The use of a finite elements based Digital Volume Correlation (FE-DVC) leads to lower measurement uncertainties in comparison to subset-based approaches. However, the associated computing time may become prohibitive when dealing with high resolution measurements. To overcome this limitation, a PGD solver was recently applied to 2D digital image correlation. In this paper, this method is extended to measure volumetric displacements from 3D digital images. In addition, a multigrid PGD algorithm is developed, which allows to use different discretizations in each term of the decomposition. Associated to a coarse graining of the digital images, this allows to avoid local minima, especially in presence of large displacements. Synthetic and practical cases are analyzed with the present approach and measurement uncertainties are compared with standard FE-DVC. Results show that such an approach reduces the computational cost (when compared to FE-DVC) while maintaining lower measurement uncertainties than standard subset-based DVC.