On particles and splines in bounded domains

Kirchhart, Matthias

ESAIM: Mathematical Modelling and Numerical Analysis , Tome 55 (2021), pp. S301-S321

We propose numerical schemes that enable the application of particle methods for advection problems in general bounded domains. These schemes combine particle fields with Cartesian tensor product splines and a fictitious domain approach. Their implementation only requires a fitted mesh of the domain’s boundary, and not the domain itself, where an unfitted Cartesian grid is used. We establish the stability and consistency of these schemes in W$$-norms, s ∈ ℝ, 1 ≤ p ≤ ∞.

Classification : 65M75, 65M85, 65D07, 65D32
Keywords: Particle methods, splines, fictitious domains, ghost penalty

R. A. Adams and J. J. F. Fournier, Sobolev Spaces, 2nd edition. In: Vol. 140 of Pure and Applied Mathematics. Elsevier (2003). | MR | Zbl

J. Bergh and J. Löfström, Interpolation Spaces. An Introduction. In: Vol. 223 of Grundlehren der mathematischen Wissenschaften. Springer (1976). | MR | Zbl

S. C. Brenner and L. R. Scott, The Mathematical Theory of Finite Element Methods, 3rd edition. In: Vol. 15 of Texts in Applied Mathematics. Springer (2008). | MR | Zbl

E. Burman, La pénalisation fantôme. C.R. Math. 348 (2010) 1217–1220. | Zbl

J.-P. Chehab, A. Cohen, D. Jennequin, J. Nieto, C. Roland and J. Roche, An adaptive Particle-In-Cell method using multi-resolution analysis, edited by S. Cordier, T. Goudon, M. Gutnic and E. Sonnendrücker. In: Vol. 7 of Numerical Methods for Hyperbolic and Kinetic Problems. IRMA Lectures in Mathematics and Theoretical Physics. European Mathematical Society (2005) 29–42. | MR | Zbl | DOI

A. Cohen and B. Perthame, Optimal approximations of transport equations by particle and pseudoparticle methods. SIAM J. Math. Anal. 32 (2000) 616–636. | MR | Zbl | DOI

G.-H. Cottet, A new approach for the analysis of vortex methods in two and three dimensions. Ann. Inst. Henri Poincaré. Anal. nonlin. 5 (1988) 227–285. | MR | Zbl | Numdam | DOI

G.-H. Cottet and P. D. Koumoutsakos, Vortex Methods. Cambridge University Press (2000). | MR | DOI

M. Crouzeix and V. Thomée, The stability in $L_p$ and $W{}_p^1$ of the $L_2$-projection onto finite element function spaces. Math. Comput. 48 (1987) 521–532. | MR | Zbl

W. Dahmen, R. A. De Vore and K. Scherer, Multi-dimensional spline approximation. SIAM J. Numer. Anal. 17 (1980) 380–402. | MR | Zbl | DOI

P. J. Davis, A construction of nonnegative approximate quadratures. Math. Comput. 21 (1967) 578–582. | MR | Zbl | DOI

R. A. De Vore and V. A. Popov, Interpolation of Besov spaces. Trans. Am. Math. Soc. 305 (1988) 397–414. | MR | Zbl | DOI

R. A. De Vore and R. C. Sharpley, Besov spaces on domains in ${ℝ}^d$. Trans. Am. Math. Soc. 335 (1993) 843–864. | MR | Zbl

J. Douglas Jr, T. Dupont and L. Wahlbin, The stability in $L^q$ of the $L^2$-projection into finite element function spaces. Numer. Math. 23 (1974) 193–197. | MR | Zbl | DOI

S. Duczek and U. Gabbert, Efficient integration method for fictitious domain approaches. Comput. Mech. 56 (2015) 725–738. | MR | Zbl | DOI

M. W. Evans and F. H. Harlow, The Particle-in-Cell Method for Hydrodynamic Calculations. Los Alamos National Lab NM (1957).

R. Fletcher, Practical Methods of Optimization, 2nd edition. Wiley 7 (2000). | MR

O. H. Hald, Convergence of vortex methods for Euler’s equations. II. SIAM J Numer. Anal. 16 (1979) 726–755. | MR | Zbl | DOI

M. Kirchhart and S. Obi, A smooth partition of unity finite element method for vortex particle regularization. SIAM J. Sci. Comput. 39 (2017) A2345–A2364. | MR | Zbl | DOI

C. Lehrenfeld, A higher order isoparametric fictitious domain method for level set domains, edited by S. P. A. Bordas, E. Burman, M. G. Larson and M. A. Olshanskii. In: Vol. 121 of Geometrically Unfitted Finite Element Methods and Applications. Lecture Notes in Computational Science and Engineering. Springer (2017) 65–92. | MR | Zbl

Y. Marichal, P. Chatelain and G. Winckelmans, Immersed interface interpolation schemes for particle–mesh methods. J. Comput. Phys. 326 (2016) 947–972. | MR | Zbl | DOI

A. Massing, M. G. Larson, A. Logg and M. E. Rognes, A stabilized Nitsche fictitious domain method for the Stokes problem. J. Sci. Comput. 61 (2014) 604–628. | MR | Zbl | DOI

P.-A. Raviart, An analysis of particle methods, edited by F. Brezzi. In: Vol. 1127 of Numerical Methods in Fluid Dynamics. Lecture Notes in Mathematics. Springer (1985) 243–324. | MR | Zbl | DOI

L. Rosenhead, The formation of vortices from a surface of discontinuity. Proc. R. Soc. London 142 (1931) 170–192. | Zbl

L. L. Schumaker, Spline Functions. Basic Theory, 3rd edition. Cambridge University Press (2007). | MR | Zbl

B. Seibold, Minimal positive stencils in meshfree finite difference methods for the poisson equation. Comput. Methods Appl. Mech. Eng. 198 (2008) 592–601. | MR | Zbl | DOI

S. L. Sobolev and V. L. Vaskevich, The Theory of Cubature Formulas, 1st edition. In: Vol. 415 of Mathematics and Its Appl. Springer (1997). | MR | Zbl

E. M. Stein, Singular Integrals and Differentiability Properties of Functions. In: Vol. 30 of Princeton Mathematical Series. Princeton University Press (1970). | MR | Zbl

V. Tchakaloff, Formules de cubatures mécaniques a coefficients non négatifs. Bull. Sci. Math. 81 (1957) 123–134. | MR | Zbl

M. W. Wilson, A general algorithm for nonnegative quadrature formulas. Math. Comput. 23 (1969) 253–258. | MR | Zbl | DOI

R. Yokota, L. A. Barba, T. Narumi and K. Yasuoka, Petascale turbulence simulation using a highly parallel fast multipole method on GPUs. Comput. Phys. Commun. 184 (2013) 445–455. | MR | DOI