As disease often alters structural and functional properties in tissue, the noninvasive measurement of material stiffness in vivo is desirable. Magnetic resonance elastography provides an approach to in vivo tissue characterization, using images of wave motion in tissue and biomechanical principles to reconstruct and quantify stiffness. Successful clinical translation of this technology requires stiffness reconstruction algorithms that are robust, easy to manage, and fast. In this paper, a reconstruction method is presented which addresses these issues by using a local compact divergence-free reconstruction kernel coupled with non-physical constraint elimination and inverse residual weighting to reliably reconstruct stiffness. The proposed technique is compared with local curl reconstructions and global stiffness-pressure reconstructions across two ground-truth phantoms as well as in vivo data sets. Sensitivity analysis is also performed, assessing the variability of reconstruction results and robustness to noise. It is shown that the proposed method can be robustly applied across data sets, is less sensitive to noise, attains comparable (or improved) accuracy, provides better correlation to anatomical features, and can be completed in short timescales.