Convergence of Finite-Element Solutions for Nonlinear PDEs

  • Xiaohua Xuan


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


Nonlinear Finite Element Nonlinear Elliptic System Finite Element Equation Strong Ellipticity Uniform Ellipticity 
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© Springer-Verlag New York, Inc. 1993

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  • Xiaohua Xuan

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