发表机构
George Mason University; Baylor University; Monash University（乔治梅森大学; 贝勒大学; 蒙纳士大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文提出并分析了一种完全混合有限元方法，用于粘弹性组织中钙信号传导的双向耦合机械化学模型，通过混合变量和弱对称性处理，建立了适定性和最优误差估计，并数值验证了收敛速率。
AI 中文摘要
我们提出并分析了一种完全混合有限元方法，用于粘弹性组织中钙信号传导的双向耦合机械化学模型。力学响应遵循Kelvin-Voigt定律，以弹性应力和粘性应力、速度以及自旋张量作为未知量，总应力的对称性通过弱形式施加。钙离子和失活的肌醇三磷酸的反应-扩散系统也以混合形式表述，引入扩散通量作为额外的未知量。耦合是双向的：钙产生主动应力，而局部膨胀反馈到钙动力学中。我们通过将耦合系统简化为积分微分问题，随后恢复混合变量，建立了全局存在性和唯一性。半离散格式针对一般协调有限元空间进行表述，并在单一inf-sup条件下证明其适定性。一个与散度交换并保持弱对称配对的插值算子产生了最优先验误差估计。我们将分析专门应用于三族弱对称有限元，推导出显式收敛速率，并对其中一族给出了位移的超收敛局部后处理。二维和三维数值测试证实了预测的收敛速率，并通过粘弹性基准和机械驱动的钙波列说明了该方法。
英文摘要
We propose and analyse a fully mixed finite element method for a two-way coupled mechanochemical model of calcium signalling in viscoelastic tissue. The mechanical response follows a Kelvin--Voigt law, with the elastic and viscous stresses, velocity, and spin tensor as unknowns, and the symmetry of the total stress imposed weakly. The reaction-diffusion system for calcium and inactivated inositol triphosphate is also formulated in mixed form, introducing the diffusive flux as an additional unknown. The coupling is bidirectional: calcium generates an active stress, while local dilation feeds back into the calcium dynamics. We establish global existence and uniqueness by reducing the coupled system to an integro-differential problem and subsequently recovering the mixed variables. The semidiscrete scheme is formulated for general conforming finite element spaces and shown to be well posed under a single inf-sup condition. An interpolation operator that commutes with the divergence and preserves the weak symmetry pairing yields optimal a priori error estimates. We specialise the analysis to three families of weakly symmetric finite elements, deriving explicit convergence rates and, for one family, a superconvergent local post-processing of the displacement. Numerical tests in two and three dimensions confirm the predicted convergence rates and illustrate the method through viscoelastic benchmarks and mechanically driven calcium wavetrains.
Comments43 pages, 8 figures
