Non-Overlapping Domain Decomposition Algorithm for the Hermite Radial Basis Function Meshless Collocation Approach: Applications to Convection Diffusion Problems
Free accessResearch articleFirst published online January, 2007
Non-Overlapping Domain Decomposition Algorithm for the Hermite Radial Basis Function Meshless Collocation Approach: Applications to Convection Diffusion Problems
In the particular case of solving large-scale boundary value problems, the computational cost derived as a result of the application of any numerical scheme represent a determinant factor in the determination of its computational efficiency. The present work studies the influence of the non-overlapping domain decomposition and higher order interpolation functions on the Hermite radial basis collocation method, as a way to improve its efficiency under high demanding numerical conditions. A series of high Péclet convection diffusion and multi-zone problems are tested. Comparison between the numerical results and its analytical solution is carried out for each of them.
References
1.
AtluriS.N. and ZhuT., New meshless local Petrov-Galerkin (MLPG) approach in computational mechanics, Comput. Mech., Vol. 22, No. 2, pp. 117–127, (1998)
2.
AtluriS.N. and ZhuT., New concepts in meshless methods. International Journal for Numerical Methods in Engineering, 47, pp. 537–556, (2000).
3.
KansaE.J., Multiquadrics- A scattared data approximation scheme with applications to computation fluid-dynamics-I: Surface approximations and partial derivatives estimates; Computers Math. Applic.19, pp 127–145, (1990)
4.
KansaE.J., Multiquadrics- A scattared data approximation scheme with applications to computation fluid-dynamics-II: Solution to parabolic, hyperbolic and elliptic partial differential equations; Computers Math. Applic.19, pp. 147–161, (1990)
5.
DubalM.R.Domain decomposition and local refinemet for multiquadric approximations. I: Second-order equations in one-dimension, Journal of Applied Science, 1, No.1, pp. 146–171 (1994).
6.
HonY. C. and MaoX.Z.. An efficient numerical scheme for Burgers' equations, Appl. Math. and Comp.95, pp. 37–50 (1998).
7.
ZerroukatM.PowerH. and ChenC.S., A numerical method for heat trasfer problems using collocation and radial basis functions, Int. J. Numer. Meth. Engng, 42, pp. 1263–1279, (1998).
8.
DubalM.R.OliveraS.R. and MatznerR.A.In Approaches to Numerical Relativity, Editors: InvernoR.d, Cambridge University Press, Cambridge UK, (1993).
9.
KansaE.J. and HonY.C., Circumventing the ill conditioning problem with multiquadric radial basis functions: Applications to elliptic partial differential equations, Advances Computational Mathematics, 39, pp. 123–137, (2000).
10.
WuY.L. and ShuC., Development of RBF-DQ method for derivative approximation and its application to simulate natural convection in concentric annuli, Computational Mechanics, 29, pp. 477–485, (2002).
11.
Mai-DuyN. and Tran-CongT., Numerical solution of Navier-Stokes equations using multiquadric radial basis function networks, International journal for numerical methods in fluids, 37, pp. 65–86, (2001).
12.
WuZ. and HonY.C., Convergence error estimate in solving free boundary diffusion problem by radial basis functions method, Engineering Analysis with Boundary Element, 27, pp. 73–79, (2003).
13.
LingLOpferR. and SchabackR.Results on meshless collocation techniques, Engineering Analysis with Boundary Elements, Vol. 30, No.4, pp. 247–253 (2006)
14.
FasshauerG.E.Solving Partial Differential Equations by Collocation with Radial Basis functions, Proceedings of Chamonix, Editors: Le MéchautéA.RabutC. and SchumakerL.L., 1–8, Vanderbilt University Press, Nashville, TN (1996).
15.
WuZ., Hermite-Birkhoff interpolation of scatterd data by radial basis functions; Approx. Theory, 8:2, pp. 1–11 (1992).
16.
