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Error analysis in $$L^ p,1\leq p\leq \infty$$, for mixed finite element methods for linear and quasi-linear elliptic problems. (English) Zbl 0698.65060
Summary: We consider the approximation by mixed finite element methods of second order elliptic problems in $${\mathbb{R}}^ 2$$. We show that error estimates in $$L^ p$$ follow from stability properties of a weighted $$L^ 2$$- projection on the divergence free vectors of the finite element space. Since we work in two dimensions, we show that this projection is related with a Ritz projection and consequently optimal $$L^ p$$ estimates for $$1<p<\infty$$ can be derived easily from the known results for the standard finite element method. Also quasi-optimal $$L^{\infty}$$ and $$L^ 1$$ estimates are obtained. Finally we analyze a quasi-linear problem obtaining similar results than in the linear case.

##### MSC:
 65N15 Error bounds for boundary value problems involving PDEs 65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs 35J25 Boundary value problems for second-order elliptic equations 35J65 Nonlinear boundary value problems for linear elliptic equations
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