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On finite element methods for nonlinear elliptic problems on domains with corners

Part of the Lecture Notes in Mathematics book series (LNM,volume 1121)

Abstract

For the variational problem \(\int\limits_\Omega {\left\{ {\tfrac{1}{p}{{\left( {K + {{\left| {\nabla u} \right|}^2}} \right)}^{p/2}} - fu} \right\}} \)dx→Min in H1,p(Ω), Ω ⊂ℝ2, p≥2, k≥0, it is shown that in the near of corners a singular expansion μ=ks+w holds, where k εℝ, s=r±t(φ) and w satisfies |w| ≤ cr±+ɛ, etc. with a small ɛ=ɛ (α,p). The pair (α,t(ϕ)) is obtained by a nonlinear eigenvalue problem. It is proved that the eigenvalue α is given by a root of a quadratic polynomial with known coefficients. The theoretical results are used for the investigation of the ordinary Finite Element Method and the Dual Singular Function Method already known from the linear case. Some numerical computations illustrate the theoretical results.

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© 1985 Springer-Verlag

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Dobrowolski, M. (1985). On finite element methods for nonlinear elliptic problems on domains with corners. In: Grisvard, P., Wendland, W.L., Whiteman, J.R. (eds) Singularities and Constructive Methods for Their Treatment. Lecture Notes in Mathematics, vol 1121. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0076264

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  • DOI: https://doi.org/10.1007/BFb0076264

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  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-15219-4

  • Online ISBN: 978-3-540-39377-1

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