Mixed Methods for Nonlinear Second-Order Elliptic Problems in Three Variables

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 L^q; 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.