In surface-based registration for image-guided interventions, the presence of missing data can be a significant issue. This often arises with real-time imaging modalities such as ultrasound, where poor contrast can make tissue boundaries difficult to distinguish from surrounding tissue. Missing data poses two challenges: ambiguity in establishing correspondences; and extrapolation of the deformation field to those missing regions. To address these, we present a novel non-rigid registration method. For establishing correspondences, we use a probabilistic framework based on a Gaussian mixture model (GMM) that treats one surface as a potentially partial observation. To extrapolate and constrain the deformation field, we incorporate biomechanical prior knowledge in the form of a finite element model (FEM). We validate the algorithm, referred to as GMM-FEM, in the context of prostate interventions. Our method leads to a significant reduction in target registration error (TRE) compared to similar state-of-the-art registration algorithms in the case of missing data up to 30%, with a mean TRE of 2.6 mm. The method also performs well when full segmentations are available, leading to TREs that are comparable to or better than other surface-based techniques. We also analyze robustness of our approach, showing that GMM-FEM is a practical and reliable solution for surface-based registration.