Multigrid methods for discrete elliptic problems on triangular surfaces
Tóm tắt
We construct and analyze multigrid methods for discretized self-adjoint elliptic problems on triangular surfaces in
$${\mathbb{R}^3}$$
. The methods involve the same weights for restriction and prolongation as in the case of planar triangulations and therefore are easy to implement. We prove logarithmic bounds of the convergence rates with constants solely depending on the ellipticity, the smoothers and on the regularity of the triangles forming the triangular surface. Our theoretical results are illustrated by numerical computations.
Tài liệu tham khảo
Aksoylu, B., Khodakovsky, A., Schröder, P.: Multilevel solvers for unstructured surface meshes. SIAM J. Sci. Comput. 26, 1146–1165 (2005)
Bank, R., Dupont, T., Yserentant, H.: The hierarchical basis multigrid method. Numer. Math. 52, 387–404 (1988)
Bornemann, F., Yserentant, H.: A basic norm equivalence in the theory of multilevel methods. Numer. Math. 64, 455–476 (1993)
Bramble, J.: Multigrid Methods Pitman Research Notes in Mathematics. Longman, Harlow (1993)
Dahmen, W., Schneider, R.: Wavelets on manifolds. I: Construction and domain decomposition. SIAM J. Numer. Anal. 31, 184–230 (1999)
Deckelnick, K., Dziuk, G., Elliott, C.: Computation of geometric partial differential equations and mean curvature flow. Acta Numer. 14, 139–232 (2005)
Deuflhard, P., Leinen, P., Yserentant, H.: Concepts of an adaptive hierarchical finite element code. IMPACT Comput. Sci. Eng. 1, 3–35 (1989)
Dziuk, G.: Finite elements for the Beltrami operator on arbitrary surfaces. In: Hildebrandt, S., Leis, R.(eds) Partial Differential Equations and Calculus of Variations, Lecture Notes in Mathematics, vol. 1357., pp. 483–490. Springer, Heidelberg (1988)
Hackbusch, W.: Multi-Grid Methods and Applications. Springer, Berlin (1985)
Hildebrandt, S., Karcher, H. (eds): Geometric Analysis and Nonlinear Partial Differential Equations. Springer, Heidelberg (2003)
Holst, M.: Adaptive numerical treatment of elliptic systems on manifolds. Adv. Comput. Math. 15, 139–191 (2001)
Kornhuber, R.: Adaptive Monotone Multigrid Methods for Nonlinear Variational Problems. Teubner, Stuttgart (1997)
Sapiro, G.: Geometric Partial Differential Equations and Image Analysis. Cambridge University Press, Cambridge (2006)
Xu, J.: Iterative methods by space decomposition and subspace correction. SIAM Rev. 34(4), 581–613 (1992)
Yserentant, H.: On the multi-level splitting of finite element spaces. Numer. Math. 49, 379–412 (1986)
Yserentant, H.: Two preconditioners based on the multilevel splitting of finite element spaces. Numer. Math. 58, 163–184 (1990)
Yserentant, H.: Old and new convergence proofs for multigrid methods. Acta Numer. 285–326 (1993)