Mixed methods of nonlinear second-order elliptic problems in three variables

Abstract

Mixed finite element methods for treating the Dirichlet problem for fully nonlinear second-order elliptic operators in divergence form are extended to cover the three-dimensional case. Existence and uniqueness of the approximation are proved, and optimal error estimates in L2 are demonstrated for both the scalar and vector functions approximated by the method. Error estimates for the pressure variable are also derived in Lq; the result is optimal in order for 2 ≤ q ≤ 6 and less than optimal for 6 < q ≤ + ∞. Newton's method can be used to solve the nonlinear algebraic equations. © 1996 John Wiley & Sons, Inc.