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Error estimates in $$L^{2}$$, $$H^{1}$$ and $$L^{\infty}$$ in covolume methods for elliptic and parabolic problems: A unified approach. (English) Zbl 0936.65127
Consider a selfadjoint second-order elliptic partial differential equation in a convex smooth domain in the plane. The basic idea of the finite volume method for this problem is to apply the divergence theorem on the elliptic operator, to convert the double integral in the boundary integral. Here, the authors propose a method to connect this approach with the finite element method analysis; the central ideal, to this end, is to apply a standard Galerkin method. After a lengthy Section 2 which describes all the necessary mathematical background, the main result, that is to say the maximum norm error estimates, can be found in the last third section.

##### MSC:
 65N15 Error bounds for boundary value problems involving PDEs 65N06 Finite difference methods for boundary value problems involving PDEs 35J25 Boundary value problems for second-order elliptic equations 65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
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