SIAM Journal on Numerical Analysis, Volume 60, Issue 3, Page 1193-1225, June 2022.
We investigate a mortar technique for mixed finite element approximations of a class of domain decomposition saddle point problems on nonmatching grids in which the variable associated with the essential boundary condition, referred to as flux, is chosen as the coupling variable. It plays the role of a Lagrange multiplier to impose weakly continuity of the variable associated with the natural boundary condition. The flux-mortar variable is incorporated with the use of a discrete extension operator. We present well-posedness and error analysis in an abstract setting under a set of suitable assumptions, followed by a nonoverlapping domain decomposition algorithm that reduces the global problem to a positive definite interface problem. The abstract theory is illustrated for Darcy flow, where the normal flux is the mortar variable used to impose continuity of pressure, and for Stokes flow, where the velocity vector is the mortar variable used to impose continuity of normal stress. In both examples, suitable discrete extension operators are developed and the assumptions from the abstract theory are verified. Numerical studies illustrating the theoretical results are presented for Darcy flow.

中文翻译：

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非匹配网格上的通量砂浆混合有限元方法

SIAM 数值分析杂志，第 60 卷，第 3 期，第 1193-1225 页，2022 年 6 月。
我们研究了一种砂浆技术，用于非匹配网格上一类域分解鞍点问题的混合有限元近似，其中与基本边界条件相关的变量（称为通量）被选为耦合变量。它起到拉格朗日乘数的作用，以强加与自然边界条件相关的变量的弱连续性。通量砂浆变量与离散扩展算子结合使用。我们在一组合适的假设下在抽象设置中提出适定性和错误分析，然后是非重叠域分解算法，将全局问题简化为正定接口问题。为达西流说明了抽象理论，其中法向通量是用于施加压力连续性的砂浆变量，对于斯托克斯流，其中速度矢量是用于施加法向应力连续性的砂浆变量。在这两个例子中，都开发了合适​​的离散扩展算子，并验证了抽象理论的假设。为达西流提供了说明理论结果的数值研究。