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A local discontinuous Galerkin scheme for Darcy flow with internal jumps

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We present a new version of the local discontinuous Galerkin method which is capable of dealing with jump conditions along a submanifold ΓLG (i.e., Henry’s Law) in instationary Darcy flow. Our analysis accounts for a spatially and temporally varying, non-linear permeability tensor in all estimates which is also allowed to have a jump at ΓLG and gives a convergence order result for the primary and the flux unknowns. In addition to this, different approximation spaces for the primary and the flux unknowns are investigated. The results imply that the most efficient choice is to choose the degree of the approximation space for the flux unknowns one less than that of the primary unknown. The only stabilization in the proposed scheme is represented by a penalty term in the primary unknown.

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A. Rupp received financial support from the DFG RU 2179 “MAD Soil - Microaggregates: Formation and turnover of the structural building blocks of soils”.

C. Dawson received support from the U.S. Department of Energy Office of Science, Office of Advanced Scientific Computing Research, Applied Mathematics program under Award Number DE-SC0009286 as part of the DiaMonD Multifaceted Mathematics Integrated Capability Center.

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Rupp, A., Knabner, P. & Dawson, C. A local discontinuous Galerkin scheme for Darcy flow with internal jumps. Comput Geosci 22, 1149–1159 (2018).

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