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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 $$L^q$$, $$2 \leq q \leq \infty$$ using some generalizations of the $$L^2$$ lemmas (proved in the previous paragraph) and Nitsche’s weighted $$L^2$$-norms.
Reviewer: K.Georgiev (Sofia)

MSC:
 65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs 65N15 Error bounds for boundary value problems involving PDEs 35J65 Nonlinear boundary value problems for linear elliptic equations
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References:
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