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Discontinuous finite volume element method for parabolic problems. (English) Zbl 1189.65203

A class of discretization methods applied to partial differential equations of parabolic type is analyzed. The integration methods are based on finite volume elements and involve the discontinuous Galerkin technique, so that they can be used when there are elements of several types and shapes and/or irregular non-matching grids.
A semi-discrete scheme (in space) is constructed by following this approach, and then a fully discrete version is obtained by considering the backward Euler method for the time-dependent part. Error estimates are given for both versions in terms of a mesh dependent norm and in the usual \(L^2\)-norm. In particular, the error estimate in the \(L^2\)-norm is suboptimal with respect to regularity of the solution and optimal with respect to the order of convergence, requiring in this case a higher regularity of the solution.

MSC:

65M08 Finite volume methods for initial value and initial-boundary value problems involving PDEs
65M60 Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs
65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs
65M20 Method of lines for initial value and initial-boundary value problems involving PDEs
65M15 Error bounds for initial value and initial-boundary value problems involving PDEs
35K20 Initial-boundary value problems for second-order parabolic equations
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