For interpretation of electroencephalography (EEG) and magnetoencephalography (MEG) data, multiple solutions of the respective forward problems are needed. In this paper, we assess performance of the mixed-hybrid finite element method (MHFEM) applied to EEG and MEG modeling. The method provides an approximate potential and induced currents and results in a system with a positive semi-definite matrix. The system thus can be solved with a variety of standard methods (e.g. the preconditioned conjugate gradient method). The induced currents satisfy discrete charge conservation law making the method conservative. We studied its performance on unstructured tetrahedral grids for a layered spherical head model as well as a realistic head model. We also compared its accuracy versus the conventional nodal finite element method ( P1{P}_{1} FEM). To avoid modeling singular sources, we completed our computations with a subtraction approach; the derived expression for the MEG response different from earlier published and involves integration of finite quantities only. We conclude that although the MHFEM is more computationally demanding than the P1{P}_{1} FEM, its use is justified for EEG and MEG modeling on low-resolution head models where P1{P}_{1} FEM loses accuracy.

中文翻译：

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非结构化网格上脑电图和脑磁图的保守有限元建模


为了解释脑电图（EEG）和脑磁图（MEG）数据，需要各个正向问题的多种解决方案。在本文中，我们评估了应用于 EEG 和 MEG 建模的混合混合有限元方法 (MHFEM) 的性能。该方法提供近似电势和感应电流，并产生具有半正定矩阵的系统。因此，可以使用多种标准方法（例如预条件共轭梯度法）对该系统进行求解。感应电流满足离散电荷守恒定律，使得该方法保守。我们研究了其在分层球形头部模型和现实头部模型的非结构化四面体网格上的性能。我们还将其精度与传统的节点有限元法 (P1{P}_{1} FEM) 进行了比较。为了避免对奇异源进行建模，我们用减法方法完成了计算； MEG 响应的导出表达式与之前发布的不同，并且仅涉及有限量的积分。我们的结论是，尽管 MHFEM 比 P1{P}_{1} FEM 的计算要求更高，但它的使用对于低分辨率头部模型的 EEG 和 MEG 建模是合理的，其中 P1{P}_{1} FEM 会失去准确性。