Discontinuous Galerkin methods (DG) use trial and test functions that are continuous within
elements and discontinuous at element boundaries. Although DG methods have been invented
in the early 1970s they have become very popular only recently.
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DG methods are very attractive for flow and transport problems in porous media since they
can be used to solve hyperbolic as well as elliptic/parabolic problems, (potentially) offer
high-order convergence combined with local mass balance and can be applied to unstructured,
non-matching grids.
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In this talk we present a discontinuous Galerkin method based on the non-symmetric interior
penalty formulation introduced by Wheeler and Rivi\`{e}re for an elliptic equation coupled to
a nonlinear parabolic/hyperbolic equation. The equations cover models for groundwater flow and
solute transport as well as two-phase flow in porous media.
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We show that the method is comparable in efficiency with the mixed finite element method for
elliptic problems with discontinuous coefficients. In the case of two-phase flow the method
can outperform standard finite volume schemes by a factor of ten for a five-spot problem and
also for problems with dominating capillary pressure.