In this paper, an adaptive computational technique is applied to solve a set of two-dimensional (2D) drift-diffusion (DD) equations together with nonlinear trap model in thin-film transistors (TFTs). Different from the conventional DD equations in metal-oxide-semiconductor field effect transistors, the nonlinear trap model depending on the potential energy accounts for the effect of grain boundary on the electrical characteristics of low temperature polycrystalline-silicon (LTPS) TFTs. Our adaptive computing technique is mainly based on Gummel's decoupling method, a finite volume (FV) approximation, a monotone iterative (MI) method, a posteriori error estimation, and an 1-irregular meshing scheme. Applying Gummel's decoupling method to the set of DD equations firstly, each decoupled partial differential equation (PDE) is then approximated with FV method over 1-irregular mesh. Instead of conventional Newton's iterative method, the corresponding system of nonlinear algebraic equations is solved with MI method. Variations of the computed solutions, such as potential and electron density are captured and a posteriori error estimation scheme is adopted to assess the quality of the computed solutions. The mesh is adaptively refined accordingly. The numerical method converges monotonically in both MI and Gummel's iteration loops, respectively. Various cases of simulation have been verified for a typical LTPS TFT to demonstrate the accuracy and robustness of the method.