Abstract
We present a new technology for generating meshes minimizing the interpolation and discretization errors or their gradients. The key element of this methodology is construction of a space metric from edge-based error estimates. For a mesh with N h triangles, the error is proportional to \(N_h^{-1}\) and the gradient of error is proportional to \(N_h^{-1/2}\) which are the optimal asymptotics. The methodology is verified with numerical experiments.
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Agouzal, A., Vassilevski, Y.: Minimization of gradient errors of piecewise linear interpolation on simplicial meshes. Submitted to SIAM J. Numer. Anal. (2008)
Agouzal, A., Lipnikov, K., Vassilevski, Y.: Adaptive generation of quasi-optimal tetrahedral meshes. East-West J. Numer. Math. 7, 223–244 (1999)
Agouzal, A., Lipnikov, K., Vassilevski, Y.: Generation of quasi-optimal meshes based on a posteriori error estimates. In: Brewer, M., Marcum, D. (eds.) Proceedings of 16th International Meshing Roundtable, pp. 139–148. Springer, Heidelberg (2007)
Agouzal, A., Lipnikov, K., Vassilevski, Y.: Hessian-free metric-based mesh adaptation via geometry of interpolation error. Comp. Math. Math. Phys. 49(11) (to appear, 2009)
Ait-Ali-Yahia, D., Habashi, W.G., Tam, A., Vallet, M.-G., Fortin, M.: Directionally adaptive methodology using an edge-based error estimate on quadrilateral grids. Inter. J. Numer. Meth. Fluids 23(7), 673–690 (1996)
Bono, G., Awruch, A.M.: An adaptive mesh strategy for high compressible flows based on nodal re-allocation. J. Brazilian Soc. Mech. Sci. Engr. 30(3), 189–196 (2008)
Buscaglia, G.C., Dari, E.A.: Anisotropic mesh optimization and its application in adaptivity. Inter. J. Numer. Meth. Engrg. 40, 4119–4136 (1997)
Chen, L., Sun, P., Xu, J.: Optimal anisotropic meshes for minimizing interpolation errors in Lp-norm. Mathematics of Computation 76, 179–204 (2007)
Ciarlet, P., Wagschal, C.: Multipoint Taylor formulas and applications to the finite element method. Numer. Math. 17, 84–100 (1971)
Ciarlet, P.: The finite element method for elliptic problems. North-Holland, Amsterdam (1978)
D’Azevedo, E.: Optimal triangular mesh generation by coordinate transformation. SIAM J. Sci. Stat. Comput. 12, 755–786 (1991)
Deuflhard, P., Leinen, P., Yserentant, H.: Concepts of an adaptive hierarchical finite element code. IMPACT 1, 3–35 (1989)
Frey, P.J., Alauzet, F.: Anisotropic mesh adaptation for CFD computations. Comput. Meth. Appl. Mech. Eng. 194, 5068–5082 (2005)
Huang, W.: Metric tensors for anisotropic mesh generation. J. Comp. Physics. 204, 633–665 (2005)
Lipnikov, K., Vassilevski, Y.: Parallel adaptive solution of 3D boundary value problems by Hessian recovery. Comput. Meth. Appl. Mech. Eng. 192, 1495–1513 (2003)
Tichomirov, V.M.: The widths of sets in functional spaces and theory of optimal approximations. Uspehi Matem. Nauk 15(3), 81–120 (1960)
Vassilevski, Y., Lipnikov, K.: Adaptive algorithm for generation of quasi-optimal meshes. Comp. Math. Math. Phys. 39, 1532–1551 (1999)
Vassilevski, Y., Agouzal, A.: An unified asymptotic analysis of interpolation errors for optimal meshes. Doklady Mathematics 72, 879–882 (2005)
Zhu, J.Z., Zienkiewicz, O.C.: Superconvergence recovery technique and a posteriori error estimators. Inter. J. Numer. Meth. Engrg. 30, 1321–1339 (1990)
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Agouzal, A., Lipnikov, K., Vassilevski, Y. (2009). Anisotropic Mesh Adaptation for Solution of Finite Element Problems Using Hierarchical Edge-Based Error Estimates. In: Clark, B.W. (eds) Proceedings of the 18th International Meshing Roundtable. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-04319-2_34
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DOI: https://doi.org/10.1007/978-3-642-04319-2_34
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