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.
KeywordsLocal discontinuous Galerkin method Instationary Darcy problem Jump condition Henry’s Law Different orders of approximation spaces
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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.
- 1.Abels, H., Garcke, H., Grün, G.: Thermodynamically consistent, frame indifferent diffuse interface models for incompressible two-phase flows with different densities. Math. Models Methods Appl. Sci. 3, 1150013–1–1150013-40 (2012)Google Scholar
- 3.Aizinger, V., Rupp, A., Schütz, J., Knabner, P.: Analysis of a mixed discontinuous Galerkin method for instationary Darcy flow. Computational Geosciences (2017), n/a–n/a. https://doi.org/10.1007/s10596-017-9682-8
- 6.Cheng, J., Shu, C.: High order schemes for CFD: A review. Jisuan Wuli/Chinese J. Comput. Phys. 26, 633–655 (2009)Google Scholar
- 7.Ciarlet, P.G., Lions, J.L.: Handbook of Numerical Analysis, vol. 2. Elsevier (1990)Google Scholar
- 10.Di Pietro, D.A., Ern, A.: Mathematical Aspects of Discontinuous Galerkin Methods. Mathmatiques et Applications. Springer, Heidelberg (2012)Google Scholar
- 15.Shu, C.: A brief survey on discontinuous Galerkin methods in computational fluid dynamics. Adv. Mech. 43, 541–554 (2013)Google Scholar
- 17.Wieners, C.: Distributed point objects. A new concept for parallel finite elements. English. Domain decomposition methods in science and engineering. In: Barth, T., Griebel, M., Keyes, D., Nieminen, R., Roose, D., Schlick, T., Kornhuber, R., Hoppe, R., Priaux, J., Pironneau, O., Widlund, O., Xu, J. (eds.) Lecture Notes in Computational Science and Engineering, vol. 40, pp 175–182. Springer, Berlin (2005), https://doi.org/10.1007/3-540-26825-1_14