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A mixed finite element method for a strongly nonlinear second-order elliptic problem. (English) Zbl 0829.65128

The authors consider the approximation of the solution of strongly nonlinear two-dimensional second-order elliptic boundary value problems in divergence form by using the mixed finite element method. The spatial domain is considered to be a bounded, convex domain with C 2 - boundary. Furthermore, the coefficient vector is assumed to have a bounded positive definite Jacobian with respect to the second argument. This assumption implies that the gradient of the solution can be locally represented as a function of the “flux”. The authors assume that such a representation is global.

Theorems for the existence and uniqueness of the approximation are proved making use of the Raviart-Thomas space of index k>0 and introducing L 2 and Raviart-Thomas projections. After that, the authors extend some previous results of F. A. Milner [Math. Comput. 44, 303-320 (1985; Zbl 0567.65079)] and derive error estimates in L q , 2q using some generalizations of the L 2 lemmas (proved in the previous paragraph) and Nitsche’s weighted L 2 -norms.

MSC:
65N30Finite elements, Rayleigh-Ritz and Galerkin methods, finite methods (BVP of PDE)
65N15Error bounds (BVP of PDE)
35J65Nonlinear boundary value problems for linear elliptic equations