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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 ${L}^{q}$, $2\le q\le \infty$ using some generalizations of the ${L}^{2}$ lemmas (proved in the previous paragraph) and Nitsche’s weighted ${L}^{2}$-norms.

##### MSC:
 65N30 Finite elements, Rayleigh-Ritz and Galerkin methods, finite methods (BVP of PDE) 65N15 Error bounds (BVP of PDE) 35J65 Nonlinear boundary value problems for linear elliptic equations