A Meshfree Method for Solving the Stokes Problem on Bounded Domains

Abstract

We investigate the application of an analytically divergence-free kernel method for solving the Stokes equations on bounded domains in this paper. We develop a new collocation method based on the divergence-free kernel trial spaces produced by radial basis functions (RBFs). An $L^{2}$ error is obtained when the testing discretization is finer than the trial discretization. The convergence rates depend on the regularity of the solutions, the smoothness of the computational domain, the selection of the scaling parameter, and the approximation of divergence-free kernel trial spaces (for the velocity field) and RBF trial spaces (for the pressure field). The convergence theory is established in Sobolev spaces, which covers stationary and non-stationary approximations. Several numerical examples are provided to demonstrate numerical efficiency.

Keywords

Stokes problem / Radial basis functions (RBFs) / Collocation method / Divergence-free kernel / Sobolev spaces / Convergence rates / 65N12 / 65N15 / 65N35

References

[1] Adams, R.A., Fournier, P.: Sobolev Spaces, 2nd edn. Pure and Applied Mathematics. Academic Press, New York (2003)

[2] Arcangeli R, De Silanes MCL, Torrens JJ. Extension of sampling inequalities to Sobolev semi-norms of fractional order and derivative data. Numer. Math., 2012, 121: 587-608.

[3] Burman E. Pressure projection stabilizations for Galerkin approximations of Stokes’ and Darcy’s problem. Numer. Methods Part. Differ. Equ., 2008, 24: 127-143.

[4] Chernih A, Le Gia QT. Multiscale methods with compactly supported radial basis functions for the Stokes problem on bounded domains. Adv. Comput. Math., 2016, 42: 1187-1208.

[5] Cheung KC, Ling L, Schaback R. $H^{2}$-convergence of least-squares kernel collocation methods. SIAM J. Numer. Anal., 2018, 56(1): 614-633.

[6] Drake KP, Fuselier EJ, Wright GB. A partition of unity method for divergence-free or curl-free radial basis function approximation. SIAM J. Sci. Comput., 2021, 43(3): A1950-A1974.

[7] Fuselier, E.J.: Refined error estimates for matrix-valued radial basis functions. Ph.D. thesis, Texas A&M University, College Station (2006)

[8] Fuselier EJ. Improved stability estimates and a characterization of the native space for matrix-valued RBFs. Adv. Comput. Math., 2008, 29: 269-290.

[9] Fuselier EJ. Sobolev-type approximation rates for divergence-free and curl-free RBF interpolation. Math. Comput., 2008, 77(263): 1407-1423.

[10] Fuselier EJ, Shankar V, Wright GB. A high-order radial basis function (RBF) Leray projection method for the solution of the incompressible unsteady Stokes equations. Comput. Fluids, 2016, 128: 41-52.

[11] Hangelbroek T, Narcowich FJ, Rieger C, Ward JD. An inverse theorem for compact Lipschitz regions in $\mathbb{R} ^d$ using localized kernel bases. Math. Comput., 2018, 87: 1949-1989.

[12] Hoff D, Wendland H. A meshfree method for a PDE-constrained optimization problem. SIAM J. Numer. Anal., 2021, 59(4): 1896-1917.

[13] Hon YC, Schaback R. On unsymmetric collocation by radial basis functions. J. Appl. Math. Comput., 2001, 119: 177-186.

[14] Kansa E. J.: Application of Hardy’s multiquadric interpolation to hydrodynamics. In: Proceedings of the 1986 Annual Simulations Conference, Vol. 4, San Diego, pp. 111–117 (1986)

[15] Keim C, Wendland H. A high-order, analytically divergence-free approximation method for the time-dependent Stokes problem. SIAM J. Numer. Anal., 2016, 54: 1288-1312.

[16] Kosec G. A local numerical solution of a fluid-flow problem on an irregular domain. Adv. Eng. Softw., 2018, 120: 136-44.

