  We consider the numerical approximation of parabolic-elliptic interface
problems by the non-symmetric coupling method of MacCamy and Suri [Quart. Appl.
Math., 44 (1987), pp. 675--690]. We establish well-posedness of this
formulation for problems with non-smooth interfaces and prove quasi-optimality
for a class of conforming Galerkin approximations in space. Therefore, error
estimates with optimal order can be deduced for the semi-discretization in
space by appropriate finite and boundary elements. Moreover, we investigate the
subsequent discretization in time by a variant of the implicit Euler method. As
for the semi-discretization, we establish well-posedness and quasi-optimality
for the fully discrete scheme under minimal regularity assumptions on the
solution. Error estimates with optimal order follow again directly. Our
analysis is based on estimates in appropriate energy norms. Thus, we do not use
duality arguments and corresponding estimates for an elliptic projection which
are not available for the non-symmetric coupling method. Additionally, we
provide again error estimates under minimal regularity assumptions. Some
numerical examples illustrate our theoretical results.
