Abstract
The paper provides a computational technique that allows to compare all linear methods for PDE solving that use the same input data. This is done by writing them as linear recovery formulas for solution values as linear combinations of the input data, and these formulas are continuous linear functionals on Sobolev spaces. Calculating the norm of these functionals on a fixed Sobolev space will then serve as a quality criterion that allows a fair comparison of all linear methods with the same inputs, including standard, extended or generalized finite–element methods, finite–difference– and meshless local Petrov–Galerkin techniques. The error bound is computable and yields a sharp worst–case bound in the form of a percentage of the Sobolev norm of the true solution. In this sense, the paper replaces proven error bounds by calculated error bounds. A number of illustrative examples is provided. As a byproduct, it turns out that a unique error–optimal method exists. It necessarily outperforms any other competing technique using the same data, e.g. those just mentioned, and it is necessarily meshless, if solutions are written “entirely in terms of nodes” (Belytschko et. al. Comput. Methods Appl. Mech. Eng., Spec. issue, 139, 3–47, 1996). On closer inspection, it turns out that it coincides with symmetric meshless collocation carried out with the kernel of the Hilbert space used for error evaluation, e.g. with the kernel of the Sobolev space used. This technique is around since at least 1998, but its optimality properties went unnoticed, so far. Examples compare the optimal method with several others listed above.
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References
Atluri, S.N.: The Meshless Method (MLPG) for Domain and BIE Discretizations. Tech Science Press, Encino (2005)
Atluri, S.N., Zhu, T.-L.: A new meshless local Petrov-Galerkin (MLPG) approach in computational mechanics. Comput. Mech. 22, 117–127 (1998)
Babuska, I., Banerjee, U., Osborn, J.E., Zhang, Q.: Effect of numerical integration on meshless methods. Comput. Mech. Appl. Mech. Eng. 198, 27–40 (2009)
Beatson, R.K., Cherrie, J.B., Mouat, C.T.: Fast fitting of radial basis functions: methods based on preconditioned GMRES iteration. Adv. Comput. Math. 11, 253–270 (1999)
Belytschko, T., Krongauz, Y., Organ, D.J., Fleming, M., Krysl, P.: Meshless methods: an overview and recent developments. Comput. Methods Appl. Mech. Eng., Spec. issue 139, 3–47 (1996)
Brown, D., Ling, L., Kansa, E.J., Levesley, J.: On approximate cardinal preconditioning methods for solving PDEs with radial basis functions. Eng. Anal. Bound. Elem. 19, 343–353 (2005)
Buhmann, M.D.: Radial Basis Functions, Theory and Implementations. Cambridge University Press (2003)
Davydov, O., Schaback, R.: Error Bounds for Kernel-Based Numerical Differentiation. Draft (2013)
Marchi, Stefano De, Schaback, R., Wendland, H.: Near-optimal data-independent point locations for radial basis function interpolation. Adv. Comput. Math. 23(3), 317–330 (2005)
Fasshauer, G.F.: Meshfree Approximation Methods with MATLAB, volume 6 of Interdisciplinary Mathematical Sciences. World Scientific Publishers, Singapore (2007)
Franke, C., Schaback, R.: Convergence order estimates of meshless collocation methods using radial basis functions. Adv. Comput. Math. 8, 381–399 (1998)
Franke, C., Schaback, R.: Solving partial differential equations by collocation using radial basis functions. Appl. Math. Comp. 93, 73–82 (1998)
Hon, Y.C., Schaback, R.: On unsymmetric collocation by radial basis functions. J. Appl. Math. Comp. 119, 177–186 (2001)
Jost, J.: Partial Differential Equations, volume 214 of Graduate Texts in Mathematics. Springer-Verlag, New York (2002). Translated and revised from the 1998 German original by the author
Kansa, E.J.: Application of Hardy’s multiquadric interpolation to hydrodynamics. In: Proceedings of 1986 Simulation Conference, vol. 4, pp. 111–117 (1986)
Kansa, E.J.: Multiquadrics - a scattered data approximation scheme with applications to computational fluid-dynamics - I: Surface approximation and partial dervative estimates. Comput. Math. Appl. 19, 127–145 (1990)
Lions, J.L., Magenes, E.: Problèmes aux limites non homogènes at applications vol. 1. Travaux et recherches mathématiques. Dunod (1968)
D. Mirzaei, R. Schaback: Direct Meshless Local Petrov-Galerkin (DMLPG) method: a generalized MLS approximation. Appl. Numer. Math. (2013). doi:10.1016/j.apnum.2013.01.002
Mirzaei, D., Schaback, R., Dehghan, M.: On generalized moving least squares and diffuse derivatives. IMA J. Numer. Anal. 32(3), 983–1000 (2012). doi:10.1093/imanum/drr030.