WuZ., Solving PDE with radial basis function and the error estimation; Advances in Computational Mathematics, Lecture Notes on Pure and Applied Mathematics, 202, Editors: ChenZ.LiY.MicchelliC.A.XuY. and DekkerM., GuangZhou (1998).
17.
SchabackR and FrankeC., ‘Covergence order estimates of meshless collocation methods using radial basis functions’, Advances Computational Mathematics, 8, Issue 4, 381–399, (1998).
18.
JumarhonB.AminiS. and ChenK., The Hermite collocation method using radial basis functions, Engineering analysis with boundary elements, 24, pp. 607–611, (2000).
19.
LeitaoV.M.A., A meshless method for Kirchoff plate bending problems, International journal for numerical methods in engineering, 52, pp. 1107–1130, (2001).
20.
PowerH. and BarracoV., A comparison analysis between unsymmethc and symmetric radial basis function collocation methods for the numerical solution of partial differential equations, Computers and mathematics, 43, pp. 551–583, (2002).
21.
LiJ. and ChenC.S., Some observations on unsymmetric radial basis function collocation methods for convection-diffusion problems, International journal for numerical methods in engineering, 57, pp. 1085–1094, (2003).
22.
GolbergM.A. and ChenC.S., Discrete projection methods for integral equations, Computation Mechanics Publications, Southampton, UK, (1997).
23.
DuchonJ., ‘Spline minimizing rotation – invariant seminorms in Sobolev spaces’, in Constractive Theory of Functions of Several Variables, Lecture Notes in Mathematics, eds. SchemppW. and ZellerK., Springer-Verlag, Berlin, 571, pp. 85–100 (1977).
24.
PowellM. J. D., ‘The uniform convergence of thin plate spline interpolation in two dimensions’, Numerische Mathematik68/1, pp. 107–128 (1994).
25.
MadychW. R. and NelsonS. A., ‘Multivariable interpolation and conditionally positive definite functions’, Mathematic and Computing, 54, pp. 211–230 (1990).
26.
MicchelliC. A., ‘Interpolation of scattered data: Distance matrices and conditionally positive definite functions’, Constr. Approx.2, pp. 11–22 (1986).
27.
SchabackR., ‘Multivariate interpolation and approximation by translates of a basis function’, Approximation Theory VIII, ChuiCharles K. and SchumakerLarry L. (Editors), 1–8, (1995).
28.
FedoseyevAIFriedmannMJKansaEJ. Improved multiquadratic method for elliptic partial differential equation via PDE collocation on the boundary. Comput. Math. Appl.2002, 43, pp. 439–455
29.
SchabackR. & FrankeC.1998, ‘Covergence order estimates of meshless collocation methods using radial basis functions’, Advances Computational Mathematics, vol.8, no.4, pp. 381–399.
30.
ChanT.F.GlowinskiR. & PériauxJ.1989, Proceedings of the second international symposium on domain decomposition methods, SIAM, Philadelphia, PA, USA.
31.
GarbeyM. & KaperH.G.1997, ‘Heterogeneous domain decomposition for singularity perturbed elliptic boundary value problems’, Journal of Numerical Analysis, vol. 34, pp. 1513–1544.
32.
GlowinskiR.PériauxJ.ShiZ.C.1996, Proceedings of the eighth international symposium on domain decomposition methods for partial differential equations, Wiley, Chichester, UK.
33.
GriebelM. & SchweitzerM.A.2000, ‘A particle-partition of unity method for the solution of elliptic, parabolic, and hyperbolic PDEs’, Journal of Science and Computing, vol. 22, pp. 853–890.
34.
KansaE.J. & Carlson1992, ‘Improved accuracy of multiquadric interpolation using variable shape parameters’, Computers & Mathematic with Application, vol. 24, pp. 99–120.
35.
CarslawH.S. and JaegerJ.C., Conduction of heat in solids, Oxford at the Clarendon press, Oxford, (1959).
