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Comparison of continuous and discontinuous Galerkin approaches for variable‐viscosity Stokes flow
Author(s) -
Lehmann Ragnar S.,
LukáčováMedvid'ová Mária,
Kaus Boris J. P.,
Popov Anton A.
Publication year - 2016
Publication title -
zamm ‐ journal of applied mathematics and mechanics / zeitschrift für angewandte mathematik und mechanik
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.449
H-Index - 51
eISSN - 1521-4001
pISSN - 0044-2267
DOI - 10.1002/zamm.201400274
Subject(s) - piecewise , discontinuous galerkin method , finite element method , mathematics , flow (mathematics) , variable (mathematics) , mathematical analysis , galerkin method , convergence (economics) , divergence (linguistics) , geometry , physics , linguistics , philosophy , economics , thermodynamics , economic growth
We describe a Discontinuous Galerkin (DG) scheme for variable‐viscosity Stokes flow which is a crucial aspect of many geophysical modelling applications and conduct numerical experiments with different elements comparing the DG approach to the standard Finite Element Method (FEM). We compare the divergence‐conforming lowest‐order Raviart‐Thomas (RT 0 P 0 ) and Brezzi‐Douglas‐Marini (BDM 1 P 0 ) element in the DG scheme with the bilinear Q 1 P 0 and biquadratic Q 2 P 1 elements for velocity and their matching piecewise constant/linear elements for pressure in the standard continuous Galerkin (CG) scheme with respect to accuracy and memory usage in 2D benchmark setups. We find that for the chosen geodynamic benchmark setups the DG scheme with the BDM 1 P 0 element gives the expected convergence rates and accuracy but has (for fixed mesh) higher memory requirements than the CG scheme with the Q 1 P 0 element without yielding significantly higher accuracy. The DG scheme with the RT 0 P 0 element is cheaper than the other first‐order elements and yields almost the same accuracy in simple cases but does not converge for setups with non‐zero shear stress. The known instability modes of the Q 1 P 0 element did not play a role in the tested setups leading to the BDM 1 P 0 and Q 1 P 0 elements being equally reliable. Not only for a fixed mesh resolution, but also for fixed memory limitations, using a second‐order element like Q 2 P 1 gives higher accuracy than the considered first‐order elements.