  In this paper, a new and efficient numerical algorithm by using weak Galerkin
(WG) finite element methods is proposed for a type of fourth order problem
arising from fluorescence tomography(FT). Fluorescence tomography is an
emerging, in vivo non-invasive 3-D imaging technique which reconstructs images
that characterize the distribution of molecules that are tagged by
fluorophores. Weak second order elliptic operator and its discrete version are
introduced for a class of discontinuous functions defined on a finite element
partition of the domain consisting of general polygons or polyhedra. An error
estimate of optimal order is derived in an $H^2$-equivalent norm for the WG
finite element solutions. Error estimates in the usual $L^2$ norm are
established, yielding optimal order of convergence for all the WG finite
element algorithms except the one corresponding to the lowest order (i.e.,
piecewise quadratic elements). Some numerical experiments are presented to
illustrate the efficiency and accuracy of the numerical scheme.
