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Convergence of Finite-Element Solutions for Nonlinear PDEs

  • Xiaohua Xuan

Abstract

The aim of this note is to prove the convergence of FEM solutions (finite-element-method solutions) in solving
$$\begin{array}{*{20}{c}} {\Delta u = b(x,u,Du)} & {in \Omega ,} \\ {u = 0} & {on \partial \Omega .} \\ \end{array}$$
(1)
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.

Keywords

Nonlinear Finite Element Nonlinear Elliptic System Finite Element Equation Strong Ellipticity Uniform Ellipticity 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

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Copyright information

© Springer-Verlag New York, Inc. 1993

Authors and Affiliations

  • Xiaohua Xuan

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