  A balancing domain decomposition by constraints (BDDC) algorithm with
adaptive primal constraints in variational form is introduced and analyzed for
high-order mortar discretization of two-dimensional elliptic problems with high
varying and random coefficients. Some vector-valued auxiliary spaces and
operators with essential properties are defined to describe the variational
algorithm, and the coarse space is formed by using a transformation operator on
each interface. Compared with the adaptive BDDC algorithms for conforming
Galerkin approximations, our algorithm is more simple, because there is not any
continuity constraints at subdomain vertices in the mortar method involved in
this paper. The condition number of the preconditioned system is proved to be
bounded above by a user-defined tolerance and a constant which is dependent on
the maximum number of interfaces per subdomain, and independent of the mesh
size and the contrast of the given coefficients. Numerical results show the
robustness and efficiency of the algorithm for various model problems.
