Expanded mixed finite element methods for quasilinear second order elliptic problems, II

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 ux the coe cients times the gradient Based on this formulation mixed nite element approximations of the quasilinear problems are established Existence and uniqueness of the solution of the mixed formulation and its discretization are demon strated Optimal order error estimates in the L and H s norms 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 pro posed here is suitable for the case where the coe cient of di erential equations is a small tensor and does not need to be inverted Introduction This is the second paper of a series in which we develop and analyze expanded mixed formulations for numerical solution of second order elliptic problems This new formula tion expands the standard mixed formulation in the sense that three variables are explicitly treated i e the scalar unknown its gradient and its ux the coe cient times the gra dient It is suitable for the case where the coe cient of di erential equations is a small tensor and does not need to be inverted It directly applies to the ow equation with low permeability and to the transport equation with small dispersion in ground water modeling and petroleum reservoir simulation In the rst paper of the series we analyzed the expanded mixed formulation for linear second order elliptic problems Optimal order and superconvergent error estimates for mixed approximations were obtained and various implementation techniques for solving the system of algebraic equations were discussed Mathematics Subject Classi cation Primary N N F