Analysis of a non-symmetric coupling of Interior Penalty DG and BEM

Heuer, Norbert; Sayas, Francisco-Javier

doi:10.1090/S0025-5718-2014-02918-9

Analysis of a non-symmetric coupling of Interior Penalty DG and BEM
HTML articles powered by AMS MathViewer

by Norbert Heuer and Francisco-Javier Sayas;

Math. Comp. 84 (2015), 581-598

DOI: https://doi.org/10.1090/S0025-5718-2014-02918-9

Published electronically: October 30, 2014

PDF | Request permission

Abstract:

We analyze a non-symmetric coupling of interior penalty discontinuous Galerkin and boundary element methods in two and three dimensions. Main results are discrete coercivity of the method, and thus unique solvability, and quasi-optimal convergence. The proof of coercivity is based on a localized variant of the variational technique from [F.-J. Sayas, The validity of Johnson-Nédeléc’s BEM-FEM coupling on polygonal interfaces, SIAM J. Numer. Anal., 47(5):3451–3463, 2009]. This localization gives rise to terms which are carefully analyzed in fractional order Sobolev spaces, and by using scaling arguments for rigid transformations. Numerical evidence of the proven convergence properties has been published previously.

References

Toufic Abboud, Patrick Joly, Jerónimo Rodríguez, and Isabelle Terrasse, Coupling discontinuous Galerkin methods and retarded potentials for transient wave propagation on unbounded domains, J. Comput. Phys. 230 (2011), no. 15, 5877–5907. MR 2804957, DOI 10.1016/j.jcp.2011.03.062
Douglas N. Arnold, An interior penalty finite element method with discontinuous elements, SIAM J. Numer. Anal. 19 (1982), no. 4, 742–760. MR 664882, DOI 10.1137/0719052
Douglas N. Arnold, Franco Brezzi, Bernardo Cockburn, and L. Donatella Marini, Unified analysis of discontinuous Galerkin methods for elliptic problems, SIAM J. Numer. Anal. 39 (2001/02), no. 5, 1749–1779. MR 1885715, DOI 10.1137/S0036142901384162
Susanne C. Brenner, Poincaré-Friedrichs inequalities for piecewise $H^1$ functions, SIAM J. Numer. Anal. 41 (2003), no. 1, 306–324. MR 1974504, DOI 10.1137/S0036142902401311
F. Brezzi and C. Johnson, On the coupling of boundary integral and finite element methods, Calcolo 16 (1979), no. 2, 189–201. MR 569615, DOI 10.1007/BF02575926
Rommel Bustinza, Gabriel N. Gatica, and Francisco-Javier Sayas, On the coupling of local discontinuous Galerkin and boundary element methods for non-linear exterior transmission problems, IMA J. Numer. Anal. 28 (2008), no. 2, 225–244. MR 2401197, DOI 10.1093/imanum/drm019
Bernardo Cockburn, Johnny Guzmán, and Francisco-Javier Sayas, Coupling of Raviart-Thomas and hybridizable discontinuous Galerkin methods with BEM, SIAM J. Numer. Anal. 50 (2012), no. 5, 2778–2801. MR 3022242, DOI 10.1137/100818339
Bernardo Cockburn and Francisco-Javier Sayas, The devising of symmetric couplings of boundary element and discontinuous Galerkin methods, IMA J. Numer. Anal. 32 (2012), no. 3, 765–794. MR 2954729, DOI 10.1093/imanum/drr019
M. Costabel, Symmetric methods for the coupling of finite elements and boundary elements (invited contribution), Boundary elements IX, Vol. 1 (Stuttgart, 1987) Comput. Mech., Southampton, 1987, pp. 411–420. MR 965328, DOI 10.1007/978-3-662-21908-9_{2}6
M. Costabel and E. P. Stephan, Coupling of finite elements and boundary elements for inhomogeneous transmission problems in $\textbf {R}^3$, The mathematics of finite elements and applications, VI (Uxbridge, 1987) Academic Press, London, 1988, pp. 289–296. MR 956900
Gabriel N. Gatica, Norbert Heuer, and Francisco-Javier Sayas, A direct coupling of local discontinuous Galerkin and boundary element methods, Math. Comp. 79 (2010), no. 271, 1369–1394. MR 2629997, DOI 10.1090/S0025-5718-10-02309-4
Gabriel N. Gatica and George C. Hsiao, Boundary-field equation methods for a class of nonlinear problems, Pitman Research Notes in Mathematics Series, vol. 331, Longman, Harlow, 1995. MR 1379331
Gabriel N. Gatica, George C. Hsiao, and Francisco-Javier Sayas, Relaxing the hypotheses of Bielak-MacCamy’s BEM-FEM coupling, Numer. Math. 120 (2012), no. 3, 465–487. MR 2890297, DOI 10.1007/s00211-011-0414-z
Gabriel N. Gatica and Francisco-Javier Sayas, An a priori error analysis for the coupling of local discontinuous Galerkin and boundary element methods, Math. Comp. 75 (2006), no. 256, 1675–1696. MR 2240630, DOI 10.1090/S0025-5718-06-01864-3
Hou De Han, A new class of variational formulations for the coupling of finite and boundary element methods, J. Comput. Math. 8 (1990), no. 3, 223–232. MR 1299224
George C. Hsiao and Wolfgang L. Wendland, Boundary integral equations, Applied Mathematical Sciences, vol. 164, Springer-Verlag, Berlin, 2008. MR 2441884, DOI 10.1007/978-3-540-68545-6
Claes Johnson and J.-Claude Nédélec, On the coupling of boundary integral and finite element methods, Math. Comp. 35 (1980), no. 152, 1063–1079. MR 583487, DOI 10.1090/S0025-5718-1980-0583487-9
William McLean, Strongly elliptic systems and boundary integral equations, Cambridge University Press, Cambridge, 2000. MR 1742312
Salim Meddahi, Francisco-Javier Sayas, and Virginia Selgás, Nonsymmetric coupling of BEM and mixed FEM on polyhedral interfaces, Math. Comp. 80 (2011), no. 273, 43–68. MR 2728971, DOI 10.1090/S0025-5718-2010-02401-9
G. Of, G. J. Rodin, O. Steinbach, and M. Taus, Coupling of discontinuous Galerkin finite element and boundary element methods, SIAM J. Sci. Comput. 34 (2012), no. 3, A1659–A1677. MR 2970268, DOI 10.1137/110848530
Francisco-Javier Sayas, The validity of Johnson-Nédélec’s BEM-FEM coupling on polygonal interfaces, SIAM J. Numer. Anal. 47 (2009), no. 5, 3451–3463. MR 2551202, DOI 10.1137/08072334X
O. Steinbach, A note on the stable one-equation coupling of finite and boundary elements, SIAM J. Numer. Anal. 49 (2011), no. 4, 1521–1531. MR 2831059, DOI 10.1137/090762701
O. C. Zienkiewicz, D. W. Kelly, and P. Bettess, The coupling of the finite element method and boundary solution procedures, Internat. J. Numer. Methods Engrg. 11 (1977), no. 2, 355–375. MR 451784, DOI 10.1002/nme.1620110210
O. C. Zienkiewicz, D. W. Kelly, and P. Bettess, Marriage à la mode—the best of both worlds (finite elements and boundary integrals), Energy methods in finite element analysis, Wiley Ser. Numer. Methods Engrg., Wiley, Chichester, 1979, pp. 81–107. MR 537001