Ferretti, Roberto; Finzi Vita, Stefano On a variational approximation of superlinear indefinite elliptic problems. (English) Zbl 0957.65101 Numer. Funct. Anal. Optimization 19, No. 7-8, 759-771 (1998). The authors study a finite element discretisation of the semilinear elliptic Neumann problem \[ -\Delta u = a(x)u^p \quad \text{in }\Omega, \qquad \frac{\partial u}{\partial \nu} = 0 \quad \text{on }\partial\Omega, \qquad u > 0 \quad \text{in } \Omega,\tag{1} \] where \(a \in W^1_\infty(\Omega)\) and \(\Omega\) is a bounded regular open set of \(\mathbb R^N\) with \(1<p<(N+2)/(N-2)\) if \(N \geq 3\) or \(p>1\) if \(N=2\). It is known that there exists a solution \(u^*\) of (1) iff the function \(a\) satisfies the condition \[ \int_\Omega a(x) dx < 0 \;\text{ and} \;a(x) \;\text{ changes\;its\;sign\;in} \;\Omega. \] The existence of \(u^*\) relies basically on the solution of the problem \[ \max_{v \in S} \Biggl\{ I(v):= \int_\Omega a(x)|v(x)|^{p+1} dx \Biggr\}, \quad S:= \Biggl\{ v \in W^1_2(\Omega), \int_\Omega |\nabla v|^2 dx =1 \Biggr\}. \] This constrained maximisation problem is discretised by linear finite elements, where for \(I(v)\) a quadrature rule is used for a further discretisation. It is shown that the discretised problem has a solution \(u^*_h\). Assuming that these solutions satisfy for a sequence of triangulations with meshsize \(h\) tending to zero the a priori estimate \[ \|u^*_h \|_\infty \leq C < \infty \tag{2} \] the authors prove the weak convergence of a subsequence in \(W^1_2(\Omega)\) and the convergence of the maxima. The convergence proof relies on an abstract convergence theorem for constrained minimisation problems [see e.g. R. D. Grigorieff and R. Reemtsen, Numer. Funct. Anal. Optimization 11, No. 7/8, 701-719 (1990; Zbl 0743.65058)]. The condition (2) is shown to hold true in the case \(N=1\). For the case \(N=1\) and \(p=2\) the authors provide a numerical example. Reviewer: R.D.Grigorieff (Berlin) MSC: 65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs 35J65 Nonlinear boundary value problems for linear elliptic equations Keywords:semilinear elliptic Neumann problem; variational method; finite element approximation; convergence; numerical examples Citations:Zbl 0743.65058 PDFBibTeX XMLCite \textit{R. Ferretti} and \textit{S. Finzi Vita}, Numer. Funct. Anal. Optim. 19, No. 7--8, 759--771 (1998; Zbl 0957.65101) Full Text: DOI References: [1] DOI: 10.1007/BF01206962 · Zbl 0809.35022 · doi:10.1007/BF01206962 [2] DOI: 10.1007/BF01210623 · Zbl 0840.35035 · doi:10.1007/BF01210623 [3] Brezzi F., Num. Math. 2 pp 36– (1980) [4] DOI: 10.1016/0362-546X(93)90147-K · Zbl 0779.35032 · doi:10.1016/0362-546X(93)90147-K [5] Ciarlet, P. 1978. ”The Finite Element Method for elliptic problems”. Amsterdam: North-Holland. · Zbl 0383.65058 [6] Daniel J. W., The approximate minimization of junctionals (1971) [7] DOI: 10.1080/01630569008816398 · Zbl 0743.65058 · doi:10.1080/01630569008816398 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.