A mixed-FEM and BEM coupling for the approximation of the scattering of thermal waves in locally non-homogeneous media
ESAIM: Modélisation mathématique et analyse numérique, Volume 40 (2006) no. 5, pp. 871-896.

This paper proposes and analyzes a BEM-FEM scheme to approximate a time-harmonic diffusion problem in the plane with non-constant coefficients in a bounded area. The model is set as a Helmholtz transmission problem with adsorption and with non-constant coefficients in a bounded domain. We reformulate the problem as a four-field system. For the temperature and the heat flux we use piecewise constant functions and lowest order Raviart-Thomas elements associated to a triangulation approximating the bounded domain. For the boundary unknowns we take spaces of periodic splines. We show how to transmit information from the approximate boundary to the exact one in an efficient way and prove well-posedness of the Galerkin method. Error estimates are provided and experimentally corroborated at the end of the work.

DOI: 10.1051/m2an:2006033
Classification: 65J05, 65N30, 65N38, 65R20
Keywords: coupling, finite elements, boundary elements, exterior boundary value problem, Helmholtz equation
Rapún, María-Luisa ; Sayas, Francisco-Javier 1

1 Dep. Matemática Aplicada, Universidad de Zaragoza, Centro Politécnico Superior, c/ María de Luna, 3–50015 Zaragoza, Spain.
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Rapún, María-Luisa; Sayas, Francisco-Javier. A mixed-FEM and BEM coupling for the approximation of the scattering of thermal waves in locally non-homogeneous media. ESAIM: Modélisation mathématique et analyse numérique, Volume 40 (2006) no. 5, pp. 871-896. doi : 10.1051/m2an:2006033. http://www.numdam.org/articles/10.1051/m2an:2006033/

[1] D.P. Almond and P.M. Patel, Photothermal science and techniques. Chapman and Hall, London (1996).

[2] J.-P. Aubin, Approximation of elliptic boundary-value problems. Wiley-Interscience, New York-London-Sydney (1972). | MR | Zbl

[3] H.T. Banks, F. Kojima and W.P. Winfree, Boundary estimation problems arising in thermal tomography. Inverse Problems 6 (1990) 897-921. | Zbl

[4] F. Brezzi and M. Fortin, Mixed and hybrid finite element methods. Springer-Verlag, New York (1991). | MR | Zbl

[5] F. Brezzi and C. Johnson, On the coupling of boundary integral and finite element methods. Calcolo 16 (1979) 189-201. | Zbl

[6] G. Chen and J. Zhou, Boundary element methods. Academic Press, London (1992). | MR | Zbl

[7] M. Costabel, Symmetric methods for the coupling of finite elements and boundary elements. Boundary elements IX, Vol. 1 (Stuttgart, 1987), Comput. Mech. (1987) 411-420. | Zbl

[8] M. Costabel and E. Stephan, A direct boundary integral equation method for transmission problems. J. Math. Anal. Appl. 106 (1985) 367-413. | Zbl

[9] M. Crouzeix and F.-J. Sayas, Asymptotic expansions of the error of spline Galerkin boundary element methods. Numer. Math. 78 (1998) 523-547. | Zbl

[10] F. Garrido and A. Salazar, Thermal wave scattering by spheres. J. Appl. Phys. 95 (2004) 140-149.

[11] G.N. Gatica and G.C. Hsiao, On the coupled BEM and FEM for a nonlinear exterior Dirichlet problem in 2 . Numer. Math. 61 (1992) 171-214. | Zbl

[12] G.N. Gatica and G.C. Hsiao, Boundary-field equation methods for a class of nonlinear problems. Pitman Research Notes in Mathematics Series 331, Longman Scientific and Technical, Harlow, UK (1995). | MR | Zbl

[13] G.N. Gatica and S. Meddahi, A dual-dual mixed formulation for nonlinear exterior transmission problems. Math. Comp. 70 (2001) 1461-1480. | Zbl

[14] V. Girault and P.-A. Raviart, Finite element methods for Navier-Stokes equations. Theory and algorithms. Springer-Verlag, New York (1986). | MR | Zbl

[15] H. Han, A new class of variational formulations for the coupling of finite and boundary element methods. J. Comput. Math. 8 (1990) 223-232. | Zbl

[16] T. Hohage, M.-L. Rapún and F.-J. Sayas, Detecting corrosion using thermal measurements. Inverse Probl. (to appear). | MR | Zbl

[17] G.C. Hsiao, The coupling of BEM and FEM - a brief review. Boundary elements X, Vol 1 (Southampton, 1988). Comput. Mech. (1988) 431-445.

