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.

References

1. [1]
Babuška I., and Aziz K., “Survey lectures on the mathematical foundations of the finite element methods, “The Mathematical Foundations of the Finite Element Method, With applications to Partial Differential Equations (edit by A.D. Aziz), Academic Press, New York and London 1972.Google Scholar
2. [2]
Bank R., “Analysis of a multilevel iterative methods for nonlinear finite element equations, ” Math. Comp., 39 (1982), 453–465.
3. [3]
Ciarlet P.G., The Finite Element Method for Elliptic Problems, North-Holland Pubi., Amsterdam. Bifurcation Theory, Vol. 2, Springer Verlag: New York, 1982.Google Scholar
4. [4]
Frehse J., “Existence and perturbation theorems for nonlinear elliptic systems, ” Nonlinear PDE and Their Applications: Collège de France Seminar Vol. IV, Pitman, New York, 1983.Google Scholar
5. [5]
Mansfield L., “On the solution of nonlinear finite element system, ” SIAM J. Numer. Anal., 17 (1980), 752–765.
6. [6]
Schatz A.H., “An observation concerning Ritz-Galerkin methods with indefinite bilinear forms, ” Math. Comp., 28 (1974), 959–962.
7. [7]
Smale S., “On the efficiency of algorithms of analysis, “Bull. Amer. Math. Soc., 13 (1985), 87–121.
8. [8]
Smale S., “Some remarks on the foundations of numerical analysis, ” SIAM Review, 32 (1990), 211–220.
9. [9]
Xuan Xiaohua, Nonlinear boundary value problems: computation, convergence and complexity, Ph. D. Dissertation, University of California, 1990.Google Scholar