论文标题
多尺寸砂浆混合有限元方法的生物弹性系统
Multiscale mortar mixed finite element methods for the Biot system of poroelasticity
论文作者
论文摘要
我们在非匹配网格的毛弹性系统上开发了一种混合有限元域分解方法。在界面上引入了位移压力矢量砂浆功能，并用作拉格朗日乘数，以施加正常应力和正常速度的弱连续性。砂浆空间可以粗糙的规模，从而导致多尺度近似。我们在适当的条件下，建立了半分化连续制剂的存在，唯一性，稳定性和误差估计值。我们进一步考虑了一种基于向后的Euler时间离散化的完全差异方法，并表明每个时间步骤的代数系统的解决方案可以减小以解决复合砂浆变量的正定界面问题。构建了多尺度应力 - 升华基础，这使得子域的数量独立于接口问题所需的迭代次数以及时间步骤的数量。我们提出了数值实验，以验证理论结果并说明该方法的多尺度能力用于异质基准问题。
We develop a mixed finite element domain decomposition method on non-matching grids for the Biot system of poroelasticity. A displacement-pressure vector mortar function is introduced on the interfaces and utilized as a Lagrange multiplier to impose weakly continuity of normal stress and normal velocity. The mortar space can be on a coarse scale, resulting in a multiscale approximation. We establish existence, uniqueness, stability, and error estimates for the semidiscrete continuous-in-time formulation under a suitable condition on the richness of the mortar space. We further consider a fully-discrete method based on the backward Euler time discretization and show that the solution of the algebraic system at each time step can be reduced to solving a positive definite interface problem for the composite mortar variable. A multiscale stress-flux basis is constructed, which makes the number of subdomain solves independent of the number of iterations required for the interface problem, as well as the number of time steps. We present numerical experiments verifying the theoretical results and illustrating the multiscale capabilities of the method for a heterogeneous benchmark problem.
