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Stable Non-symmetric Coupling of the Finite Volume Method and the Boundary Element Method for Convection-Dominated Parabolic-Elliptic Interface Problems

Christoph Erath and Robert Schorr ORCID logo


Many problems in electrical engineering or fluid mechanics can be modeled by parabolic-elliptic interface problems, where the domain for the exterior elliptic problem might be unbounded. A possibility to solve this class of problems numerically is the non-symmetric coupling of finite elements (FEM) and boundary elements (BEM) analyzed in [H. Egger, C. Erath and R. Schorr, On the nonsymmetric coupling method for parabolic-elliptic interface problems, SIAM J. Numer. Anal. 56 2018, 6, 3510–3533]. If, for example, the interior problem represents a fluid, this method is not appropriate since FEM in general lacks conservation of numerical fluxes and in case of convection dominance also stability. A possible remedy to guarantee both is the use of the vertex-centered finite volume method (FVM) with an upwind stabilization option. Thus, we propose a (non-symmetric) coupling of FVM and BEM for a semi-discretization of the underlying problem. For the subsequent time discretization we introduce two options: a variant of the backward Euler method which allows us to develop an analysis under minimal regularity assumptions and the classical backward Euler method. We analyze both, the semi-discrete and the fully-discrete system, in terms of convergence and error estimates. Some numerical examples illustrate the theoretical findings and give some ideas for practical applications.

Funding source: Deutsche Forschungsgemeinschaft

Award Identifier / Grant number: GSC 233

Funding statement: The research of Robert Schorr was supported by the Excellence Initiative of the German Federal and State Governments and the Graduate School of Computational Engineering at TU Darmstadt.


[1] M. Augustin, A. Caiazzo, A. Fiebach, J. Fuhrmann, V. John, A. Linke and R. Umla, An assessment of discretizations for convection-dominated convection-diffusion equations, Comput. Methods Appl. Mech. Engrg. 200 (2011), no. 47–48, 3395–3409. 10.1016/j.cma.2011.08.012Search in Google Scholar

[2] M. Aurada, M. Ebner, M. Feischl, S. Ferraz-Leite, T. Führer, P. Goldenits, M. Karkulik, M. Mayr and D. Praetorius, HILBERT—A MATLAB implementation of adaptive 2D-BEM, Numer. Algorithms 67 (2014), no. 1, 1–32. 10.1007/s11075-013-9771-2Search in Google Scholar

[3] R. E. Bank and D. J. Rose, Some error estimates for the box method, SIAM J. Numer. Anal. 24 (1987), no. 4, 777–787. 10.1137/0724050Search in Google Scholar

[4] P. Chatzipantelidis, R. D. Lazarov and V. Thomée, Error estimates for a finite volume element method for parabolic equations in convex polygonal domains, Numer. Methods Partial Differential Equations 20 (2004), no. 5, 650–674. 10.1002/num.20006Search in Google Scholar

[5] S.-H. Chou and Q. Li, Error estimates in L2,H1 and L in covolume methods for elliptic and parabolic problems: A unified approach, Math. Comp. 69 (2000), no. 229, 103–120. 10.1090/S0025-5718-99-01192-8Search in Google Scholar

[6] P. G. Ciarlet, The Finite Element Method for Elliptic Problems, Stud. Math. Appl. 4, North-Holland, Amsterdam, 1978. 10.1115/1.3424474Search in Google Scholar

[7] M. Costabel, Boundary integral operators on Lipschitz domains: Elementary results, SIAM J. Math. Anal. 19 (1988), no. 3, 613–626. 10.1137/0519043Search in Google Scholar

[8] H. Egger, C. Erath and R. Schorr, On the nonsymmetric coupling method for parabolic-elliptic interface problems, SIAM J. Numer. Anal. 56 (2018), no. 6, 3510–3533. 10.1137/17M1158276Search in Google Scholar

[9] C. Erath, Coupling of the finite volume method and the boundary element method – Theory, analysis, and numerics, PhD thesis, University of Ulm, 2010. Search in Google Scholar

[10] C. Erath, Coupling of the finite volume element method and the boundary element method: An a priori convergence result, SIAM J. Numer. Anal. 50 (2012), no. 2, 574–594. 10.1137/110833944Search in Google Scholar

[11] C. Erath, A new conservative numerical scheme for flow problems on unstructured grids and unbounded domains, J. Comput. Phys. 245 (2013), 476–492. 10.1016/ in Google Scholar

[12] C. Erath, G. Of and F.-J. Sayas, A non-symmetric coupling of the finite volume method and the boundary element method, Numer. Math. 135 (2017), no. 3, 895–922. 10.1007/s00211-016-0820-3Search in Google Scholar

[13] C. Erath and D. Praetorius, Adaptive vertex-centered finite volume methods with convergence rates, SIAM J. Numer. Anal. 54 (2016), no. 4, 2228–2255. 10.1137/15M1036701Search in Google Scholar

[14] C. Erath and D. Praetorius, Céa-type quasi-optimality and convergence rates for (adaptive) vertex-centered FVM, Finite Volumes for Complex Applications VIII—Methods and Theoretical Aspects, Springer Proc. Math. Stat. 199, Springer, Cham (2017), 215–223. 10.1007/978-3-319-57397-7_14Search in Google Scholar

[15] L. C. Evans, Partial Differential Equations, 2nd ed., Grad. Stud. Math. 19, American Mathematical Society, Providence, 2010. Search in Google Scholar

[16] R. E. Ewing, T. Lin and Y. Lin, On the accuracy of the finite volume element method based on piecewise linear polynomials, SIAM J. Numer. Anal. 39 (2002), no. 6, 1865–1888. 10.1137/S0036142900368873Search in Google Scholar

[17] W. Hackbusch, On first and second order box schemes, Computing 41 (1989), no. 4, 277–296. 10.1007/BF02241218Search in Google Scholar

[18] C. Johnson and J.-C. Nédélec, On the coupling of boundary integral and finite element methods, Math. Comp. 35 (1980), no. 152, 1063–1079. 10.1090/S0025-5718-1980-0583487-9Search in Google Scholar

[19] R. C. MacCamy and M. Suri, A time-dependent interface problem for two-dimensional eddy currents, Quart. Appl. Math. 44 (1987), no. 4, 675–690. 10.1090/qam/872820Search in Google Scholar

[20] W. McLean, Strongly Elliptic Systems and Boundary Integral Equations, Cambridge University, Cambridge, 2000. Search in Google Scholar

[21] H. G. Roos, M. Stynes and L. Tobiska, Numerical Methods for Singularly Perturbed Differential Equations, 2nd ed., Springer, Berlin, 2008. Search in Google Scholar

[22] O. Steinbach and W. L. Wendland, On C. Neumann’s method for second-order elliptic systems in domains with non-smooth boundaries, J. Math. Anal. Appl. 262 (2001), no. 2, 733–748. 10.1006/jmaa.2001.7615Search in Google Scholar

[23] F. Tantardini, Quasi-optimality in the backward Euler–Galerkin method for linear parabolic problems, PhD thesis, Università degli Studi di Milano, Milan, 2014. Search in Google Scholar

Received: 2018-09-28
Revised: 2019-02-19
Accepted: 2019-03-22
Published Online: 2019-05-07
Published in Print: 2020-04-01

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