We apply a lumped mass finite element to approximate Dirichlet problems for nonsmooth elliptic equations. It is proved that the lumped mass FEM approximation error in energy norm is the same as that of standard piecewise linear finite element approximation. Under the quasi-uniform mesh condition and the maximum angle condition, we show that the operator in the finite element problem is diagonally isotone and off-diagonally antitone. Therefore, some monotone convergent algorithms can be used. As an example, we prove that the nonsmooth Newton-like algorithm is convergent monotonically if Gauss-Seidel iteration is used to solve the Newton's equations iteratively. Some numerical experiments are presented. 1. Introduction In this paper, we consider a lumped mass finite element method (FEM) to the following Dirichlet problem for a nonsmooth elliptic equation: where is a bounded convex domain with a Lipschitz continuous boundary , is a constant, is a given smooth function, and . The above nonsmooth elliptic problem has many applications. For instance, it can arise from the MHD equilibria, thin stretched membranes problems, or reaction-diffusion problems (see, e.g., [1–4]). In order to solve problem (1), firstly, it is generally discretized by a finite element (volume) method or a finite difference method, and then various numerical algorithms are constructed to solve the corresponding discrete problems (see, e.g., [1, 4–8] and the references therein). In this paper, we apply a lumped mass finite element to approximate problem (1) by introducing the lumping domain for each node to deal with the nonsmooth term. We refer to [9–12] for such schemes of lumped mass type. For nonsmooth problem (1), one advantage of the lumped mass FEM is that one can calculate the nonsmooth terms in the finite element equations very easily and numerical quadrature algorithms are no longer needed. Another advantage of this method is that the operator in the finite element problem is diagonally isotone and off-diagonally antitone and thereby some monotone convergent algorithms can be applied (see, e.g., [13, 14]). In this paper, we will prove that the FEM error in energy norm is the same as that of the standard FEM. Since the finite element problem is a nonsmooth equation, we will apply nonsmooth Newton-like algorithm to solve it and focus our attention on the monotone convergence property of the algorithm for some special inner iterators. Throughout this paper, we adopt the standard notations for Sobolev spaces on with norm and seminorm . We denote by and and let be the subspace of
A. K. Aziz, A. B. Stephens, and M. Suri, “Numerical methods for reaction-diffusion problems with non-differentiable kinetics,” Numerische Mathematik, vol. 53, no. 1-2, pp. 1–11, 1988.
X. Chen, Z. Nashed, and L. Qi, “Smoothing methods and semismooth methods for nondifferentiable operator equations,” SIAM Journal on Numerical Analysis, vol. 38, no. 4, pp. 1200–1216, 2001.
F. Kikuchi, K. Nakazato, and T. Ushijima, “Finite element approximation of a nonlinear eigenvalue problem related to MHD equilibria,” Japan Journal of Applied Mathematics, vol. 1, no. 2, pp. 369–403, 1984.
L. Chang, W. Gong, and N. Yan, “Finite element method for a nonsmooth elliptic equation,” Frontiers of Mathematics in China, vol. 5, no. 2, pp. 191–209, 2010.
X. Chen, N. Matsunaga, and Y. Yamamoto, “Smoothing Newton methods for nonsmooth Dirichlet problems,” in Reformulation: Nonsmooth, Piecewise Smooth, Semismooth and Smoothing Methods, M. Fukushima and L. Qi, Eds., pp. 65–79, Kluwer Academic, Norwell, Mass, USA, 1998.
F. Kikuchi, “Finite element analysis of a nondifferentiable nonlinear problem related to MHD equilibria,” Journal of The Faculty of Science, The University of Tokyo IA, vol. 35, pp. 77–101, 1988.
X. Zhou, Y. Song, and L. Wang, “A two-step SOR-Newton method for nonsmooth equations,” Nonlinear Analysis, Theory, Methods and Applications, vol. 71, no. 10, pp. 4387–4395, 2009.
K. H. Hoffmann and J. Zou, “Parallel solution of variational inequality problems with nonlinear source terms,” IMA Journal of Numerical Analysis, vol. 16, no. 1, pp. 31–45, 1996.
K. Ishihara, “Monotone explicit iterations of the finite element approximations for the nonlinear boundary value problem,” Numerische Mathematik, vol. 43, no. 3, pp. 419–437, 1984.
O. Pironneau and M. Tabata, “Stability and convergence of a Galerkin-characteristics finite element scheme of lumped mass type,” International Journal for Numerical Methods in Fluids, vol. 64, no. 10–12, pp. 1240–1253, 2010.
J. Zeng and S. Zhou, “Schwarz algorithm for the solution of variational inequalities with nonlinear source terms,” Applied Mathematics and Computation, vol. 97, no. 1, pp. 23–35, 1998.
M. Ulbrich, Nonsmooth Newton-like methods for variational inequalities and constrained optimization problems in function spaces [Ph.D. thesis], Technische Universit？t München, Fakult？t für Mathematik, München, Germany, 2002.
Y. Jiang and J. Zeng, “An -error estimate for finite element solution of nonlinear elliptic problem with a source term,” Applied Mathematics and Computation, vol. 174, no. 1, pp. 134–149, 2006.
J. Martínez and L. Qi, “Inexact Newton methods for solving nonsmooth equations,” Journal of Computational and Applied Mathematics, vol. 60, no. 1-2, pp. 127–145, 1995.
P. N. Brown, P. S. Vassilevski, and C. S. Woodwar, “On mesh-independent convergence of an inexact Newton-multigrid algorithm,” SIAM Journal on Scientific Computing, vol. 25, no. 2, pp. 570–590, 2003.
J. Xu, “An introduction to multigrid convergence theory,” in Iterative Methods in Scientific Computing, R. Chan, T. Chan, and G. Golub, Eds., pp. 169–241, Springer, Singapore, 1997.