An edge-based scheme on polyhedral meshes for vector advection-reaction equations

Cantin, Pierre; Ern, Alexandre

ESAIM: Mathematical Modelling and Numerical Analysis , Tome 51 (2017) no. 5, pp. 1561-1581.

We devise and analyze an edge-based scheme on polyhedral meshes to approximate a vector advection-reaction problem. The well-posedness of the discrete problem is analyzed first under the classical positivity hypothesis of Friedrichs’ systems that requires a lower bound on the lowest eigenvalue of some tensor depending on the model parameters. We also prove stability when the lowest eigenvalue is null or even slightly negative if the mesh size is small enough. A priori error estimates are established for solutions in ${𝐖}^{1,q}\left(\Omega \right)$ with q ∈ ((3/2),2]. Numerical results are presented on three-dimensional polyhedral meshes.

R. Abraham, J.E. Marsden and T. Ratiu, Manifolds, tensor analysis, and applications. Vol. 75 of Appl. Math. Sci. Springer-Verlag, New York, 2nd edition (1988). | MR | Zbl

C. Amrouche, C. Bernardi, M. Dauge and V. Girault, Vector potentials in three-dimensional non-smooth domains. Math. Methods Appl. Sci. 21 (1998) 823–864. | DOI | MR | Zbl

J. Bonelle, Compatible Discrete Operator schemes on polyhedral meshes for elliptic and Stokes equations. Ph.D. thesis, Université Paris Est (2014).

J. Bonelle, D. Di Pietro and A. Ern, Low-order reconstruction operators on polyhedral meshes: application to compatible discrete operator schemes. Comput. Aided Geom. Design 35/36 (2015) 27–41. | DOI | MR | Zbl

J. Bonelle and A. Ern, Analysis of compatible discrete operator schemes for elliptic problems on polyhedral meshes. ESAIM: M2AN 48 (2014) 553–581. | DOI | Numdam | MR | Zbl

J. Bonelle and A. Ern, Analysis of compatible discrete operator schemes for stokes problems on polyhedral meshes. IMA J. Numer. Anal. 34 (2014) 553–581. | DOI | Numdam | MR | Zbl

A. Bossavit, Extrusion, contraction: their discretization via Whitney forms. In Selected papers from: 10th International IGTE Symposium on Numerical Field Computation, Graz, 2002 . COMPEL 22 (2003) 470–480. | MR | Zbl

P. Cantin and A. Ern, Vertex-based compatible discrete operator schemes on polyhedral meshes for advection-diffusion equations. Comput. Meth. Appl. Math. 16 (2016) 187–212. | DOI | MR | Zbl

S.H. Christiansen, A construction of spaces of compatible differential forms on cellular complexes. Math. Models Methods Appl. Sci. 18 (2008) 739–757. | DOI | MR | Zbl

L. Codecasa, R. Specogna and F. Trevisan, A new set of basis functions for the discrete geometric approach. J. Comput. Phys. 229 (2010) 7401–7410. | DOI | MR | Zbl

P. Deuring, R. Eymard and M. Mildner, L${}^2$-stability independent of diffusion for a Finite Element-Finite Volume discretization of a linear convection-diffusion equation. SIAM J. Numer. Anal. 53 (2015) 508–526. | DOI | MR | Zbl

A. Devinatz, R. Ellis and A. Friedman. The asymptotic behavior of the first real eigenvalue of second order elliptic operators with a small parameter in the highest derivatives. II. Indiana Univ. Math. J. 23 (1973-1974) 991–1011. | DOI | MR | Zbl

A. Ern and J.-L. Guermond, Theory and practice of finite elements. Vol. 159 of Applied Mathematical Sciences. Springer-Verlag, New York (2004). | MR | Zbl

A. Ern and J.-L. Guermond, Discontinuous Galerkin methods for Friedrichs’ systems. I. General theory. SIAM J. Numer. Anal. 44 (2006) 753–778. | DOI | MR | Zbl

A. Ern and J.-L. Guermond, Finite element quasi-interpolation and best approximation. ESAIM: M2AN 51 (2017) 1367–1385. | DOI | MR | Zbl

M. Gerritsma, An introduction to a compatible spectral discretization method. Mech. Adv. Mater. Struct. 19 (2012) 48–67. | DOI

V. Girault, The Navier-Stokes Equations Theory and Numerical Methods. In: Proc. of a Conference held at Oberwolfach, 1988. Springer Berlin Heidelberg (1990) 201–218. | Zbl

H. Heumann, Eulerian and Semi-Lagrangian Methods for Advection-Diffusion of Differential Forms. Ph.D. thesis, ETH Zürich (2011).

H. Heumann, R. Hiptmair and C. Pagliantini, Stabilized Galerkin for Transient Advection of Differential Forms. Research report, SAM, ETH Zürich (2015). | MR

C. Johnson and J. Pitkäranta, An analysis of the discontinuous Galerkin method for a scalar hyperbolic equation. Math. Comput. 46 (1986) 1–26. | DOI | MR | Zbl

J. Kreeft, A. Palha and M. Gerritsma, Mimetic framework on curvilinear quadrilaterals of arbitrary order. Preprint (2011). | arXiv

P. Lesaint and P.-A. Raviart, On a finite element method for solving the neutron transport equation. In Mathematical Aspects of Finite Elements in Partial Differential Equations. Academic Press (1974) 89–123 | MR | Zbl

P. Mullen, A. Mckenzie, D. Pavlov, L. Durant, Y. Tong, E. Kanso, J.E. Marsden and M. Desbrun, Discrete Lie advection of differential forms. Found. Comput. Math. 11 (2011) 131–149. | DOI | MR | Zbl

A. Palha, High order mimetic discretization. PhD thesis, Technische Universiteit Delft (2013).

S. Zaglmayr, High Order Finite Element Methods for Electromagnetic Field Computation. Ph.D. thesis, Johannes Kepler University (2006).

G.M. Ziegler, Lectures on polytopes. Graduate texts in mathematics. Springer, New York (1995). | MR | Zbl