Narcowich, F.J., Ward, J.D., Wendland, H.: Sobolev bounds on functions with scattered zeros, with applications to radial basis function surface fitting. Math. Comput. 74, 743–763 (2005)
Partridge, P.W., Brebbia, C.A., Wrobel, L.C.: The dual reciprocity boundary element method. CMP/Elsevier (1992)
Šarler, B.: From global to local radial basis function collocation method for transport phenomena. In: Advances in meshfree techniques, Computer Methods Application Science, vol. 5, pp. 257–282. Springer, Dordrecht (2007)
Schaback, R.: Approximation by radial basis functions with finitely many centers. Constr. Approx. 12, 331–340 (1996)
Schaback, R.: Reconstruction of multivariate functions from scattered data (1997). Manuscript, available via http://www.num.math.uni-goettingen.de/schaback/research/group.html
Schaback, R.: Unsymmetric meshless methods for operator equations. Numer. Math. 114, 629–651 (2010)
Schaback, R.: Kernel–based meshless methods. Lecture Note, Göttingen (2011). http://num.math.uni-goettingen.de/schaback/teaching/AV2.pdf
Schaback, R.: Direct discretizations with applications to meshless methods for PDEs. Dolomites Res. Notes Approx. Proc. DWCAA12 6, 37–51 (2013)
Schaback, R., Wendland, H.: Adaptive greedy techniques for approximate solution of large RBF systems. Numer. Algoritm. 24(3), 239–254 (2000)
Shen, Q.: Local RBF-based differential quadrature collocation method for the boundary layer problems. Eng. Anal. Bound. Elem. 34(3), 213–228 (2010)
Shu, C., Ding, H., Yeo, K. S.: Computation of incompressible Navier-Stokes equations by local RBF-based differential quadrature method. CMES Comput. Model. Eng. Sci. 7(2), 195–205 (2005)
Stevens, D., Power, H., Lees, M., Morvan, H.: The use of PDE centres in the local RBF Hermitian method for 3D convective-diffusion problems. J. Comput. Phys. 228(12), 4606–4624 (2009)
Vertnik, R., Šarler, B.: Local collocation approach for solving turbulent combined forced and natural convection problems. Adv. Appl. Math. Mech. 3(3), 259–279 (2011)
Wendland, H.: Scattered Data Approximation. Cambridge University Press (2005)
Wu, Z.: Convergence of interpolation by radial basis functions. Chinese Ann. Math. Ser. A 14, 480–486 (1993)
Yao, G.M., Šarler, B., Chen, C.S.: A comparison of three explicit local meshless methods using radial basis functions. Eng. Anal. Bund. Elem. 35(3), 600–609 (2011)
Yao, G.M., Siraj ul Islam, Šarler, B.: A comparative study of global and local meshless methods for diffusion-reaction equation. CMES Comput. Model. Eng. Sci. 59(2), 127–154 (2010)
Yao, G.M., Siraj ul Islam, Šarler, B.: Assessment of global and local meshless methods based on collocation with radial basis functions for parabolic partial differential equations in three dimensions. Eng. Anal. Bund. Elem. 36(11), 1640–1648 (2012)
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Communicated by: Leslie Greengard
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Schaback, R. A computational tool for comparing all linear PDE solvers. Adv Comput Math 41, 333–355 (2015). https://doi.org/10.1007/s10444-014-9360-5
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DOI: https://doi.org/10.1007/s10444-014-9360-5
Keywords
- Sobolev spaces
- Finite elements
- Meshless methods
- Finite differences
- Error bounds
- Radial basis functions
- Kernels
- Partial differential equations
- Discretization