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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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