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Zbl 1189.65203
Bi, Chunjia; Geng, Jiaqiang
Discontinuous finite volume element method for parabolic problems. (English)
[J] Numer. Methods Partial Differ. Equations 26, No. 2, 367-383 (2010). ISSN 0749-159X; ISSN 1098-2426/e

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.
[Fernando Casas (Castellon)]

MSC 2000:: *65M08
65M60 Finite numerical methods (IVP of PDE)
65M12 Stability and convergence of numerical methods (IVP of PDE)
65M20 Method of lines (IVP of PDE)
65M15 Error bounds (IVP of PDE)
35K20 Second order parabolic equations, boundary value problems

Keywords: finite volume element methods; discontinuous Galerkin methods; parabolic problems; semidiscrete scheme; fully discrete scheme; $H^{1}$-error estimate; $L^{2}$-error estimate; backward Euler method; convergence

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