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Homogenization for a non-local coupling model. (English) Zbl 1154.35307

Summary: In [P. Deuflhard and R. Hochmuth, On the thermoregulation in the human microvascular system, Proc. Appl. Math. Mech. 3, 378–379 (2003); Math. Meth. Appl. Sci. 27, 971–989 (2004; Zbl 1062.92015); Models Methods Appl. Sci. 14, No. 11, 1621–1634 (2004; Zbl 1063.80006)], homogenization techniques are applied to derive an anisotropic variant of the bio-heat transfer equation as asymptotic result of boundary value problems providing a microscopic description for microvascular tissue. In view of a future application on treatment planning in hyperthermia, we investigate here the homogenization limit for a coupling model, which takes additionally into account the influence of convective heat transfer in medium-size blood vessels. This leads to second-order elliptic boundary value problems with non-local boundary conditions on parts of the boundary. Moreover, we present asymptotic estimates for first-order correctors.

MSC:

35B27 Homogenization in context of PDEs; PDEs in media with periodic structure
80A20 Heat and mass transfer, heat flow (MSC2010)
92C50 Medical applications (general)
35J25 Boundary value problems for second-order elliptic equations
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