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Numerical Solutions to Nonsmooth Dirichlet Problems Based on Lumped Mass Finite Element Discretization

DOI: 10.1155/2014/549305

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Abstract:

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

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