Abstract
A novel approach for implicit residual-type error estimation in meshfree methods is presented. This allows to compute upper and lower bounds of the error in energy norm with the ultimate goal of obtaining bounds for outputs of interest. The proposed approach precludes the main drawbacks of standard residual type estimators circumventing the need of flux-equilibration and resulting in a simple implementation that avoids integrals on edges/sides of a domain decomposition (mesh). This is especially interesting for mesh-free methods.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
M. Ainsworth and J.T. Oden, A posteriori error estimation in finite element analysis, John Wiley & Sons, Chichester, 2000.
R. Becker, Mesh adaptation for Dirichlet flow control via Nitsche's method, Commun. Numer. Methods Eng. 18 (2002), no. 9, 669–680.
J. Bonet, A. Huerta, and J. Peraire, The effcient computation of bounds for functionals of finite element solutions in large strain elasticity, Comput. Methods Appl. Mech. Engrg. 191 (2002), no. 43, 4807–4826.
C. Carstensen and S.A. Funken, Fully reliable localized error control in the FEM, SIAM J. Sci. Comput. 21 (1999/00), no. 4, 1465–1484.
H.J. Chung and T. Belytschko, An error estimate in the EFG method, Comput. Mech. 21 (1998), no. 2, 91–100.
P. Diez, N. Pares, and A. Huerta, Recovering lower bounds of the error by postprocessing implicit residual a posteriori error estimates, Internat. J. Numer. Methods Engrg. 56 (2003), no. 10, 1465–1488.
C.A. Duarte and J.T. Oden, An h-p adaptive method using clouds, Comput. Methods Appl. Mech. Eng. 139 (1996), no. 1–4, 237–262.
S. Fernandez-Mendez, J. Bonet, and A. Huerta, Continuous blending of SPH with finite elements, Comput. Struct. 83 (2005), no. 17–18, 1448–1458.
S. Fernandez-Mendez and A. Huerta, Imposing essential boundary conditions in mesh-free methods, Comput. Methods Appl. Mech. Eng. 193 (2004), no. 12–14, 1257–1275.
L. Gavete, J.L. Cuesta, and A. Ruiz, A procedure for approximation of the error in the EFG method, Int. J. Numer. Methods Eng. 53 (2002), 667–690.
M. Griebel and M.A. Schweitzer, A Particle-Partition of Unity Method. Part V: Boundary Conditions, Geometric Analysis and Nonlinear Partial Differential Equations, (S. Hildebrandt and H. Karcher, eds.), Springer, 2002, pp. 517–540.
A. Huerta and S. Fernandez-Mendez, Enrichment and coupling of the finite element and meshless methods, Int. J. Numer. Methods Eng. 48 (2000), no. 11, 1615–1636.
A. Huerta, S. Fernandez-Mendez, and W.K. Liu, A comparison of two formulations to blend finite elements and mesh-free methods, Comput. Methods Appl. Mech. Eng. 193 (2004), no. 12–14, 1105–1117.
P. Ladevèze and D. Leguillon, Error estimate procedure in the finite element method and applications, SIAM J. Numer. Anal. 20 (1983), no. 3, 485–509.
W.K. Liu, S. Jun, D.T. Sihling, Y. Chen, and W. Hao, Multiresolution reproducing kernel particle method for computational fluid dynamics, Int. J. Numer. Methods Fluids 24 (1997), 1391–1415.
L. Machiels, Y. Maday, and A.T. Patera, A “flux-free” nodal Neumann subproblem approach to output bounds for partial differential equations, C. R. Acad. Sci. Paris Sér. I Math. 330 (2000), no. 3, 249–254.
Y. Maday, A.T. Patera, and J. Peraire, A general formulation for a posteriori bounds for output functionals of partial differential equations; application to the eigenvalue problem, C. R. Acad. Sci. Paris Sér. I Math. 328 (1999), no. 9, 823–828.
P. Morin, R.H. Nochetto, and K.G. Siebert, Local problems on stars: a posteriori error estimators, convergence, and performance, Math. Comp. 72 (2003), no. 243, 1067–1097.
J. Nitsche, über ein variations zur lösung von dirichlet-problemen bei verwendung von teilräumen die keinen randbedingungen unterworfen sind, Abh. Math. Se. Univ. 36 (1970), 9–15.
J.T. Oden and S. Prudhomme, Goal-oriented error estimation and adaptivity for the finite element method, Comput. Math. Appl. 41 (2001), no. 5–6, 735–756.
M. Paraschivoiu, J. Peraire, and A.T. Patera, A posteriori finite element bounds for linear-functional outputs of elliptic partial differential equations, Comput. Methods Appl. Mech. Eng. 150 (1997), no. 1–4, 289–312.
N. Pares, J. Bonet, A. Huerta, and J. Peraire, The computation of bounds for linear-functional outputs of weak solutions to the two-dimensional elasticity equations, Comput. Methods Appl. Mech. Eng. 194 (2006), no. 4–6, 406–429.
N. Pares, P. Diez, and A. Huerta, Subdomain-based flux-free a posteriori error estimators, Comput. Methods Appl. Mech. Eng. 195 (2006), no. 4–6, 297–323.
A.T. Patera and J. Peraire, A general Lagrangian formulation for the computation of a posteriori finite element bounds, Error estimation and adaptive discretization methods in computational fluid dynamics, Lect. Notes Comput. Sci. Eng., vol. 25, Springer, Berlin, 2003, pp. 159–206.
S. Prudhomme and J.T. Oden, On goal-oriented error estimation for elliptic problems: application to the control of pointwise errors, Comput. Methods Appl. Mech. Eng. 176 (1999), no. 1–4, 313–331.
A.M. Sauer-Budge, J. Bonet, A. Huerta, and J. Peraire, Computing bounds for linear functionals of exact weak solutions to Poisson's equation, SIAM J. Numer. Anal. 42 (2004), no. 4, 1610–1630.
R. Stenberg, On some techniques for approximating boundary conditions in the finite element method, J. Comput. Appl. Math. 63 (1995), no. 1–3, 139–148.
G.J. Wagner and W.K. Liu, Application of essential boundary conditions in mesh-free methods: a corrected collocation method, Int. J. Numer. Methods Eng. 47 (2000), no. 8, 1367–1379.
Z.C. Xuan, N. Pares, and J. Peraire, Computing upper and lower bounds for the J-integral in two-dimensional linear elasticity, Comput. Methods Appl. Mech. Eng. 195 (2006), no. 4–6, 430–443.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2007 Springer
About this chapter
Cite this chapter
Vidal, Y., Huerta, A. (2007). Goal Oriented Error Estimation for the Element Free Galerkin Method. In: Griebel, M., Schweitzer, M.A. (eds) Meshfree Methods for Partial Differential Equations III. Lecture Notes in Computational Science and Engineering, vol 57. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-46222-4_16
Download citation
DOI: https://doi.org/10.1007/978-3-540-46222-4_16
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-46214-9
Online ISBN: 978-3-540-46222-4
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)