Numerical treatment of elliptic problems nonlinearly coupled through the interface. (English) Zbl 1282.65165
Summary: This work is devoted to the study of the numerical treatment of linear elliptic problems in adjoined domains nonlinearly coupled at the interface. The problem arises in semi-discretization of mass diffusion problems typically when an osmotic effect is taken into account. Convergence of both the conjugate gradient and the fixed point method are considered and compared.
MSC:
| 65N55 | Multigrid methods; domain decomposition for boundary value problems involving PDEs |
References:
| [1] | Bresch, D., Koko, J.: An optimization-based domani decomposition method for nonlinear wall laws in coupled systems. M3AS 14(7), 1085-1101 (2004) · Zbl 1185.65223 |
| [2] | Calabrò, F., Zunino, P.: Analysis of parabolic problems on partitioned domains with nonlinear conditions at the interface, applications to mass transfer through semi-permeable, membranes. M3AS 16(4), 1-23 (2006) · Zbl 1101.35044 |
| [3] | Canuto, C., Urban, K.: Adaptive optimization of convex functionals in Banach spaces. SIAM J. Numer. Anal. 42(5), 2043-2075 (2005) · Zbl 1081.65053 · doi:10.1137/S0036142903429730 |
| [4] | Du, Q.: Optimization based nonoverlapping domain decomposition algorithms and their convergence. SIAM J. Numer. Anal. 39(3), 1056-1077 (2001) (electronic). MR MR1860457 (2003b:65117) · Zbl 1004.65132 |
| [5] | Du, Q., Gunzburger, M.D.: A gradient method approach to optimization-based multidisciplinary simulations and nonoverlapping domain decomposition algorithms. SIAM J. Numer. Anal. 37(5), 1513-1541 (2000) (electronic). MR MR1759905 (2001d:65162) · Zbl 0964.65142 |
| [6] | Gervasio, P., Lions, J.-L., Quarteroni, A.: Heterogeneous coupling by virtual control methods. Numer. Math. 90(2), 241-264 (2001) · Zbl 1002.65133 · doi:10.1007/s002110100303 |
| [7] | Gunzburger, M.D., Peterson, J.S., Kwon, H.: An optimization based domain decomposition method for partial differential equations. Comput. Math. Appl. 37(10), 77-93 (1999) · Zbl 0941.65123 · doi:10.1016/S0898-1221(99)00127-3 |
| [8] | Gunzburger, M.D., Lee, H.K.: An optimization-based domain decomposition method for the Navier-Stokes equations. SIAM J. Numer. Anal. 37(5), 1455-1480 (2000) (electronic) · Zbl 1003.76024 |
| [9] | Hecht, F., Bernardi, D., Ohtsuka, K., Pironneau, O., Morice, J., Le Hyaric, A.: http://www.freefem.org/ff++/ · Zbl 1131.35350 |
| [10] | Hetzer, G., Meir, A.J.: On an interface problem with a nonlinear jump condition, numerical approximation of solutions. Int. J. Numer. Anal. Model. 4(3-4), 519-530 (2007) · Zbl 1131.35350 |
| [11] | Hron, J., Neuss-Radu, M., Pustějovská, P.: Mathematical modeling and simulation of flow in domains separated by leaky semipermeable membrane including osmotic effect. Appl. Math. 56(1), 51-68 (2011) · Zbl 1224.74062 · doi:10.1007/s10492-011-0009-0 |
| [12] | Kedem, O., Katchalsky, A.: Thermodynamic analysis of the permeability of biological membranes to non-electrolytes. Biochimica et Biophysica Acta. 37, 229-246 (1958) |
| [13] | Koko, J.: Uzawa conjugate gradient domain decomposition methods for coupled stokes flows. J. Sci. Comput. 26(2), 195-216 (2006) · Zbl 1203.76116 · doi:10.1007/s10915-005-4933-6 |
| [14] | Koko, Jonas: Convergence analysis of optimization-based domain decomposition methods for a bonded structure. Appl. Numer. Math. 58(1), 69-87 (2008) · Zbl 1131.65053 · doi:10.1016/j.apnum.2006.10.005 |
| [15] | Lee, H.K.: An optimization-based domain decomposition method for a nonlinear problem. Appl. Math. Comput. 113(1), 23-42 (2000) · Zbl 1023.65129 · doi:10.1016/S0096-3003(99)00079-X |
| [16] | Lui, S.H.: On linear monotone iteration and Schwarz methods for nonlinear elliptic PDEs. Numer. Math. 93(1), 109-129 (2002) · Zbl 1010.65052 |
| [17] | Neuss-Radu, M., Jäger, W.: Effective transmission conditions for reaction-diffusion processes in domains separated by an interface. SIAM J. Math. Anal. 39(3), 687-720 (2007) (electronic) · Zbl 1145.35017 |
| [18] | Pao, C.V.: Nonlinear Parabolic and Elliptic Equations. Plenum Press, New York (1992) · Zbl 0777.35001 |
| [19] | Quarteroni, A., Sacco, R., Saleri, F.: Numerical Mathematics, 2nd edn, Texts in Applied Mathematics, vol. 37. Springer, Berlin (2007) · Zbl 1136.65001 |
| [20] | Quarteroni, A., Valli, A.: Numerical Approximation of Partial Differential Equations, Springer Series in Computational Mathematics, vol. 23. Springer, Berlin (1994) · Zbl 0803.65088 |
| [21] | Quarteroni, A., Veneziani, A., Zunino, P.: Mathematical and numerical modeling of solute dynamics in blood flow and arterial walls. SIAM J. Numer. Anal. 39(5), 1488-1511 (2002) · Zbl 1022.76059 · doi:10.1137/S0036142900369714 |
| [22] | van der Zee, K.G., Verhoosel, C.V.: Isogeometric analysis-based goal-oriented error estimation for free-boundary problems. Finite Elem. Anal. Des. 47(6), 600-609 (2011) · doi:10.1016/j.finel.2010.12.013 |
| [23] | Vurro, M., Castellano, L.: Numerical treatments of the interface discontinuity in solid-water mass transfer problems. Comput. Math. Appl. 45(4-5), 785-788 (2003) · Zbl 1133.65307 · doi:10.1016/S0898-1221(03)00040-3 |
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