  Elliptic PDE-constrained optimal control problems with $L^1$-control cost
($L^1$-EOCP) are considered. To solve $L^1$-EOCP, the primal-dual active set
(PDAS) method, which is a special semismooth Newton (SSN) method, used to be a
priority. However, in general solving Newton equations is expensive. Motivated
by the success of alternating direction method of multipliers (ADMM), we
consider extending the ADMM to $L^1$-EOCP. To discretize $L^1$-EOCP, the
piecewise linear finite element (FE) is considered. However, different from the
finite dimensional $l^1$-norm, the discretized $L^1$-norm does not have a
decoupled form. To overcome this difficulty, an effective approach is utilizing
nodal quadrature formulas to approximately discretize the $L^1$-norm and
$L^2$-norm. It is proved that these approximation steps will not change the
order of error estimates. To solve the discretized problem, an inexact
heterogeneous ADMM (ihADMM) is proposed. Different from the classical ADMM, the
ihADMM adopts two different weighted inner product to define the augmented
Lagrangian function in two subproblems, respectively. Benefiting from such
different weighted techniques, two subproblems of ihADMM can be efficiently
implemented. Furthermore, theoretical results on the global convergence as well
as the iteration complexity results $o(1/k)$ for ihADMM are given. In order to
obtain more accurate solution, a two-phase strategy is also presented, in which
the primal-dual active set (PDAS) method is used as a postprocessor of the
ihADMM. Numerical results not only confirm error estimates, but also show that
the ihADMM and the two-phase strategy are highly efficient.
