitätstheorie. Der Hohlkegel unter Einzellast als Beispiel. Z. Angew. Math. Mech. 14, 203–212 (1934)
Nguyen, V.P., Rabczuk, T., Bordas, S., Duflot, M.: Meshless methods: a review and computer implementation aspects. Math. Comput. Simul. 79(3), 763–813 (2008)
Ogata, H., Katsurada, M.: Convergence of the invariant scheme of the method of fundamental solutions for two-dimensional potential problems in a Jordan region. Japan J. Indust. Appl. Math. 31, 231–262 (2014)
Papkovich, P.F.: The representation of the general integral of the fundamental equations of elasticity theory in terms of harmonic functions (in Russian). Izv. Akad. Nauk. SSSR Ser. Mat. 10, 1425–1435 (1932)
Pozrikidis, C.: Boundary integral and singularity methods for linearized viscous flows. Cambridge University Press, New-York (1992)
Schaback, R.: Error estimates and condition numbers for radial basis function interpolation. Adv. Comp. Math 3, 251–264 (1995)
Sincich, E., Sarler, B.: Non-singular method of fundamental solutions based on Laplace decomposition for 2D Stokes flow problems. CMES Comput. Model. Eng. Sci. 99(5), 393–415 (2014)
Tran-Cong, T., Blake, J.R.: General solutions of the Stokes’ flow equations. J. Math. Anal. Appl. 90(1), 72–84 (1982)
Tsai, C.C., Young, D.L., Fan, C.M., Chen, C.W.: MFS With time-dependent fundamental solutions for unsteady Stokes equations. Eng. Anal. Bound. Elem. 30(10), 897–908 (2006)
Varnhorn, W.: Boundary value problems and integral equations for the stokes resolvent in bounded and exterior domains of \(\mathbb {R}^n\), pp 206–224. World Scientific Publishing, New York (1998)
Young, D.L., Lin, Y.C., Fan, C.M., Chiu, C.L.: The method of fundamental solutions for solving incompressible Navier-Stokes problems. Eng. Anal. Bound. Elem. 33(8-9), 1031–1044 (2009)
Funding
C. J. S. Alves and A. L. Silvestre acknowledge the financial support of the Portuguese FCT - Fundação para a Ciência e a Tecnologia, through the projects UIDB/04621/2020 and UIDP/04621/2020 of CEMAT/IST-ID.
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Communicated by: Gianluigi Rozza
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Alves, C.J.S., Martins, N.F.M. & Silvestre, A.L. Numerical methods with particular solutions for nonhomogeneous Stokes and Brinkman systems. Adv Comput Math 48, 44 (2022). https://doi.org/10.1007/s10444-022-09937-3
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DOI: https://doi.org/10.1007/s10444-022-09937-3
Keywords
- Meshfree method
- Fundamental solutions
- Nonhomogeneous Brinkman equations
- Nonhomogeneous Stokes equations
- Particular solutions
- Density theorems