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A vertex-centered, dual discontinuous Galerkin method. (English) Zbl 1091.65098
Summary: This note introduces a new version of the discontinuous Galerkin method for discretizing first-order hyperbolic partial differential equations. The method uses piecewise polynomials that are continuous on a macroelement surrounding the nodes in the unstructured mesh but discontinuous between the macroelements. At lowest order, the method reduces to a vertex-centered finite-volume method with control volumes based on a dual mesh, and the method can be implemented using an edge-based data structure. The method provides therefore a strategy to extend existing vertex-centered finite-volume codes to higher order using the discontinuous Galerkin method. Preliminary tests on a model linear hyperbolic equation in two-dimensional indicate a favorable qualitative behavior for nonsmooth solutions and optimal convergence rates for smooth solutions.

65M60Finite elements, Rayleigh-Ritz and Galerkin methods, finite methods (IVP of PDE)
35L45First order hyperbolic systems, initial value problems
Full Text: DOI
[1] Blazek, J.: Computational fluid dynamicsprinciples and applications. (2001) · Zbl 0995.76001
[2] B. Cockburn, G.E. Karniadakis, C.-W. Shu, Discontinuous Galerkin Methods. Theory, Computation and Applications, vol. 11, Lecture Notes in Computational Science and Engineering, Springer, Berlin, 2000.
[3] P. Eliasson, EDGE, a Navier -- Stokes solver, for unstructured grids, Technical Report FOI-R-0298-SE, Swedish Defence Research Agency, SE-172 90 Stockholm, November 2001.
[4] T. Gerhold, O. Friedrich, J. Evans, M. Galle, Calculation of complex three-dimensional configurations employing the DLR-TAU code, AIAA Paper 97-0167, 1997.
[5] Johnson, C.: Numerical solutions of partial differential equations by the finite element method. (1987) · Zbl 0628.65098
[6] Johnson, C.; Pitk√§ranta, J.: An analysis of the discontinuous Galerkin method for a scalar hyperbolic equation. Math. comp. 46, No. 173, 1-26 (1986) · Zbl 0618.65105
[7] Leveque, R. J.: Finite volume methods for hyperbolic problems. (2002) · Zbl 1010.65040
[8] Peterson, T. E.: A note on the convergence of the discontinuous Galerkin method for a scalar hyperbolic equation. SIAM J. Numer. anal. 28, No. 1, 133-140 (1991) · Zbl 0729.65085