  In this article we extend the adaptive cross approximation (ACA) method known
for the efficient approximation of discretisations of integral operators to a
block-adaptive version. While ACA is usually employed to assemble hierarchical
matrix approximations having the same prescribed accuracy on all blocks of the
partition, for the solution of linear systems it may be more efficient to adapt
the accuracy of each block to the actual error of the solution as some blocks
may be more important for the solution error than others. To this end, error
estimation techniques known from adaptive mesh refinement are applied to
automatically improve the block-wise matrix approximation. This allows to
interlace the assembling of the coefficient matrix with the iterative solution.
