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Convergence of Finite-Element Solutions for Nonlinear PDEs

In: From Topology to Computation: Proceedings of the Smalefest

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  • Xiaohua Xuan

Abstract

The aim of this note is to prove the convergence of FEM solutions (finite-element-method solutions) in solving 1 $$\begin{array}{*{20}{c}} {\Delta u = b(x,u,Du)} & {in \Omega ,} \\ {u = 0} & {on \partial \Omega .} \\ \end{array}$$ Here the domain Ω ⊂ ℝ n (n = 2) is a bounded convex polygon and the righthand side b(x, z, p), smooth, has at most quadratic gradient growth.1 Indeed, the proof below will also apply to second-order quasilinear boundary value problems in the divergence form $$\begin{array}{*{20}{c}} { - div A(x,u, Du) = b(x,u, Du)} & {in \Omega ,} \\ {u = 0} & {on \partial \Omega ,} \\ \end{array}$$ with the uniform ellipticity of the operator A. Such convergence is important for the study of feasibility and complexity of finite-element methods for nonlinear boundary value problems.

Suggested Citation

  • Xiaohua Xuan, 1993. "Convergence of Finite-Element Solutions for Nonlinear PDEs," Springer Books, in: Morris W. Hirsch & Jerrold E. Marsden & Michael Shub (ed.), From Topology to Computation: Proceedings of the Smalefest, chapter 20, pages 196-200, Springer.
    • Handle: RePEc:spr:sprchp:978-1-4612-2740-3_20
    DOI: 10.1007/978-1-4612-2740-3_20
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