An implicit A Posteriori framework to compute the upper and lower bounds for outputs of finite element solutions is extended for adaptive mesh refinement in three space dimensions. To motivate the characteristics of this framework, termed the bound method, a two dimensional heat transfer problem in a multi-material electronic components array is analyzed. The bound method calculates very sharp lower and upper bounds for the temperature of the hottest component which is assumed to be the engineering output of interest. For this two-dimensional problem, the bound method can yields more than eighty-fold reduction in simulation time over a fine mesh calculation (330,050 d.o.f.) while still maintaining quantitative control over the accuracy of the engineering output of interest. For three dimensional problems, the bound method evaluates large bound gaps (i.e., the difference between upper and lower bounds). To achieve a desired bound gap at the lowest cost, an adaptive mesh refinement technique is used to refine the subdomain mesh only where needed. An optimal stabilization parameter is also applied to improve the sharpness of the bound gap. These techniques are applied to an output of a heat transfer problem in a rectangular duct with a given velocity field. The average temperature at one section of the duct is bounded for given inlet temperature and heat flux. For this problem, the adaptive mesh refinement strategy uses half the number of subdomain elements required by an uniform mesh refinement strategy to calculate the same bound gap.