  We present a new diffuse interface model for the dynamics of inextensible
vesicles in a viscous fluid. A new feature of this work is the implementation
of the local inextensibility condition in the diffuse interface context. Local
inextensibility is enforced by using a local Lagrange multiplier, which
provides the necessary tension force at the interface. To solve for the local
Lagrange multiplier, we introduce a new equation whose solution essentially
provides a harmonic extension of the local Lagrange multiplier off the
interface while maintaining the local inextensibility constraint near the
interface. To make the method more robust, we develop a local relaxation scheme
that dynamically corrects local stretching/compression errors thereby
preventing their accumulation. Asymptotic analysis is presented that shows that
our new system converges to a relaxed version of the inextensible sharp
interface model. This is also verified numerically. Although the model does not
depend on dimension, we present numerical simulations only in 2D. To solve the
2D equations numerically, we develop an efficient algorithm combining an
operator splitting approach with adaptive finite elements where the
Navier-Stokes equations are implicitly coupled to the diffuse interface
inextensibility equation. Numerical simulations of a single vesicle in a shear
flow at different Reynolds numbers demonstrate that errors in enforcing local
inextensibility may accumulate and lead to large differences in the dynamics in
the tumbling regime and differences in the inclination angle of vesicles in the
tank-treading regime. The local relaxation algorithm is shown to effectively
prevent this accumulation by driving the system back to its equilibrium state
when errors in local inextensibility arise.
