Expanded Mixed Finite Element Methods for
Quasilinear Second-Order Elliptic Problems, II

Zhangxin Chen
Department of Mathematics, Box 156
Southern Methodist University
Dallas, Texas 75275--0156, USA.

Abstract

A new mixed formulation recently proposed for linear problems is extended to quasilinear second-order elliptic problems. This new formulation expands the standard mixed formulation in the sense that three variables are explicitly treated; i.e., the scalar unknown, its gradient, and its flux (the coefficient times the gradient). Based on this formulation, mixed finite element approximations of the quasilinear problems are established. Existence and uniqueness of the solution of the mixed formulation and its discretization are demonstrated. Optimal order error estimates in Lp and H-s are obtained for the mixed approximations. A postprocessing method for improving the scalar variable is analyzed, and superconvergent estimates are derived. Implementation techniques for solving the systems of algebraic equations are discussed. Comparisons between the standard and expanded mixed formulations are given both theoretically and experimentally. The mixed formulation proposed here is suitable for the case where the coefficient of differential equations is a small tensor and does not need to be inverted.

This paper will appear in RAIRO Modèl. Math. Anal. Numér.


Contributed June 23, 1997.