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Zbl 0761.65079
Barrett, J.W.
Finite element approximation of a non-Lipschitz nonlinear eigenvalue problem. (English)
[J] RAIRO, Modélisation Math. Anal. Numér. 26, No.5, 627-656 (1992). ISSN 0764-583X

The author considers the semilinear elliptic eigenvalue problem $-\Delta u=\lambda f(u)$ on $\Omega\subset\bbfR\sp 2$, $u=0$ on $\partial\Omega$, where $f(t)=t\sp p$, $t\ge 0$, and $f(t)=0$ otherwise, $p\in (0,1)$. If $\Omega\sp h$ is a triangulation, with triangles of diameter $\le h$, and $\text{dist}(\partial\Omega,\partial\Omega\sp h)\le Ch\sp 2$, then the Galerkin approximation with piecewise linear basis functions satisfies $$\Vert u-u\sp h\Vert\sb{0,\infty,\Omega}+h\Vert u-u\sp h\Vert\sb{1,\infty,\Omega}\le Ch\sp{2-\varepsilon}.$$ This result is for exact integration. For numerical integration, $$\Vert u-\hat u\sp h\Vert\sb{0,\infty,\Omega}+\Vert u-\hat u\sp h\Vert\sb{0,\infty,\Omega}\le \cases Ch\sp{p-\varepsilon}, & p\in [1/2,1),\\Ch\sp{3/2+p-\varepsilon}, & p\in (0,1/2].\endcases.$$
[J.R.Kuttler (Laurel)]

MSC 2000:: *65N25 Numerical methods for eigenvalue problems (BVP of PDE)
65N30 Finite numerical methods (BVP of PDE)
35P30 Nonlinear eigenvalue problems for PD operators
35P15 Estimation of eigenvalues for PD operators

Keywords: finite elements; Galerkin method; semilinear elliptic eigenvalue problem

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