[18] G.C. Hsiao, P. Kopp and W.L. Wendland, A Galerkin collocation method for some integral equations of the first kind. Computing 25 (1980) 89-130. | Zbl

[19] G.C. Hsiao, P. Kopp and W.L. Wendland, Some applications of a Galerkin-collocation method for boundary integral equations of the first kind. Math. Method. Appl. Sci. 6 (1984) 280-325. | Zbl

[20] C. Johnson and J.-C. Nédélec, On the coupling of boundary integral and finite element methods. Math. Comp. 35 (1980) 1063-1079. | Zbl

[21] R.E. Kleinman and P.A. Martin, On single integral equations for the transmission problem of acoustics. SIAM J. Appl. Math 48 (1988) 307-325. | Zbl

[22] R. Kress, Linear integral equations. Second edition. Springer-Verlag, New York (1999). | MR | Zbl

[23] R. Kress and G.F. Roach, Transmission problems for the Helmholtz equation. J. Math. Phys. 19 (1978) 1433-1437. | Zbl

[24] M. Lenoir, Optimal isoparametric finite elements and error estimates for domains involving curved boundaries. SIAM J. Num Anal. 23 (1986) 562-580. | Zbl

[25] A. Mandelis, Photoacoustic and thermal wave phenomena in semiconductors. North-Holland, New York (1987).

[26] A. Mandelis, Diffusion-wave fields. Mathematical methods and Green functions. Springer-Verlag, New York (2001). | MR | Zbl

[27] A. Márquez, S. Meddahi and V. Selgas, A new BEM-FEM coupling strategy for two-dimensional fluid-solid interaction problems. J. Comput. Phys. 199 (2004) 205-220. | Zbl

[28] W. Mclean, Strongly elliptic systems and boundary integral equations. Cambridge University Press, Cambridge (2000). | MR | Zbl

[29] S. Meddahi, A mixed-FEM and BEM coupling for a two-dimensional eddy current problem. Numer. Funct. Anal. Optim. 22 (2001) 127-141.

[30] S. Meddahi and A. Márquez, A combination of spectral and finite elements for an exterior problem in the plane. Appl. Numer. Math. 43 (2002) 275-295. | Zbl

[31] S. Meddahi and F.-J. Sayas, A fully discrete BEM-FEM for the exterior Stokes problem in the plane. SIAM J. Numer. Anal. 37 (2000) 2082-2102. | Zbl

[32] S. Meddahi and F.-J. Sayas, Analysis of a new BEM-FEM coupling for two-dimensional fluid-solid interaction. Numer. Methods Partial Differ. Equ. 21 (2005) 1017-1042. | Zbl

[33] S. Meddahi and V. Selgas, A mixed-FEM and BEM coupling for a three-dimensional eddy current problem. ESAIM: M2AN 37 (2003) 291-318. | Numdam | Zbl

[34] S. Meddahi, J. Valdés, O. Menéndez and P. Pérez, On the coupling of boundary integral and mixed finite element methods. J. Comput. Appl. Math. 69 (1996) 113-124. | Zbl

[35] S. Meddahi, A. Márquez and V. Selgas, Computing acoustic waves in an inhomogeneous medium of the plane by a coupling of spectral and finite elements. SIAM J. Numer. Anal. 41 (2003) 1729-1750. | Zbl

[36] S.G. Mikhlin, Mathematical Physics, an advanced course. North-Holland, Amsterdam-London (1970). | MR | Zbl

[37] L. Nicolaides and A. Mandelis, Image-enhanced thermal-wave slice diffraction tomography with numerically simulated reconstructions. Inverse problems 13 (1997) 1393-1412. | Zbl

[38] M.-L. Rapún, Numerical methods for the study of the scattering of thermal waves. Ph.D. Thesis, University of Zaragoza, (2004). In Spanish.

[39] M.-L. Rapún and F.-J. Sayas, Boundary integral approximation of a heat diffusion problem in time-harmonic regime. Numer. Algorithms 41 (2006) 127-160. | Zbl

[40] F.-J. Sayas, A nodal coupling of finite and boundary elements. Numer. Methods Partial Differ. Equ. 19 (2003) 555-570. | Zbl

[41] J.M. Terrón, A. Salazar and A. Sánchez-Lavega, General solution for the thermal wave scattering in fiber composites. J. Appl. Phys. 91 (2002) 1087-1098.

[42] R.H. Torres and G.V. Welland, The Helmholtz equation and transmission problems with Lipschitz interfaces. Indiana Univ. Math. J. 42 (1993) 1457-1485. | Zbl

[43] T. Von Petersdorff, Boundary integral equations for mixed Dirichlet, Neumann and transmission problems. Math. Methods Appl. Sci. 11 (1989) 185-213. | Zbl

[44] A. Ženišek, Nonlinear elliptic and evolution problems and their finite element approximations. Academic Press, London (1990). | MR | Zbl

[45] M. Zlámal, Curved elements in the finite element method I. SIAM J. Numer. Anal. 10 (1973) 229-240. | Zbl

[46] M. Zlámal, Curved elements in the finite element method II. SIAM J. Numer. Anal. 11 (1974) 347-362. | Zbl

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