Abstract
In this paper, we develop and analyze an efficient multigrid method to solve the finite element systems from elliptic obstacle problems on two dimensional adaptive meshes. Adaptive finite element methods (AFEMs) based on local mesh refinement are an important and efficient approach when the solution is non-smooth. An optimality theory on AFEM for linear elliptic equations can be found in [8]. To achieve optimal complexity, an efficient solver for the discretization is indispensable.
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Bibliography
L. Badea, X.-C. Tai, and J. Wang. Convergence rate analysis of a multiplicative schwarz method for variational inequalities. SIAM J. Numer. Anal., 41(3):1052–1073, 2003.
J.H. Bramble, J.E. Pasciak, and A.H. Schatz. The construction of preconditioners for elliptic problems by substructuring, I. J. Comput. Math., 47:103–134, 1986.
L. Chen, R.H. Nochetto, and J. Xu. Local multilevel methods on graded bisection grids for H 1 system. Submitted to J. Comput. Math., 2009.
L. Chen and C.-S. Zhang. A coarsening algorithm and multilevel methods on adaptive grids by newest vertex bisection. Submitted to J. Comput. Math., 2009.
C.W. Cryer. Successive overrelaxation methods for solving linear complementarity problems arising from free boundary problems. Proceedings of Intensive Seminary on Free Boundary Problems, Pavie, Ed. Magenes, 1979.
C. Graser and R. Kornhuber. Multigrid methods for obstacle problems. J. Comput. Math., 27(1):1–44, 2009.
R. Kornhuber. Adaptive Monotone Multigrid Methods for Nonlinear Variational Problems. Advances in Numerical Mathematics B. G. Teubner, Stuttgart, 1997.
R.H. Nochetto, K.G. Siebert, and A. Veeser. Theory of adaptive finite element methods: an introduction. In R.A. DeVore and A. Kunoth, editors, Multiscale, Nonlinear and Adaptive Approximation. Springer, Heidelberg, 2009.
K.G. Siebert and A. Veeser. A unilaterally constrained quadratic minimization with adaptive finite elements. SIAM J. Optim., 18(1):260–289, 2007.
X.-C. Tai. Rate of convergence for some constraint decomposition methods for nonlinear variational inequalities. Numer. Math., 93(4):755–786, 2003.
X.-C. Tai, B. Heimsund, and J. Xu. Rate of convergence for parallel subspace correction methods for nonlinear variational inequalities. In Domain Decomposition Methods in Science and Engineering (Lyon, 2000), Theory Eng. Appl. Comput. Methods, pp. 127–138. Internat. Center Numer. Methods Eng. (CIMNE), Barcelona, 2002.
X.-C. Tai and J. Xu. Global convergence of subspace correction methods for convex optimization problems. Math. Comput., 71(237):105–124, 2002.
J. Xu, L. Chen, and R.H. Nochetto. Optimal multilevel methods for H(grad), H(curl), and H(div) systems on adaptive and unstructured grids. In R.A. DeVore and A. Kunoth, editors, Multiscale, Nonlinear and Adaptive Approximation. Springer, Heidelberg, 2009.
J. Xu. Iterative methods by space decomposition and subspace correction. SIAM Rev., 34:581–613, 1992.
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Chen, L., Nochetto, R.H., Zhang, CS. (2011). Multigrid Methods for Elliptic Obstacle Problems on 2D Bisection Grids. In: Huang, Y., Kornhuber, R., Widlund, O., Xu, J. (eds) Domain Decomposition Methods in Science and Engineering XIX. Lecture Notes in Computational Science and Engineering, vol 78. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-11304-8_25
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DOI: https://doi.org/10.1007/978-3-642-11304-8_25
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