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Finite element approximation of nonlinear elliptic problems with discontinuous coefficients. (English) Zbl 0712.65097

The authors’ abstract states:
“The paper presents a detailed theory of the finite element solution of second-order nonlinear elliptic equations with discontinuous coefficients in a general nonpolygonal domain \(\Omega\) with nonhomogeneous mixed Dirichlet-Neumann boundary conditions. In the discretization of the problem we proceed in the usual way: the domain \(\Omega\) is approximated by a polygonal one, conforming piecewise linear triangular elements are used and the integrals are evaluated by numerical quadratures. We prove the solvability of the discrete problem and study the convergence of the method both in strongly monotone and pseudomonotone cases under the only assumption that the exact solution \(u\in H^ 1(\Omega)\). Provided u is piecewise of class \(H^ 2\) and the problem is strongly monotone, we get the error estimate O(h).”
This paper draws heavily from three previous publications of the first author [Numer. Math. 50, 655-684 (1987; Zbl 0646.76085)] and of the first author and A. Zeníšek [ibid. 50, 451-475 (1987; Zbl 0637.65107) and ibid. 52, 147-163 (1987; Zbl 0642.65075)]. The authors consider that the problem coefficients are piecewise smooth and that the finite element triangulation respects the pieces in the sense that no subdomain boundary of the continuous problem crosses a triangle’s side. The proofs rest on estimates of behavior near internal and external boundaries which appeared in the earlier papers, with estimates over \(\Omega\) being sums over the subdomains. The authors carefully distinguish discretization errors from integration errors.
Reviewer: Myron Sussman

MSC:

65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
65N15 Error bounds for boundary value problems involving PDEs
35J65 Nonlinear boundary value problems for linear elliptic equations
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References:

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