There are many works dealing with the dynamics of bone remodeling, proposing increasingly complex and complete models. In the recent years, the efforts started to focus on developing models that not only reproduce the temporal evolution, but also include the spatial aspects of this phenomenon. In this work, we propose the spatial extension of an existing model that includes the dynamics of osteocytes. The spatial dependence is modeled in terms of a linear diffusion, as proposed in previous works dealing with related problems. The resulting model is then written in its variational form, and fully discretized using the well‐known finite element method and a combination of the implicit and explicit Euler schemes. The numerical algorithm is then analyzed, proving some a priori error estimates and its linear convergence. Finally, we extend the examples already published for the temporal model to one and two dimensions, showing the dynamics of the solution in the spatial domain.

中文翻译：

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骨重建动力学模型的空间扩展及其有限元分析

有许多工作涉及骨骼重塑的动力学，提出了越来越复杂和完整的模型。近年来，人们开始致力于开发不仅可以再现时间演变，而且还包括这种现象的空间方面的模型。在这项工作中，我们提出了包括骨细胞动力学在内的现有模型的空间扩展。正如先前处理相关问题的工作中所提出的那样，空间依赖性是根据线性扩散建模的。然后将所得模型以其变分形式编写，并使用众所周知的有限元方法以及隐式和显式欧拉方案的组合进行完全离散化。然后分析数值算法，证明一些先验误差估计及其线性收敛。最后，