[17] Li, J., Lin, X., Chen, Z.: Finite Volume Methods for the Incompressible Navier-Stokes Equations. Springer, New York (2022)

[18] Liu Z, Xu Q. $L^{2}$ error estimates of unsymmetric RBF collocation for second order quasilinear elliptic equations. Commun. Nonlinear Sci. Numer. Simulat., 2023, 127107563

[19] Liu Z, Xu Q. Meshfree method for solving the elliptic Monge-Amp$\grave{\rm e }$re equation with Dirichlet boundary. Comput. Math. Appl., 2026, 201: 53-64.

[20] Lowitzsch, S.: Approximation and interpolation employing divergence-free radial basis functions with applications. Ph.D. thesis, Texas A&M University, College Station (2002)

[21] Lowitzsch S. A density theorem for matrix-valued radial basis functions. Numer. Algorithms, 2005, 39: 253-256.

[22] Lowitzsch S. Error estimates for matrix-valued radial basis function interpolation. J. Approx. Theory, 2005, 137: 234-249.

[23] Lowitzsch S. Matrix-valued radial basis functions: stability estimates and applications. Adv. Comput. Math., 2005, 23: 299-315.

[24] Mitrano, A.A.: Properties of divergence-free kernel methods for approximation and solution of partial differential equation. Ph.D. thesis, Arizona State University (2016)

[25] Mohammadi V, Dehghan M. A divergence-free generalized moving least squares approximation with its application. Appl. Numer. Math., 2021, 162: 374-404.

[26] Mohammadi V, Dehghan M. Error estimates of divergence-free generalized moving least squares (Div-Free GMLS) derivatives approximations in Sobolev spaces. Appl. Numer. Math., 2023, 192: 373-388.

[27] Najafi, M., Dehghan, M., Šarler, B., Kosec, G., Mavrič, B.: Divergence-free meshless local Petrov-Galerkin method for Stokes flow. Eng. Comput. 38, 5359–5377 (2022)

[28] Narcowich FJ, Ward JD. Generalized Hermite interpolation via matrix-valued conditionally positive definite functions. Math. Comput., 1994, 63: 661-687.

[29] Narcowich FJ, Ward JD, Wendland H. Sobolev error estimates and a Bernstein inequality for scattered data interpolation via radial basis functions. Constr. Approx., 2006, 24: 175-186.

[30] Quarteroni A, Valli ANumerical Approximation of Partial Differential Equations, 1994, Berlin: Springer

[31] Schaback R. Convergence of unsymmetric kernel-based meshless collocation methods. SIAM J. Numer. Anal.2007, 45(1): 333-351.

[32] Schaback R, Wendland H. Inverse and saturation theorems for radial basis function interpolation. Math. Comput., 2002, 71(238): 669-681.

[33] Schräder, D.: Analytically divergence-free discretization methods for Darcy’s problem. Ph.D. thesis, University of Sussex (2010)

[34] Teman, R.: Navier-Stokes Equations: Theory and Numerical Analysis. North-Holland Pub. Co., Amsterdam (1977)

[35] Trask N, Maxey M, Hu X. A compatible high-order meshless method for the Stokes equations with applications to suspension flows. J. Comput. Phys., 2018, 355: 310-326.

[36] Veiga LB, Lipnikov K, Manzini GThe Mimetic Finite Difference Method for Elliptic Problems2014SwitzerlandSpringer

[37] Ward JP. $L^{p}$ Bernstein inequalities and inverse theorems for RBF approximation on $\mathbb{R} ^{d}$. J. Approx. Theory, 2012, 164: 1577-1593.

[38] Wendland, H.: Scattered Data Approximation. In: Cambridge Monogr. Appl. Comput. Math. Cambridge University Press, Cambridge (2005)

[39] Wendland H. Divergence-free kernel methods for approximating the Stokes problem. SIAM J. Numer. Anal., 2009, 47: 3158-3179.

[40] Wenzel T. Sharp inverse statements for kernel interpolation. Math. Comput.2025

[41] Wilbrandt UStokes-Darcy Equations: Analytic and Numerical Analysis2019SwitzerlandSpringer

[42] Wloka JPartial Differential Equations1987CambridgeCambridge University Press

Rights & permissions

Shanghai University