Crack jump conditions for elliptic problems
by A. Wiegmann, Z. Li, and R. J. LeVeque Applied Mathematics Letters, 12,6 (1999), pp. 81-86

Abstract. We derive jump conditions for a potential function and its derivatives across a crack. A crack is a "thin" region of very different conductivity, for example a fracture in otherwise homogeneous material. Such a sharp change of material properties introduces a discontinuity in the coefficient of the elliptic equation governing the potential. The crack cannot be neglected, because it substantially alters the behavior of the potential. Numerically, it is very difficult to resolve the potential near the crack. A strategy is to treat the crack as a lower dimensional interface (hypersurface). Jump conditions across the crack for the potential and its derivatives are necessary for the development of numerical schemes for this approach. Besides the jump conditions, we also give an analytic example of their validity.

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