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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:
65M08Finite volume methods (IVP of PDE)
65M60Finite elements, Rayleigh-Ritz and Galerkin methods, finite methods (IVP of PDE)
65M12Stability and convergence of numerical methods (IVP of PDE)
65M20Method of lines (IVP of PDE)
65M15Error bounds (IVP of PDE)
35K20Second order parabolic equations, initial boundary value problems