Abstract
In this paper, we study the finite element method for a non-smooth elliptic equation. Error analysis is presented, including a priori and a posteriori error estimates as well as superconvergence analysis. We also propose two algorithms for solving the underlying equation. Numerical experiments are employed to confirm our error estimations and the efficiency of our algorithms.
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Aziz A K, Stephens A B, Suri M. Numerical methods for reaction-diffusion problems with non-differentiable kinetics. Numer Math, 1988, 51: 1–11
Bergounioux M, Ito K, Kunisch K. Primal-dual strategy for constrained optimal control problems. SIAM J Control Optim, 1999, 37: 1176–1194
Chen C. Finite Element Method and High Accuracy Analysis. Changsha: Hunan Science Press, 1982 (in Chinese)
Chen X. First order conditions for discretized nonsmooth constrained optimal control problems. SIAM J Control Optim, 2004, 42: 2004–2015
Chen X, Nashed Z, Qi L. Smoothing methods and semismooth methods for nondifferentiable operator equations. SIAM J Numer Anal, 2000, 38: 1200–1216
Ciarlet P G. The Finite Element Methods for Elliptic Problems. Amsterdam: North Holland, 1978
Clément Ph. Approximation by finite element functions using local regularization. RAIRO Anal Numer, 1975, 9: 77–84
Hinze M. A variational discretization concept in control constrained optimization: the linear-quadratic case. J Computational Optimization and Applications, 2005, 30: 45–63
Kikuchi F. Finite element analysis of a nondifferentiable nonlinear problem related to MHD equilibria. J Fac Sci Univ Tokyo, Sect IA Math, 1988, 35: 77–101
Kikuchi F, Nakazato K, Ushijima T. Finite element approximation of a nonlinear eigenvalue problem related to MHD equilibria. Japan J Appl Math, 1984, 1: 369–403
Křížek M. Superconvergence results for linear triangular elements. Lecture Notes in Mathematics, 1986, 1192: 315–320
Lin Q, Yan N. Structure and Analysis for Efficient Finite Element Methods. Baoding: Publishers of Hebei University, 1996 (in Chinese)
Liu W B, Yan N. Adaptive Finite Element Methods for Optimal Control Governed by PDEs. Beijing: Science Press, 2008
Scott L R, Zhang S. Finite element interpolation of nonsmooth functions satisfying boundary conditions. Math Comp, 1990, 54: 483–493
Oganesyan L A, Rukhovetz L A. Study of the rate of convergence of variational difference scheme for second order elliptic equations in two-dimensional field with a smooth boundary. USSR Comput Math Math Phy, 1969, 9: 158–183
Verfürth R. A Review of A Posteriori Error Estimation and Adaptive Mesh Refinement. London: Wiley-Teubner, 1996
Yan N. Superconvergence analysis and a posteriori error estimation in finite element method. Beijing: Science Press, 2008
Zhu Q, Lin Q. Superconvergence Theory of Finite Element Methods. Changsha: Hunan Science Press, 1989 (in Chinese)
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Chang, L., Gong, W. & Yan, N. Finite element method for a nonsmooth elliptic equation. Front. Math. China 5, 191–209 (2010). https://doi.org/10.1007/s11464-010-0001-0
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DOI: https://doi.org/10.1007/s11464-010-0001-0
Keywords
- Finite element method
- nonsmooth elliptic equation
- a priori error estimate
- a posteriori error estimate
- superconvergence analysis
- active set method