An Adaptive Finite Element Method for the Infinity Laplacian
We construct a finite element method (FEM) for the infinity Laplacian. Solutions of this problem are well known to be singular in nature so we have taken the opportunity to conduct an a posteriori analysis of the method deriving residual based estimators to drive an adaptive algorithm. It is numerically shown that optimal convergence rates are regained using the adaptive procedure.
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- 1.G. Aronsson, Construction of singular solutions to the p-harmonic equation and its limit equation for p = ∞. Manuscr. Math. 56(2), 135–158 (1986)Google Scholar
- 2.G. Awanou, Pseudo time continuation and time marching methods for Monge–Ampère type equations (2012). In revision – tech report available on http://www.math.niu.edu/~awanou/
- 4.L.A. Caffarelli, X. Cabré, Fully Nonlinear Elliptic Equations. American Mathematical Society Colloquium Publications, vol. 43 (American Mathematical Society, Providence, 1995)Google Scholar
- 5.M.G. Crandall, H. Ishii, P.-L. Lions, User’s guide to viscosity solutions of second order partial differential equations. Bull. Am. Math. Soc. (N.S.) 27(1), 1–67 (1992)Google Scholar
- 7.A. Dedner, T. Pryer, Discontinuous Galerkin methods for nonvariational problems (2013). Submitted – tech report available on ArXiV http://arxiv.org/abs/1304.2265
- 8.S. Esedoglu, A.M. Oberman, Fast semi-implicit solvers for the infinity laplace and p-laplace equations, Arxiv, 2011Google Scholar
- 14.A.M. Oberman, A convergent difference scheme for the infinity Laplacian: construction of absolutely minimizing Lipschitz extensions. Math. Comput. 74(251), 1217–1230 (2005). (Electronic)Google Scholar